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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be an \(n\)-dimensional complex vector space and \({\mathbb G}_{d,n}\) the Grassmannian of \(d\)-dimensional linear subspaces of \(V\). The authors consider the \(T\)-equivariant integral cohomology ring \(\mathcal H\) of \({\mathbb G}_{d,n}\). It is known that the equivariant Schubert classes form a basis for \(\mathcal H\) over the \(T\)-equivariant cohomology ring of a point. The main result of the article is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subvariety in \(\mathcal H\). To prove this result, the authors use Gröbner degeneration technique. As a corollary, an equivariant version of Giambelli formula is obtained. Schubert variety; equivariant cohomology; Gröbner degeneration Lakshmibai, V., Raghavan, K.N., Sankaran, P.: Equivariant Giambelli and determinantal restriction formulas for the Grassmannian. Pure Appl. Math. Q. \textbf{2}(3) (2006) (Special Issue: In honor of Robert D. MacPherson. Part 1, 699-717) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Equivariant Giambelli and determinantal restriction formulas for the Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Earlier work of Kapranov suggested that the bounded derived category of coherent sheaves \(D^b(X)\) of any projective homogeneous variety \(X\) of a split semisimple linear algebraic group \(G\) over a field of characteristic zero, admits a full exceptional collection of vector bundles.
In this paper the authors answer certain variants of this question regarding the existence of exceptional collections of line bundles when \(G\) is of rank 2. By restricting their attention to line bundles, the authors are able to use the combinatorics of the weight lattice to prove existence and non-existence results.
The paper has two main results. The first result concerns the full flag variety \(X\) of a split semisimple linear algebraic group \(G\) of rank 2 over a field of characteristic zero. The authors use a combinatorial algorithm to explicitly construct \(\mathcal P\)-exceptional collections of line bundles of the expected length for \(D^b(X)\), where \(\mathcal P\) is a partial order isomorphic to the left weak Bruhat order on the Weyl group of \(G\).
The second part of the paper answers the question: does there exist an exceptional collection of line bundles on each projective homogeneous variety \(X\) for every split semisimple group \(G\) of rank \(\leq 2\) over an arbitrary field? The most difficult cases are the three non-trivial projective homogeneous varieties for simple algebraic groups of type \(G_2\), for which the authors show that there are no exceptional collections of line bundles of the expected length. derived category; exceptional collection; projective homogeneous variety Ananyevskiy, A.; Auel, A.; Garibaldi, S.; Zainoulline, K.: Exceptional collections of line bundles on projective homogeneous varieties, Adv. math. 236, 111-130 (2013) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Root systems, Linear algebraic groups over arbitrary fields, Grothendieck groups, \(K\)-theory and commutative rings Exceptional collections of line bundles on projective homogeneous varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ``Let \(A=\begin{pmatrix} 2&-2\\-2&2 \end{pmatrix}\), \({\mathfrak g}(A)\) the associated Kac-Moody Lie-algebra and G(A) \((=\hat SL_ 2\)) the associated Kac-Moody group. Let P be a (maximal) parabolic subgroup of G(A). Let W (resp. \(W_ P)\) be the Weyl group of \(G(A)\) (resp. P). For \(\tau \in W/W_ P\), let \(X(\tau)\) be the Schubert variety in G(A)/P associated to \(\tau\). We construct explicit bases for \(H^ 0(X(\tau),L^ m)\), \(m\in {\mathbb{Z}}^+\), in terms of ``standard monomials'' where L denotes the tautological line bundle on \(P^ N\) [as well as its restriction to \(X(\tau)\)] for some canonical projective embedding \(X(\tau)\hookrightarrow P^ N\). As a consequence, we obtain similar results for Schubert varieties in \(\hat SL_ 2/B\). Kac-Moody Lie-algebra; ŜL\({}_ 2\); Kac-Moody group; parabolic subgroup; Weyl group; Schubert variety; standard monomials; tautological line bundle V. Lakshmibai and C. S. Seshadri, ?Thèorie monomiale standard pour \(\widehat{\mathfrak{s}\mathfrak{l}}_2 \) ,? C. R. Acad. Sci. Paris Ser. I,305, 183-185 (1987). Infinite-dimensional Lie groups and their Lie algebras: general properties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) Théorie monomiale standard pour \(\hat SL_ 2\). (Standard monomial theory for \(\hat SL_ 2)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author gives results on \(1\)-dimensional Schubert problems over the moduli space \(\overline{\mathcal M_{0,r}}\) whose real geometry is described by orbits of Schützenberger promotion and a related operation involving Young and jeu de taquin tableau evacuation. Actually, the author works on Schubert problems with respect to flags osculating the rational normal curve and the results of this paper shows that the real points of solution curves are smooth. Also, the author finds a new identity involving first order \(K\)-theoretic Littlewood-Richardson coefficients.
Now let us explain osculating flags. Let \(\mathbb P^k\) be \(k\)-dimensional real projective space. Let \(f:\mathbb P^1 \rightarrow \mathbb P^{n-1}\) be the Veronese embedding \(t\rightarrow (t, t^2, \ldots, t^{n-1})\). At each point \(f(p)\in \mathbb P^{n-1}\), there is an osculating flag \(\mathcal F (p)\) of planes intersecting \(f(\mathbb P^1)\) at \(f(p)\) with the highest possible multiplicity. So the author considers this type Schubert conditions with resoect to such flags and works in the Grassmannian
\[
G(k,\mathbb C^n) = G(k, H^0 (\mathcal O_{\mathbb P^1}(n-1)),
\]
of linear series \(V\) of rank \(k\) and degree \(n-1\) on \(\mathbb P^1\).
Let \(|\underline{\overline{\text{-}|\text{-}|\text{-}}}|\) denote \(k\times (n-k)\) rectangular partition. For a partition \(\lambda \subset |\underline{\overline{\text{-}|\text{-}|\text{-}}}|\), \(\Omega(\lambda, p)\) denotes Schubert variety for \(\lambda\) with respect to \(\mathcal F (p)\). For a collection of distinct points \(p_{\bullet}=(p_1, p_2, \ldots, p_r)\) and partitions \(\lambda_{\bullet} = (\lambda_1, \lambda_2, \ldots, \lambda_r)\), let
\[
S(\lambda_{\bullet}, p_{\bullet}) = \bigcap_{i=1}^r \Omega (\lambda_i, p_i).
\]
Note that codimension of \(\Omega(\lambda, p_i)\) is \(|\lambda|\). We call the quantity
\[
\rho(\lambda_{\bullet})= k(n-k) - \sum |\lambda_i|
\]
expected dimension of \( S(\lambda_{\bullet}, p_{\bullet})\). Schubert calculus; stable curves; Shapiro-Shapiro conjecture; jeu de taquin; growth diagram; promotion Levinson, J.: One-dimensional Schubert problems with respect to osculating flags. Canadian J. Math. (2016). 10.4153/CJM-2015-061-1 Classical problems, Schubert calculus One-dimensional Schubert problems with respect to osculating flags | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G : = G (1,n,q)\) denote the Grassmannian of lines in \(PG(n,q)\), embedded as a point-set in \(PG(N,q)\) with \(N := \binom{n+1}{2}-1\). For \(n=2\) or 3 the characteristic function \(\chi(\overline G)\) of the complement of \(G\) is contained in the linear code generated by characteristic functions of complements of \(n\)-flats in \(PG(N,q)\). In this paper we prove this to be true for all cases \((n,q)\) with \(q=2\) and we conjecture this to be true for all remaining cases \((n,q)\). We show that the exact polynomial degree of \(\chi(\overline G)\) is \((q-1)(\binom{n}{2}-1+\delta)\) for \(\delta : = \delta(n,q) = 0\) or 1, and that the possibility \(\delta = 1\) is ruled out if the above conjecture is true. The result \(\deg(\chi(\overline G)) = \binom{n}{2}-1\) for the binary cases \((n,2)\) can be used to construct quantum codes by intersecting \(G\) with subspaces of dimension at least \(\binom{n}{2}\). Polynomial degree; Geometric codes; Quantum codes Glynn D.G., Maks J.G., Casse L.R.A.: The polynomial degree of the Grassmannian G(1, n, q)of lines in finite projective space PG(n, q). Des. Codes Cryptogr. 40, 335--341 (2006) Grassmannians, Schubert varieties, flag manifolds, Linear codes and caps in Galois spaces, Geometric methods (including applications of algebraic geometry) applied to coding theory The polynomial degree of the Grassmannian \(G(1,n,q)\) of lines in finite projective space \(PG(n,q)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For arbitrary quantizable compact Kähler manifolds, relations between the geometry given by the coherent states based on the manifold and the algebraic (projective) geometry realized via the coherent state mapping into projective space, are studied. Polar divisors, formulas relating the scalar products of coherent vectors on the manifold with the corresponding scalar products on projective space (Cauchy formulas), two-point, three-point and more generally cyclic \(m\)-point functions are discussed. The three-point function is related to the shape invariant of geodesic triangles in projective. two-point, three-point and more generally cyclic \(m\)-point functions; quantizable compact Kähler manifolds; algebraic (projective) geometry Bercenau, St., Schlichenmaier, M.: Coherent state embedding, polar divisor and Cauchy formulas. J. Geom. Phys. \textbf{34}, 336-358 (2000) Coherent states, Global differential geometry of Hermitian and Kählerian manifolds, Embeddings in algebraic geometry, Geometry and quantization, symplectic methods, Geometric quantization Coherent state embeddings, polar divisors and Cauchy formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The purpose of this note is to sketch an amplification of the duality theorem of \textit{H. Hecht, D. Miličić, W. Schmid} and \textit{J. Wolf}. Their main theorem in [Invent. Math. 90, 297-332 (1987; Zbl 0699.22022)]\ establishes a natural duality between Harish-Chandra modules constructed in two different ways: the \(\mathcal D\)-module construction on a flag variety (due to Beilinson and Bernstein) on the one hand and the cohomological induction from a Borel subalgebra (due to Zuckerman) on the other hand. This also follows from a result of Bernstein, which gives a \({\mathcal D}\)- module construction of Zuckerman's functors.
In this note the author introduces a similar \(\widetilde{D}\)-module construction on a generalized flag variety. Following Hecht, Miličić, Schmid and Wolf, this construction is then dual to the cohomological induction from a parabolic subalgebra (when containing a \(\sigma\)-stable Levi factor). As an application of this result, a different proof of a result by \textit{B. Speh} and \textit{D. Vogan} [Acta Math. 145, 227-299 (1980; Zbl 0457.22011)], which asserts the irreducibility of certain standard Zuckerman modules, is given. duality theorem; Harish-Chandra modules; cohomological induction; \({\mathcal D}\)-module construction; generalized flag variety; Zuckerman modules [C] Chang, J.T.: Remarks on localization and standard modules: The duality theorem on a generalized flag variety. Proc. Am. Math. Soc.117, 585-591 (1993) Semisimple Lie groups and their representations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Linear algebraic groups over the reals, the complexes, the quaternions Remarks on localization and standard modules: The duality theorem on a generalized flag variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let A be a generalized Cartan matrix. The author constructs a so called ind-group scheme G associated to A. There is also a notion of Borel subgroup scheme B, and the quotient G/B exists with \(G\to G/B\) locally trivial. He proves that the ``Schubert variety'' \(S_{w\lambda}\) associated to a Weyl group element w and an integral dominant weight \(\lambda\) is normal (and projectively normal with respect to its natural embedding). Finally he proves the PRV-conjecture (for symmetrizable Kac- Moody algebras over a field of characteristic 0). An important ingredient in the proofs is the techniques of Frobenius splitting in characteristic p.
\textit{P. Polo} proved the PRV-conjecture for finite dimensional Lie algebras of type A [Variétés de Schubert et excellentes filtrations, Astérisque 173-174, 281-311 (1989; Zbl 0733.20021)] while \textit{S. Kumar} handled the general case of semisimple Lie algebras [Invent. Math. 93, 117-130 (1988; Zbl 0668.17008)]. projective normality; generalized Cartan matrix; ind-group scheme; Borel subgroup scheme; Schubert variety; PRV-conjecture; symmetrizable Kac- Moody algebras; Frobenius splitting de Cataldo, M.A., Migliorini, L., Mustaţă, M.: The combinatorics and topology of proper toric maps. Crelle's J (to appear) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Infinite-dimensional Lie groups and their Lie algebras: general properties, Group schemes Construction d'un groupe de Kac-Moody et applications. (Construction of a Kac-Moody group and applications) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We relate the counting of honeycomb dimer configurations on the cylinder to the counting of certain vertices in Kirillov-Reshetikhin crystal graphs. We show that these dimer configurations yield the quantum Kostka numbers of the small quantum cohomology ring of the Grassmannian, i.e., the expansion coefficients when multiplying a Schubert class repeatedly with different Chern classes. This allows one to derive sum rules for Gromov-Witten invariants in terms of dimer configurations. dimers; crystal graphs; quantum cohomology Enumeration in graph theory, Grassmannians, Schubert varieties, flag manifolds Dimers, crystals and quantum Kostka numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective.
We prove a formula for Whittaker functions of a real semisimple group as an integral over a geometric crystal in the sense of Berenstein-Kazhdan. We explain the connections of this formula to the program of mirror symmetry of flag varieties developed by Givental and Rietsch; in particular, the integral formula proves the equivariant version of Rietsch's mirror symmetry conjecture.
We also explain the idea that Whittaker functions should be thought of as geometric analogues of irreducible characters of finite-dimensional representations. Lam, Th.: Whittaker functions, geometric crystals, and quantum Schubert calculus (2013). arXiv:1308.5451 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Whittaker functions, geometric crystals, and quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(W\) be a finite Coxeter group generated by reflections in a finite-dimensional complex vector space \(\mathfrak h\). The rational Cherednik algebras associated to \(W\) is a family of associative algebras \(\{H_c(W)\}\) parametrized by the set of \(W\)-invariant complex multiplicities \(c\colon R\to\mathbb{C}\) on the system of roots \(R\subset{\mathfrak h}^*\) of \(W\). Specifically, for a fixed \(c\colon\alpha\to c_\alpha\), the algebra \(H_c=H_c(W)\) is generated by the vectors of \(\mathfrak{h,h}^*\), and the elements of \(W\) subject to the following relations
\[
wxw^{-1}=w(x),\quad wyw^{-1}=w(y),\quad\forall y\in{\mathfrak h},\;x\in{\mathfrak h}^*,\;w\in W;
\]
\[
x_1x_2=x_2x_1,\quad y_1y_2=y_2y_1,\quad\forall y_1,y_2\in{\mathfrak h},\;x_1,x_2\in{\mathfrak h}^*;
\]
\[
yx-xy=\langle y,x\rangle-\sum_{\alpha\in R_+}c_\alpha\langle y,\alpha\rangle\langle\alpha^\vee,x\rangle s_\alpha,\quad\forall y\in{\mathfrak h},\;x\in{\mathfrak h}^*.
\]
Here, as usual, one writes \(\alpha^\vee\in{\mathfrak h}\) for the coroot, \(s_\alpha\in W\) for the reflection corresponding to the root \(\alpha\in R\), \(R_+\subset R\) for a choice of `positive' roots in \(R\), and \(\langle\cdot,\cdot\rangle\) for the canonical pairing between \(\mathfrak h\) and \({\mathfrak h}^*\). Furthermore, each \(H_c\) contains a distinguished subalgebra \(B_c:=eH_ce\), where \(e:=\tfrac 1{|W|}\sum w\) is the symmetrizing idempotent in \(\mathbb{C} W\subset H_c\). We call \(B_c=B_c(W)\) the spherical algebra associated to \((W,c)\).
In this paper, the authors develop representation theory of the rational Cherednik algebras \(H_c\). The authors show that, for integral values of \(c\), the algebra \(H_c\) is simple and Morita equivalent to \({\mathcal D}({\mathfrak h})\#W\), the cross product of \(W\) with the algebra of polynomial differential operators on \(\mathfrak h\).
For the algebra, \(Q_c\), of quasi-invariant polynomials on \(\mathfrak h\) introduced by \textit{O. A. Chalykh, M. Feigin} and \textit{A. P. Veselov} [Commun. Math. Phys. 126, No. 3, 597-611 (1990; Zbl 0746.47025) and Int. Math. Res. Not. 2002, No. 10, 521-545 (2002; Zbl 1009.20044)] the authors prove that the algebra \({\mathcal D}(Q_c)\) of differential operators on quasi-invariants is a simple algebra, Morita equivalent to \({\mathcal D}(\mathfrak h)\). The subalgebra \({\mathcal D}(Q_c)^W\subset{\mathcal D}(Q_c)\) of \(W\)-invariant operators turns out to be isomorphic to the spherical subalgebra \(B_c\). The authors show that \({\mathcal D}(Q_c)\) is generated, as an algebra, by \(Q_c\) and its ``Fourier dual'' \(Q_c^b\), and that \({\mathcal D}(Q_c)\) is a rank-one projective \((Q_c\otimes Q_c^b)\)-module. Cherednik algebras; Hecke algebras; Coxeter groups; root systems; algebras of differential operators; algebras of quasi-invariants; spherical algebras Berest, Yu., Etingof, P., Ginzburg, V.: Cherednik algebras and differential operators on quasi-invariants. Duke Math. J. \textbf{118}(2), 279-337 (2003), corrected version arXiv:math/0111005v6 Rings of differential operators (associative algebraic aspects), Noncommutative algebraic geometry, Simple, semisimple, reductive (super)algebras, Hecke algebras and their representations, Reflection and Coxeter groups (group-theoretic aspects), Geometric invariant theory Cherednik algebras and differential operators on quasi-invariants. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(W\) be a finite Coxeter group generated by reflections in a finite-dimensional complex vector space \(\mathfrak h\). The rational Cherednik algebras associated to \(W\) is a family of associative algebras \(\{H_c(W)\}\) parametrized by the set of \(W\)-invariant complex multiplicities \(c\colon R\to\mathbb{C}\) on the system of roots \(R\subset{\mathfrak h}^*\) of \(W\). Specifically, for a fixed \(c\colon\alpha\to c_\alpha\), the algebra \(H_c=H_c(W)\) is generated by the vectors of \(\mathfrak{h,h}^*\), and the elements of \(W\) subject to the following relations
\[
wxw^{-1}=w(x),\quad wyw^{-1}=w(y),\quad\forall y\in{\mathfrak h},\;x\in{\mathfrak h}^*,\;w\in W;
\]
\[
x_1x_2=x_2x_1,\quad y_1y_2=y_2y_1,\quad\forall y_1,y_2\in{\mathfrak h},\;x_1,x_2\in{\mathfrak h}^*;
\]
\[
yx-xy=\langle y,x\rangle-\sum_{\alpha\in R_+}c_\alpha\langle y,\alpha\rangle\langle\alpha^\vee,x\rangle s_\alpha,\quad\forall y\in{\mathfrak h},\;x\in{\mathfrak h}^*.
\]
Here, as usual, one writes \(\alpha^\vee\in{\mathfrak h}\) for the coroot, \(s_\alpha\in W\) for the reflection corresponding to the root \(\alpha\in R\), \(R_+\subset R\) for a choice of `positive' roots in \(R\), and \(\langle\cdot,\cdot\rangle\) for the canonical pairing between \(\mathfrak h\) and \({\mathfrak h}^*\). Furthermore, each \(H_c\) contains a distinguished subalgebra \(B_c:=eH_ce\), where \(e:=\tfrac 1{|W|}\sum w\) is the symmetrizing idempotent in \(\mathbb{C} W\subset H_c\). We call \(B_c=B_c(W)\) the spherical algebra associated to \((W,c)\).
In this paper the authors classify the rational Cherednik algebras \(H_c(W)\) (and their spherical subalgebras) up to isomorphism and Morita equivalence in case when \(W\) is a symmetric group and \(c\) is a generic parameter value. Their main results can be encapsulated into the following two theorems.
Theorem 2. If \(c\notin\overline\mathbb{Q}\), where \(\overline\mathbb{Q}\) is the algebraic closure of the field of rational numbers in \(\mathbb{C}\), the algebras \(H_c\) and \(H_{c'}\) are (a) isomorphic if and only if \(c=\pm c'\), (b) \(\mathbb{C}\)-linearly Morita equivalent if and only if \(c\pm c'\in\mathbb{Z}\).
Theorem 3. If \(c\notin\overline\mathbb{Q}\), the algebras \(B_c\) and \(B_{c'}\) are (a) isomorphic if and only if \(c=c'\) or \(c=-c'-1\), (b) \(\mathbb{C}\)-linearly Morita equivalent if and only if \(c\pm c'\in\mathbb{Z}\). Cherednik algebras; Morita equivalences; spherical algebras; Hecke algebras; Coxeter groups; symmetric groups; root systems; algebras of differential operators; algebras of quasi-invariants Berest Y., Etingof P., Ginzburg V.: Morita equivalence of Cherednik algebras. J. Reine Angew. Math. 568, 81--98 (2004) Rings of differential operators (associative algebraic aspects), Module categories in associative algebras, Noncommutative algebraic geometry, Simple, semisimple, reductive (super)algebras, Hecke algebras and their representations, Reflection and Coxeter groups (group-theoretic aspects), Representations of finite symmetric groups, Geometric invariant theory Morita equivalence of Cherednik algebras. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a hyperkähler variety with an anti-symplectic involution \(\iota\). According to Beauville's conjectural ``splitting property'', the Chow groups of \(X\) should split in a finite number of pieces such that the Chow ring has a bigrading. The Bloch-Beilinson conjectures predict how \(\iota\) should act on certain of these pieces of the Chow groups. We verify part of this conjecture for a 19-dimensional family of hyperkähler sixfolds that are ``double EPW cubes'' (in the sense of Iliev-Kapustka-Kapustka-Ranestad [\textit{A. Iliev} et al., J. Reine Angew. Math. 748, 241--268 (2019; Zbl 1423.14220)]). This has interesting consequences for the Chow ring of the quotient \(X/\iota\), which is an ``EPW cube'' (in the sense of Iliev-Kapustka-Kapustka-Ranestad [loc. cit.]). algebraic cycles; Bloch-Beilinson filtration; Bloch's conjecture; Chow groups; (double) EPW cubes; hyperkähler varieties; \(K3\) surfaces; motives; multiplicative Chow-Künneth decomposition; non-symplectic involution; splitting property (Equivariant) Chow groups and rings; motives, Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects) Algebraic cycles and EPW cubes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\widehat {GT}\) denote the profinite Grothendieck-Teichmüller group defined by \textit{V. G. Drinfel'd} [Leningr. Math. J. 2, No. 4, 829--860 (1991); translation from Algebra Anal. 2, No. 4, 149--181 (1990; Zbl 0728.16021)]. Let \(G_{\mathbb Q} = Gal({\overline {\mathbb Q}}/ {\mathbb Q}) \subset \widehat {GT}\) and \((\lambda , f): \widehat{GT} \hookrightarrow {\widehat {\mathbb Z}}\times \widehat{F}_2\) be the standard parametrization of \(\widehat{GT}, f\) being a cocycle. \textit{P. Lochak} and \textit{L. Schneps} introduced new cocycles \(g, h: \widehat{GT} \to \widehat{F}_2\) [Invent. Math. 127, No. 3, 571--600 (1997; Zbl 0883.20016)]. The authors give new equations expressing \(g, h\) on \(G_{\mathbb Q}\) in terms of \(f\). This is done by considering the orbifold fundamental groups of some quotient lines of \({\mathbb P}^1 - \{0,1,\infty \}\) by the \(S_3\)-symmetry together with the Galois transformations of certain standard chains appearing there. Galois group; Grothendieck-Teichmüller group and , Harmonic and equianharmonic equations in the Grothendieck--Teichmüller group II, preprint (2003). Coverings of curves, fundamental group, Arithmetic theory of algebraic function fields, Galois theory, Solvable groups, supersolvable groups Harmonic and equianharmonic equations in the Grothendieck-Teichmüller group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A function \(h:[n]\to [n]\) is a Hessenberg function if \(h(i)\geq i\) for \(i=1,\dots,n\), and \(h(i+1)\geq h(i)\) for \(i=1,\dots,n-1\). Given an \(n\times n\) matrix \(A\) and a Hessenberg function \(h\), the Hessenberg variety \(\mathrm{Hess}(A,h)\) in the flag variety of \(\mathbb C^n\) parametrizes flags \(\{ V_i\}\) such that \(AV_i\subseteq V_{h(i)}\) for all \(i=1,\dots n\). The authors consider the case where \(A\) is regular nilpotent, and study the cohomology ring \(H^*(\mathrm{Hess}(A,h))\) of the associated regular nilpotent Hessenberg varieties.
The authors find a new set \(\{f_j,g_j\}\) of explicit generators of the ideal of a presentation of \(H^*(\mathrm{Hess}(A,h))\) as a quotient of \(\mathbb Q[x_1,\dots,x_n]\), which satisfies some recursive properties. The new set of generators determines a natural filtration of \(H^*(\mathrm{Hess}(A,h))\), which points out some interesting aspects of the cohomology ring. Indeed the authors use the filtration to determine recursive formulas for the Poincaré polynomials of regular nilpotent Hessenberg varieties. The authors also provide a natural basis of \(H^*(\mathrm{Hess}(A,h))\) as a \(\mathbb Q\)-vector space, and give an algorithm for deriving a basis for the set of linear relations satisfied by the images of the Schubert classes in the cohomology ring of \(\mathrm{Hess}(A,h)\). flag varieties Grassmannians, Schubert varieties, flag manifolds, Classical real and complex (co)homology in algebraic geometry A filtration on the cohomology rings of regular nilpotent Hessenberg varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(S\) be the affine plane \({\mathbb C^2}\) together with an appropriate \({\mathbb T = \mathbb C^*}\) action. Let \(S ^{[m,m+1]}\) be the incidence Hilbert scheme. Parallel to the authors' work [``Incidence Hilbert schemes and infinite dimensional Lie algebras'', in: Proceedings of Fourth International Congress of Chinese Mathematicians, Vol. II, 408--441, Hangzhou (2007)], we construct an infinite dimensional Lie algebra that acts on the direct sum
\[
\widetilde {\mathbb H}_{\mathbb T} = \bigoplus_{m=0}^{+\infty}H^{2(m+1)}_{\mathbb T}(S^{[m,m+1]})
\]
of the middle-degree equivariant cohomology group of \(S ^{[m,m+1]}\). The algebra is related to an infinite dimensional Heisenberg algebra. In addition, we study the transformations among three different linear bases of \({\widetilde {\mathbb H}_{\mathbb T}}\). Our results are applied to the ring structure of the ordinary cohomology of \(S ^{[m,m+1]}\) and to the ring of symmetric functions in infinitely many variables. Parametrization (Chow and Hilbert schemes), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Infinite-dimensional Lie (super)algebras Equivariant cohomology of incidence Hilbert schemes and infinite dimensional Lie algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In a previous paper, the authors introduced the notions of geometric and unipotent crystals corresponding to a semi-simple algebraic group \(G\) [see \textit{A. Berenstein} and \textit{D. Kazhdan}, in: GAFA 2000. Visions in mathematics -- Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25--September 3, 1999. Part I. Basel: Birkhäuser, 188--236 (2000; Zbl 1044.17006)]. These geometric and unipotent crystals are algebro-geometric analogues of Kashiwara's combinatorial notions of crystals and of crystal bases.
The present article has two aims. The first is to introduce unipotent bicrystals as regular versions of geometric and unipotent crystals. The second is to construct a large class of Kashiwara's crystal bases, or the combinatorial crystals associated to suitable modules, via tropicalization of positive unipotent bicrystals. The framework of unipotent bicrystals allows, among other things, to associate directly crystals to \(G^{\vee}\)-modules without passing to quantum deformations. Here \(G^{\vee}\) is the Langlands dual of \(G\). In particular they provide an explicit construction of the crystal for the coordinate ring of the dual flag variety. The exposition is rather complete and ends with a list of open problems and conjectures. crystal bases Berenstein, A.; Kazhdan, D., \textit{Quantum Groups}, 433, Geometric and unipotent crystals II: from unipotent bicrystals to crystal bases, 13-88, (2007), American Mathematical Society, pp, Providence, RI Group actions on varieties or schemes (quotients), Quantum groups (quantized enveloping algebras) and related deformations, Semisimple Lie groups and their representations Geometric and unipotent crystals. II: From unipotent bicrystals to crystal bases | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a semisimple Lie group, \(\mathfrak g\) its Lie algebra. Consider on the flag variety a Schubert cell \(U\) of maximal dimension. It is known that in the function space on \(U\) one can realize a family of representations of \(\mathfrak g\), contragredient to Verma modules [see \textit{A. Beilinson}, Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 699--710 (1984; Zbl 0571.20032)].
In this note the above construction is generalized to the central extension of the algebra \(\mathfrak g\otimes \mathbb C((t))\). For \(\mathfrak{sl}_ n\) the authors define these representations with concrete formulas, which generalize the one obtained by \textit{M. Wakimoto} for \(\mathfrak{sl}_ 2\) [see Commun. Math. Phys. 104, 605--609 (1986; Zbl 0587.17009)]. affine Lie algebra; flag variety; Schubert cell; Verma modules; central extension; representations Feĭgin, Boris L.; Frenkel, Edward V.: A family of representations of affine Lie algebras. Uspekhi mat. Nauk 43, No. 5263, 227-228 (1988) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Cohomology of Lie (super)algebras, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Classical groups (algebro-geometric aspects) A family of representations of affine Lie algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G^ m\) be the Grassmannian of lines of \({\mathbb{P}}^ m\) over k, algebraically closed. The space of naive n-gons of \({\mathbb{P}}^ m\) is the subvariety of \(({\mathbb{P}}^ m)^ n\times (G^ m)^ n(n\geq 3)\) consisting of points and lines satisfying the incidence relations which define an n-gon in \({\mathbb{P}}^ m\). When \(n=3\) and \(m=2\), a special natural smooth model \({\mathbb{S}}\) of this variety, the Schubert space of triangles, has been studied by several authors. A good smooth model \(Poly_ n(({\mathbb{P}}^ m)\) is here constructed and studied for arbitrary n and m. Its Chow ring \(A^*(Poly_ n({\mathbb{P}}^ m))\) is computed as a quotient of some polynomial ring; the relations have geometric interpretation. A description of \(Poly_ n({\mathbb{P}}^ m)\) as a composition of blow-ups of smooth varieties with smooth centers is also given.
When \(n=2\), the known results are again obtained by considerably simpler methods. incidence varities; Grassmannian; Chow ring Keel, S.: Intersection theory of incidence varieties and polygon spaces. Comm. algebra 18, No. 11, 3647-3670 (1990) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Incidence structures embeddable into projective geometries, Grassmannians, Schubert varieties, flag manifolds Intersection theory of incidence varieties and polygon spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The survey deals mainly with multiplicities of Verma modules, Kazhdan- Lusztig polynomials, the conjectures by Jantzen and Kazhdan-Lusztig about them, and the proofs of the conjectures. The author outlines in some detail his results on parabolic analogues and explicit calculation of the Kazhdan-Lusztig polynomials [see for instance J. Algebra 111, 483-506 (1987; Zbl 0656.22007) and Geom. Dedicata 36, 95-119 (1990; Zbl 0716.17015)]. A list of 160 references to relevant articles concludes the survey. multiplicities of Verma modules; Kazhdan-Lusztig polynomials Vinay Deodhar, A brief survey of Kazhdan-Lusztig theory and related topics, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 105 -- 124. Representation theory for linear algebraic groups, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Other geometric groups, including crystallographic groups, Universal enveloping (super)algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds A brief survey of Kazhdan-Lusztig theory and related topics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For \(p+q=n\) let \(\mathbf{L}_{p,q}\subset\mathbf{SL}_n=\mathbf{SL}_n(\mathbb C)\) be the subgroup of block-diagonal matrices \(\begin{pmatrix}A&0\\
0&B\end{pmatrix}\) with \(A\in\mathbf{GL}_p\), \(B\in\mathbf{GL}_q\), \(\det(A)\det(B)=1\). The quotient space \(\mathbf{E}_{p,q} =\mathbf{SL}_n/\mathbf{L}_{p,q}\) fibers over the Grassmannian \(\mathbf{Gr}(p,n)\) with the fibers isomorphic to the affine space consisting of \(p\times q\) matrices. The integral cohomology and the Chow ring of the space \(\mathbf{E}_{p,q}\) are isomorphic to those of the Grassmannian. The basis is given by classes of the inverse images of the Schubert varieties. On the other hand the decomposition of \(\mathbf{E}_{p,q}\) into orbits of the Borel subgroup \(\mathbf{B}_n\subset\mathbf{SL}_n\) is more complicated. The set of orbits is ordered by the straight forward generalization of the Bruhat order given by the inclusion of closures. The combinatorics of the orbits is governed by \emph{signed involutions} or alternatively by \((p,q)\)-\emph{clans}. The inverse images of a Schubert cell is a union of \(\mathbf{B}_n\)-orbits. The orbits projecting to the same Schubert cell is called a sect. The \emph{big sect} corresponding to the open Schubert cell is a maximal upper order ideal in the Bruhat poset. For \(p=q\) there is a poset isomorphism between the big sect and the rook monoid parameterizing \(\mathbf{B}_p\times\mathbf{B}_p\) orbits in the space of \(p\times p\) matrices.
Some results are stated in a greater generality, for a semisimple group instead of \(\mathbf{SL}_n\). flag variety; Levi subgroup; fibre bundle; rook monoid Homogeneous spaces and generalizations, (Equivariant) Chow groups and rings; motives, Algebraic monoids Sects | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth complete complex algebraic variety. A connective \(K\)-theory \(K(X,\beta)\) of \(X\) is, roughly speaking, an interpolation of the cohomology theory and the \(K\)-theory of \(X\), in the sense that \(K(X,0)=H^*(X)\), the ordinary singular cohomology ring of \(X\), and \(K(X,1)=K(X)\), the ordinary Grothendieck ring of \(X\).
To say it with a slogan, the paper under review is concerned with the quantum equivariant connective \(K\)-theory, \(qh^*_n:=qh^*(X;\beta)\), of the Grassmann variety \(X:=G(n,N)\), parametrizing \(n\)-dimensional subspaces of \(\mathbb{C}^N\).
The ring \(qh^*_n\) can be seen as a multiparameter deformation of the classical cohomology ring of \(X\). The involved deformation parameters \((t_1,\ldots,t_n)\), \(q\) and \(\beta\) play different roles. The first are the equivariant parameters related with the action of the algebraic torus \(\mathbb{T}:=(\mathbb{C}^*)^n\), induced by the diagonal action on \(\mathbb{P}^{N-1}\), \(q\) is the quantum deformation parameter and \(\beta\) is a parameter that connects the generalized (i.e. quantum, equivariant) \(K\)-theory of \(X\) (for \(\beta=1\)) to the quantum, equivariant cohomology ring \(QH^*_{\mathbb{T}}(X)\) (for \(\beta=0\)).
The main result of this paper is without doubt the description of the ring \(qh^*_n\). Its impact is described in another main result, named Theorem 1.1. in the introduction, where three different specializations of \(qh^*_n\), obtained by setting to zero some of the deformation parameters, are considered. It is so shown that \(qh^*_n\) generalizes all the presentations known so far, relying on one hand on the classical Schubert Calculus, ruled by Giambelli's and Pieri's formula and, on the other, on important work appeared along the last couple of decades, due to \textit{D. Peterson} [``Quantum Cohomology of \(G/P\)'', Lecture Notes, M.I.T. (1997)], \textit{B. Kostant} and \textit{S. Kumar} [J. Differ. Geom. 32, No. 2, 549--603 (1990; Zbl 0731.55005)] and more recently to \textit{A. S. Buch} and \textit{L. C. Mihalcea} [Duke Math. J. 156, No. 3, 501--538 (2011; Zbl 1213.14103)].
In particular, Theorem 1.1. shows that i) setting \(\beta=0\) one recovers the presentation due to \textit{L. C. Mihalcea} [Adv. Math. 203, No. 1, 1--33 (2006; Zbl 1100.14045)] of the quantum cohomology ring \(QH^*_{\mathbb{T}}(X)\); ii) setting \(\beta=1\) and \((t_1,\ldots, t_N)=(0,\ldots,0)\), one obtains Buch's quantum \(K\) theory \(KQ(X)\) and that iii) for \(\beta=-1\), \(q=0\) and \(t_j\) equal to certain expressions involving generators of the character ring of \({\mathfrak gl}(N)\), recovers \(K_{\mathbb{T}}(X)\), the equivariant \(K\)-functor. The proof of iii) above is certainly the most intriguing, as it involves a generalization of the celebrated Goresky-Kottwitz-MacPherson theory [\textit{M. Goresky} et al., Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)] and the localized Schubert classes are identified with certan polynomials related to the Bethe ansatz of quantum integrable models, which is another part of the story told in this impressive article.
As a matter of fact, the topic faced in the paper is so wide and important that it is hard to bound it within the narrow borders of conventional subject classifications. Indeed, the ring \(qh^*_n\), the main character of the paper, allows the authors to dig up a breath taking relationship between Schubert Calculus and certain quantum integrable systems that in statistical mechanics are known as \textsl{asymmetric six-vertex model}, invented to describe the physics of anti-ferroelectric materials.
This very well written paper is rather long and dense but the authors put a special effort not to loose the readers by clearly segmenting it in sections, with the aim to provide pre-requisites with graduality. Although combinatorial tools are inspired by the Yang Baxter equations as well as the six vertex models in statistical mechanics, a preliminary knowledge of the latter is not necessary to follow the mathematical content of the paper, which candidates itself to be a must for all mathematicians interested in the cohomological theories of homogeneous varieties. quantum cohomology; quantum \(K\)-theory; enumerative combinatorics; exactly solvable models; Bethe ansatz; Yang Baxter equations; statistical mechanics V. Gorbounov and C. Korff. ''Quantum integrability and generalised quantum Schubert calculus''. Adv. Math. 313 (2017), pp. 282--356.DOI. Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Bordism and cobordism theories and formal group laws in algebraic topology, Symmetric functions and generalizations, Exactly solvable models; Bethe ansatz, Equivariant \(K\)-theory Quantum integrability and generalised quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we describe a certain kind of \(q\)-connections on a projective line, namely \(Z\)-twisted \((G,q)\)-opers with regular singularities using the language of generalized minors. In part one we explored the correspondence between these \(q\)-connections and \(QQ\)-systems/Bethe Ansatz equations. Here we associate to a \(Z\)-twisted \((G,q)\)-oper a class of meromorphic sections of a \(G\)-bundle, satisfying certain difference equations, which we refer to as \((G,q)\)-Wronskians. Among other things, we show that the \(QQ\)-systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells. Bethe ansatz; Langlands correspondence Exactly solvable models; Bethe ansatz, Geometric Langlands program (algebro-geometric aspects), Geometric Langlands program: representation-theoretic aspects, Groups and algebras in quantum theory and relations with integrable systems \(q\)-opers, \(QQ\)-systems, and Bethe ansatz. II: Generalized minors | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [Russ. Math. Surv. 30, No. 5, 1--75 (1975; Zbl 0343.58001)], \textit{V. I. Arnol'd} discovered a strange duality between the 14 exceptional unimodal singularities. Somewhat later an extension of Arnold's duality was described for bimodal hypersurface singularities and isolated complete intersections of dimension two in \(\mathbb C^4\) (see [\textit{W. Ebeling} and \textit{C. T. C. Wall}, Compos. Math. 56, 3--77 (1985; Zbl 0586.14033)]). The paper under review aims to derive this duality using the mirror symmetry and the operation of transposition of invertible polynomials defined in [\textit{P. Berglund} and \textit{T. Hübsch}, Nucl. Phys., B 393, No. 1--2, 377--391 (1993; Zbl 1245.14039)]. Among other things, the authors explain some features of their construction in terms of Dolgachev and Gabrielov numbers, Coxeter-Dynkin diagrams for distinguished bases of vanishing cycles, Milnor lattices, and so on. virtual singularities; hypersurface singularities; isolated complete intersections; mirror symmetry; invertible polynomial; strange duality; Dolgachev numbers; Gabrielov numbers; Coxeter-Dynkin diagrams; Milnor lattices Mirror symmetry (algebro-geometric aspects), Global theory of complex singularities; cohomological properties, Deformations of complex singularities; vanishing cycles, Group actions on varieties or schemes (quotients) Strange duality between hypersurface and complete intersection singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main object of the authors' interest is the multigraded ring \(R(X)\) of a Schubert variety \(X\) in the flag variety \(G/B\), \(G\) denoting a semisimple simply connected Chevalley group defined over a field \(k\), and \(B\) denoting a Borel subgroup containing a maximal \(k\)-split torus \(T\). \(R(X)\) is the direct sum of \(H^ 0 (X,L)\), \(L\) running over all positive line bundles. The main question is whether \(R(X)\) is Cohen-Macaulay. -- The paper gives a positive answer in the case \(G = SL(n)\). The result does not pretend to be new, the general case being treated by \textit{A. Ramanathan} [Invent. Math. 80, 283-294 (1985; Zbl 0541.14039)], \textit{S. Ramanan} and \textit{A. Ramanathan} [Invent. Math. 79, 217-224 (1985; Zbl 0553.14023)].
The paper under review is based on a quite different approach. The main idea is to deform \(R(X)\) into a simpler algebra (by successive flat deformations using the explicit basis of \(R(X))\) and to prove that the deformed algebra is Cohen-Macaulay by methods of \textit{M. Hochster} and \textit{J. A. Eagon} [Am. J. Math. 93, 1020-1058 (1971; Zbl 0244.13012)]. The methods are announced to be applicable to other varieties; for example, a forthcoming paper by the second named author treats the case of the variety consisting of pairs of rectangular matrices satisfying \(AB = BA = 0\). Cohen-Macaulayness; straightening law; Schubert variety; flag variety; flat deformations C. Huneke and V. Lakshmibai,Degeneracy of Schubert varieties, Contemporary Math.139 (1992), 181--235. Grassmannians, Schubert varieties, flag manifolds, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Rings with straightening laws, Hodge algebras, Deformations and infinitesimal methods in commutative ring theory Degeneracy of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the first part of this paper [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] the authors studied the cone \(\text{BDRY}(n)\) which is the set of triples of weakly decreasing \(n\)-tuples \((\lambda,\mu,\nu)\in ({\mathbb R}^n)^3\) satisfying the three conditions (1) regarding \(\lambda,\mu,\nu\) as spectra of \(n\times n\) Hermitian matrices, there exist three Hermitian matrices with those spectra whose sum is the zero matrix; (2) regarding \(\lambda,\mu,\nu\) as dominant weights of \(\text{GL}_n({\mathbb C})\), the tensor product \(V_{\lambda}\otimes V_{\mu}\otimes V_{\nu}\) of the corresponding irreducible modules contains a \(\text{GL}_n({\mathbb C})\)-invariant vector; (3) regarding \(\lambda,\mu,\nu\) as possible boundary data on a honeycomb, there exist ways to complete it to a honeycomb.
These conditions were proved to be equivalent. A sufficient list of inequalities for this cone was given due to the efforts of several authors: Klyachko, Helmke and Rosenthal, Totaro, and Belkale in terms of Schubert calculus on Grassmannians. In the present, second, part of the paper the authors introduce new combinatorial objects called puzzles, which are certain kinds of diagrams in the triangular lattice in the plane, composed from unit equilateral triangles and unit rhombi, with edges labeled by 0 and 1. Puzzles are used to compute Grassmannian Schubert calculus, and have much interest in their own right. In particular, the authors get new, puzzle-theoretic, proofs of results of Horn and the above-mentioned authors.
The authors also characterize ``rigid'' puzzles and use them to prove a conjecture of Fulton which states that if the irreducible module \(V_\nu\) appears exactly once in \(V_\lambda \otimes V_\mu\), then for all \(N\in{\mathbb N}\), \(V_{N\lambda}\) appears exactly once in \(V_{N\lambda}\otimes V_{N\mu}\). honeycombs; symmetric functions; Littlewood-Richardson rule; puzzles; Hermitian matrix; eigenvalue problems; Schubert calculus; Grassmannian A. Knutson, T. Tao and C. Woodward, The honeycomb model of GLn tensor products II: Puzzles determine facets of the Littlewood-Richardson cone. \textit{Journal of the American Mathematical Society }17 (2004), 19--48. arXiv:math/0107011.Zbl 1043.05111 MR 2015329 Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Inequalities involving eigenvalues and eigenvectors, Representation theory for linear algebraic groups, Special polytopes (linear programming, centrally symmetric, etc.), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Combinatorial aspects of representation theory The honeycomb model of \(\text{GL}_n({\mathbb C})\) tensor products. II: Puzzles determine facets of the Little\-wood-Richardson cone | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We explain that the Plücker relations provide the defining equations of the thick flag manifold associated to a Kac-Moody algebra. This naturally transplants the result of Kumar-Mathieu-Schwede about the Frobenius splitting of thin flag varieties to the thick case. As a consequence, we provide a description of the space of global sections of a line bundle of a thick Schubert variety as conjectured in Kashiwara-Shimozono [13]. This also yields the existence of a compatible basis of thick Demazure modules and the projective normality of the thick Schubert varieties. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Frobenius splitting of thick flag manifolds of Kac-Moody algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, the authors develops a Hilbert-Kunz theory for binoids and solve several questions typically asociated to this functions in a combinatorial setting.
Consider a local ring \((R,\mathfrak{m})\) containing a field \(K\) of characteristic \(p>0\). For an ideal \(I\) of \(R\) and \(q\) a power of \(p\), define \(I^{[q]}=\langle a^q\ |\ a\in I\rangle\). Take \(I\) an \(\mathfrak{m}\)-primary ideal and \(M\) a finite \(R\)-module and define the Hilbert-Kunz function of \(M\) with respect to \(I\) as \[HKF(I,M)(q)=\mathrm{length}(M/I^{[q]}M).\]
The classical Hilbert-Kunz function was defined for \(I=\mathfrak{m}\) and was introduced by \textit{E. Kunz} [Am. J. Math. 91, 772--784 (1969; Zbl 0188.33702)] as a measure of the regularity of the ring: for example, Kunz proved that \(R\) is regular (this is, the number of generators of \(\mathfrak{m}\) is equal to \(\dim R\)) if and only if \(HKF(\mathfrak{m},R)(q)=q^{\dim R}\). \textit{P. Monsky} [Math. Ann. 263, 43--49 (1983; Zbl 0509.13023)] introduced the Hilbert-Kunz multiplicity as the limit \[e_{HK}(I,M):=\lim_{q\rightarrow\infty}\frac{HKF(I,M)(q)}{q^{\dim R}}.\]
Then, \(e_{HK}(\mathfrak{m},R)=1\) if and only if \(R\) is regular. Despite the relation of the Hilbert-Kunz function with other Hilbert functions, its behaviour is far from well-understood: in fact, there is no general method to compute it and the attempt to do it has led or has implemented several \(p\)-methods for study singularities in local rings. Just to mention one of the several issues that this function has, the multiplicity (longly believed to be a rational) can be an irrational number and it is still not known the relation between those cases where \(e_{HK}(I,M)\) is rational (for example, regular local rings, complete local domains of dimension 1, normal affine semigroup rings, among others). For a brief history and methods involved in Hilbert-Kunz theory, see for example [\textit{C. Huneke}, in: Commutative algebra. Expository papers dedicated to David Eisenbud on the occasion of his 65th birthday. New York, NY: Springer. 485--525 (2013; Zbl 1275.13012)] or [\textit{C-Y. Jean Chan}, ``The shape of Hilbert-Kunz functions'', Preprint, \url{arXiv:2106.14053}]
In this context, the authors offer a combinatorial setting to compute Hilbert-Kunz function and multiplicity of a family of algebras called binoid algebras.
A binoid \((N,+,0,\infty)\) is a monoid \((N,+,0)\) with an absorbent element \(\infty\in N\) such that \(a+\infty=\infty\) for all \(a\in N\). We say that the binoid is integral if \(N\setminus\{\infty\}\) is a monoid.
Despite \(\mathbb{N}_0\cup\{\infty\}\) being the simplest example of a binoid, many concepts from ring theory can be easily translate to a binoid by considering the ring with identity \((R,+,\cdot,0,1)\) as the binoid \((R,\cdot,1,0)\). In this way, for a binoid \(N\) we can define an ideal \(I\) as a subset of \(N\) such that for any \(a\in I\) and for any \(b\in N\), \(a+b\in I\). An ideal is prime if for any \(f+g\in I\) we have \(f\in I\) or \(g\in I\). The radical of \(I\) is the ideal \(\langle f\in N\ |\ f^n\in I\text{ for some } n\rangle\) and \(I\) is \(N_+\)-primary if its radical is \(N_+\) (the ideal of non-units of \(N\)). The (combinatorial) dimension of \(N\), denoted \(\dim N\), is the supremum of the length of a chain of prime ideals of \(N\).
Analogous to modules, for binoids we have \(N\)-sets. Given a binoid \(N\), an \(N\)-set is a set \(S\) with a distinguished element \(p\in S\) and an operation \(+:N\times S\rightarrow S\) such that
\begin{itemize}
\item For all \(n,m\in N\) and \(s\in S\), \((n+m)+s=n+(m+s)\).
\item For all \(s\in S\), \(0+s=s\).
\item For all \(s\in S\), \(\infty+s=p\).
\item For all \(n\in N\), \(a+p=p\).
\end{itemize}
Finally, the authors make a connection between \(N\)-set and algebras.
Given a commutative ring \(K\) and a binoid \(N\), the binoid algebra associated to \(N\) is the quotient algebra \[K[N]:=K[X^n\ | \ n\in N]/\langle X^\infty\rangle.\]
Therefore, two monomials \(aX^n\) and \(bX^m\) has as product \(abX^{n+m}\) if \(n+m\neq\infty\) and \(0\) otherwise.
With all these, the authors define the Hilbert-Kunz function for binoids in the following way.
Let \(N\) be finitely generated, semipositive (this is, the number of units of \(N\) is finite) binoid, \(T\) a finitely generated \(N\)-set and \(\mathfrak{n}\) an \(N_+\)-primary ideal of \(N\). Then the Hilbert-Kunz function of \(\mathfrak{n}\) on \(T\) at \(q\) is \[HKF^N(\mathfrak{n},T,q)=|T/([q]\mathfrak{n}+T)|-1,\]
\noindent where for an ideal \(I\), \([q]I=\langle qf\ |\ f\in I\rangle\) for \(q>0\). The Hilbert-Kunz multiplicity of \(\mathfrak{n}\) on \(T\) is defined by \[e_{HK}(\mathfrak{n},T):=\lim_{q\rightarrow\infty}\frac{HKF^N(\mathfrak{n},T,q)}{q^{\dim N}}.\]
These two functions are well-behaved with operations of binoids. More interesting, the computation of these functions can be reduced to one case (namely, to integral cancellative torsion-free binoids). Studying this case, the authors are able to prove their main theorem, that says that for a broad family of binoids \(N\) and for an \(N_+\)-primary ideal \(\mathfrak{n}\), \(e_{HK}(\mathfrak{n},N)\) always exists and it is a rational number.
\noindent Theorem 7.6 Let \(N\) be a finitely generated, semipositive, cancallative, reduced binoid and \(\mathfrak{n}\) be an \(N_+\)-primary ideal of \(N\). Then \(e_{HK}(\mathfrak{n},N)\) exists and is rational.
Finally, in order to relate the Hilbert-Kunz function for binoids and the Hilbert-Kunz function for rings, given that \(K[N]\) is not necessarily local, the authors give the definition of a Hilbert-Kunz function for semilocal (since we are in a commutative context, this means that the ring has finitely maximal ideals) Noetherian rings. The definition is the same as before but taking \(I\) as an \(\mathfrak{m}\)-primary ideal. Through this, it can be defined a proper Hilbert-Kunz function for binoid algebras \(K[N]\) of \(K[S]\) where \(S\) is an \(N\)-set with respect to \(K[\mathfrak{n}]\), where \(\mathfrak{n}\) is an \(N_+\)-primary ideal of \(N\), denoted by \(HKF^{K[N]}(K[\mathfrak{n}],K[S],q)\). This definition is purely algebraic, however we have that when \(N\) is a finitely generated, semipositive binoid and \(K\) is a field, \[HKF^{K[N]}(K[\mathfrak{n}],K[S],q)=HKF^N(\mathfrak{n},S,q)\] and the same happens with the respective multiplicities. This implies that the multiplicity for these binoid algebras always exists, it is independent of the characteristic of \(K\) and if \(N\) is one of the binoids used in Theorem 7.6, then the multiplicity is a rational number. binoid; Hilbert-Kunz multiplicity; Hilbert-Kunz function Combinatorial aspects of commutative algebra, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Toric varieties, Newton polyhedra, Okounkov bodies, Regular local rings Hilbert-Kunz multiplicity of binoids | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0717.00010.]
The Grothendieck ring of the flag manifold for \(Gl(\mathbb{C}^ n)\) is a quotient of the ring of polynomials generated by the classes of the so- called tautological line bundles \(a_ 1,\ldots,a_ n\). \textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surv. 28, No. 3, 1-26 (1973); translation from Usp. Mat. Nauk 28, No. 3 (171), 3-26 (1973; Zbl 0286.57025)] and \textit{M. Demazure} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)] have used Bott-Samelson geometric construction to reduce computations to the case of \(\mathbb{P}^ 1\)-bundles; \textit{W. Fulton} [Duke Math. J. 65, No.\ 3, 381-420 (1991)] gives a simpler method using correspondences. The \(\mathbb{P}^ 1\)-bundles correspond to isobaric divided differences
\[
\pi_ i:f(\ldots,a_ i,a_{i+1},\ldots)\to(a_ if(\ldots,a_ i,a_{i+1},\ldots)-a_{i+1} f(\ldots,a_{i+1},a_ i\ldots))/(a_ i- a_{i+1}).
\]
Products of such operators are in bijection with permutations. Applying them to the class of a point: \((1-1/a_ 1)^{n- 1}\cdots(1-1/a_ n)^ 0\), one obtains by definition the Grothendieck polynomials \(G_ \mu\), \(\mu\in{\mathfrak S}(n)\). These polynomials are representatives of the structure sheaves of Schubert varieties.
The article under review deals with combinatorial properties of Grothendieck polynomials and shows how to express with them the Demazure character formula, the Pieri formula for the intersection of Schubert varieties with a hyperplane section, the Riemann-Roch theorem, and the postulation of line bundles. A similar combinatoric for the cohomology ring of the flag variety can be found in ``Notes on Schubert polynomials'', Publ. LACIM (Montréal 1991) by \textit{I. G. Macdonald}. Grothendieck ring; structure sheaves of Schubert varieties; Grothendieck polynomials; cohomology ring of the flag variety Lascoux, A., Anneau de Grothendieck de la variété de drapeaux, (The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, (1990), Birkhäuser Boston Boston, MA), 1-34 Grassmannians, Schubert varieties, flag manifolds, Grothendieck groups (category-theoretic aspects) Anneau de Grothendieck de la variété de drapeaux. (Grothendieck rings of a flag variety) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give the construction of weighted Lagrangian Grassmannians \(w\mathrm{LGr}(3,6)\) and weighted partial \(A_3\) flag variety \(w\mathrm{FL}_{1,3}\) coming from the symplectic Lie group \(\mathrm{Sp}(6,\mathbb{C})\) and the general linear group \(\mathrm{GL}(4,\mathbb{C})\) respectively. We give general formulas for their Hilbert series in terms of Lie theoretic data. We use them as key varieties (Format) to construct some families of polarized 3-folds in codimension 7 and 9. Finally, we list all the distinct weighted flag varieties in codimension \(4 \leq c\leq 10\).
For part I, see [\textit{M. I. Qureshi} and \textit{B. Szendrői}, Bull. Lond. Math. Soc. 43, No. 4, 786--798 (2011; Zbl 1253.14046)]. Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/ Grassmannians, Schubert varieties, flag manifolds, Classical groups (algebro-geometric aspects), \(3\)-folds, Calabi-Yau manifolds (algebro-geometric aspects), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Constructing projective varieties in weighted flag varieties. II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Due to the earlier fundamental work of the first author, there is a bijective correspondence between Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties and certain reflexive polyhedra [cf. \textit{V. V. Batyrev}, J. Algebr. Geom. 3, No. 3, 493-535 (1994; Zbl 0829.14023)]. It was conjectured, in this context, that the polar duality of reflexive polyhedra induces the mirror symmetry for the corresponding Calabi-Yau hypersurfaces. This conjecture has been generalized, by both authors of the present paper, to Calabi-Yau complete intersections in Gorenstein Fano varieties and, in the sequel, to generalized Calabi-Yau manifolds.
As one necessarily has to work with singular Calabi-Yau varieties, the first author was then led to introduce the so-called ``string-theoretic Hodge numbers'' for such varieties, which coincide with the usual ones in the smooth case [cf. \textit{V. V. Batyrev} and \textit{D. I. Dais}, Topology 35, No. 4, 901-929 (1996; Zbl 0864.14022)]. In this framework, the mirror symmetry conjecture (predicted by physicists) has to be modified as follows:
Let \((V,W)\) a mirror pair of singular \(n\)-dimensional Calabi-Yau varieties. Then the string-theoretic Hodge numbers are related by the duality \(h^{p,q}_{\text{string}} (V)= h^{n- p,q}_{\text{string}} (W)\) for \(0\leq p\), \(q\leq n\).
The present paper provides an affirmative answer to this mirror symmetry conjecture for the string-theoretic Hodge numbers of Calabi-Yau complete intersections in Gorenstein toric Fano varieties. -- The proof of this important result is based on combinatorial properties of certain polynomials arising from the intersection homology of those varieties and its associated mixed Hodge structure. Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties; mirror symmetry; string-theoretic Hodge numbers; Calabi-Yau complete intersections Victor V. Batyrev and Lev A. Borisov. Mirror duality and string-theoretic Hodge numbers. \(Invent. Math.\), 126(1):183-203, 1996. Fano varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Complete intersections, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Applications of compact analytic spaces to the sciences Mirror duality and string-theoretic Hodge numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author proves a positive, geometric rule for expressing the structure constants of the cohomology ring of two-step flag varieties in terms of their Schubert basis. A corollary is a positive, geometric rule for expressing the structure constants of the small quantum cohomology ring of Grassmannians. He also shows a similar rule for computing the cohomology class of intersections of projections of Schubert varieties in partial flag manifolds.
The method of the paper is a combinatorial record-keeping of codimension 1 degenerations of subvarieties of Grassmannians and flag varieties. Degeneration methods to study the geometry of these varieties date back to at least Pieri. The challenge of this method is finding natural and canonical degeneration orders and finding the combinatorial objects to keep track of these. The paper under review contains such a choice (a new one), resulting in simple and effective rules. The combinatorial objects the author uses are called Mondrian tableau. They supply a convenient tool for recording the rank data for the intersection of two flags.
The author suggests that the algorithms presented in the paper apply in even more general settings than those shown in the paper. Schubert calculus; Littlewood-Richardson rule; two-step flag variety; quantum cohomology; Mondrian tableau Coşkun, I., A Littlewood-Richardson rule for two-step flag varieties, Invent. Math., 176, 325-395, (2009) Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Homogeneous complex manifolds, Enumerative problems (combinatorial problems) in algebraic geometry A Littlewood-Richardson rule for two-step flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild character varieties described as moduli spaces of points on \(\mathbb{P}^1\) by Sibuya. Furthermore, Boalch noticed that these varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which are attached to the quiver \(\Gamma_n\) on two vertices and \(n\) equioriented arrows. In this article, we go a step further by unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modelled on the quiver \(\Gamma_n\). We prove that the Poisson structure carried by these varieties is induced, via the Kontsevich-Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver \(\Gamma_n\) such that its noncommutative multiplicative moment map is given in terms of Euler continuants. This result generalises the Hamiltonian double quasi-Poisson algebra associated with the quiver \(\Gamma_1\) by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to \(\Gamma_n\) admits a factorisation in terms of \(n\) copies of the algebra attached to \(\Gamma_1\). Euler continuants; character varieties; Boalch algebra; nonommutative quasi-Poisson geometry; quasi-Poisson algebras Representations of quivers and partially ordered sets, Poisson algebras, Noncommutative algebraic geometry, Symplectic structures of moduli spaces, Momentum maps; symplectic reduction Euler continuants in noncommutative quasi-Poisson geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a Pieri-type formula for the sum of \(K\)-\(k\)-Schur functions \(\sum _{\mu \le \lambda } g^{(k)}_{\mu }\) over a principal order ideal of the poset of \(k\)-bounded partitions under the strong Bruhat order, whose sum we denote by \(\widetilde{g}^{(k)}_{\lambda }\). As an application of this, we also give a \(k\)-rectangle factorization formula \(\widetilde{g}^{(k)}_{R_t\cup \lambda }=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_{\lambda }\) where \(R_t=(t^{k+1-t})\), analogous to that of \(k\)-Schur functions \(s^{(k)}_{R_t\cup \lambda }=s^{(k)}_{R_t}s^{(k)}_{\lambda }\). \(K\)-theoretic \(K\)-Schur functions; Pieri rule; Coxeter groups; affine symmetric groups Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Classical problems, Schubert calculus, Reflection and Coxeter groups (group-theoretic aspects), Symmetric groups A Pieri formula and a factorization formula for sums of \(K\)-theoretic \(K\)-Schur functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A classical problem in invariant theory is to find a precise description of the subring of the coordinate ring for the \(n\times r\) matrices \(k[M_{nr}]\), \(r<n\), consisting of all the invariant elements with respect to the natural right action of the special linear group \(\text{SL}_r(k)\), where \(k\) is an algebraically closed field of characteristic \(0\). In this paper the authors present a solution for the analogous problem in the quantum case. Here \(k[M_{nr}]\) is replaced by the quantum matrix bialgebra and \(\text{SL}_r(k)\) by the quantum special linear group. This leads to the formulation of the first and second fundamental theorem for the quantum special linear group. Similarly to what happens for the commutative case, the first theorem of quantum coinvariant theory states that the ring of quantum coinvariants coincides with the ring generated by certain quantum minors in the quantum matrix bialgebra. This is precisely the ring of the so-called quantum Grassmannian. Using the results in the first author's earlier paper they are able to give a presentation of the ring of quantum coinvariants in terms of generators and relations. This is the content of the second fundamental theorem of quantum coinvariant theory. Both the first and the second theorem of quantum coinvariant theory reduce to the corresponding classical results when the indeterminate \(q\) is specialized to \(1\). At the end they use the given presentation of the quantum Grassmannian to define quantum Schubert varieties and to show that they are quantum homogeneous spaces; that is, they admit a coaction by a suitable quantum group. quantum special linear groups; quantum matrix bialgebras; rings of quantum coinvariants; quantum Grassmannians; presentations; quantum Schubert varieties; coactions R. Fioresi and C. Hacon, Quantum coinvariant theory for the quantum special linear group and quantum Schubert varieties. J. Algebra 242 (2001), 433-446. Quantum groups (quantized function algebras) and their representations, Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds, Actions of groups and semigroups; invariant theory (associative rings and algebras), Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Quantum groups (quantized enveloping algebras) and related deformations Quantum coinvariant theory for the quantum special linear group and quantum Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this nice article the authors prove that Schubert varieties are globally \(F\)-regular. As a consequence they obtain that local rings of Schubert varieties are strongly \(F\)-regular, and thereby \(F\)-rational, and that the local rings of varieties that can be identified with open subsets of Schubert varieties, like determinantal varieties, are strongly \(F\)-regular. The term globally \(F\)-regularity was introduced by Karen Smith, and means that the section ring \(S({\mathcal L}) =\bigoplus_{n\geq 0}H^0(X,{\mathcal L}^n)\) of an ample line bundle \(\mathcal L\) on a projective algebraic variety \(X\) over an algebraically closed field of positive characteristic, is strongly \(F\)-regular in the sense of Hochster and Huneke.
Let \(X\) denote a flag variety and \(Y\) a Schubert variety. As a consequence of recent results by \textit{M. Blickle} [Math. Ann. 328, No. 3, 425--450 (2004; Zbl 1065.14006)] the authors prove that the simple objects in the category of equivariant and holonomic \({\mathcal D}_X\)-modules are precisely the local cohomology sheaves \({\mathcal H}_Y^c({\mathcal O}_X)\), where \(c\) is the codimension of \(Y\) in \(X\). Using a local Grothendieck-Cousin complex [\textit{M. Kashiwara} and \textit{N. Lauritzen}, C. R., Math., Acad. Sci. Paris 335, No.~12, 993--996 (2002; Zbl 1016.14009)] they prove that the decomposition of the local cohomology modules with support in Bruhat cells is multiplicity free. globally \(F\)-regular; strongly \(F\)-regular; Schubert varieties; \(F\)-regular; \(F\)-rational; determinantal variety; flag variety; \(\mathcal D\)-module; Bruhat cell; Verma module Niels Lauritzen, Ulf Raben-Pedersen, and Jesper Funch Thomsen, Global \?-regularity of Schubert varieties with applications to \?-modules, J. Amer. Math. Soc. 19 (2006), no. 2, 345 -- 355. Grassmannians, Schubert varieties, flag manifolds, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Local cohomology and algebraic geometry Global \( F\)-regularity of Schubert varieties with applications to \( \mathcal{D}\)-modules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors present a conjecture for the small \(J\) -function and its small linear \(q-\)difference equation expressed linearly in terms of Gopakumar-Vafa invariants. More explicitly, they conjecture that the small \(J\) function of the quintic 3-fold is expressed linearly in terms of GV-invariants by
\[
\frac{1}{1-q}J(Q,q,0)=1+x^{2}\sum_{d,r\geq 1}a(d,r,q^{r})\mbox{GV}_{d}\mbox{Q}^{dr}+x^{3}\sum_{d,r\geq 1}b(d,r,q^{r})\mbox{GV}_{d}\mbox{Q}^{dr}.
\]
The authors also study the consequence of the conjecture and its relations to a conjecture by \textit{H. Jockers} and \textit{P. Mayr} [J. High Energy Phys. 2019, No. 11, Paper No. 11, 21 p. (2019; Zbl 1429.81090)]. quantum K-theory; quantum cohomology; quintic; Calabi-Yau manifolds; Gromov-Witten invariants; Gopakumar-Vafa invariants; \(q\)-difference equations; \(q\)-Frobenius method; \(J\)-function; reconstruction; gauged linear \(\sigma\) models; 3d-3d correspondence; Chern-Simons theory; \(q\)-holonomic functions Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Difference equations, scaling (\(q\)-differences), Relations of \(K\)-theory with cohomology theories On the quantum \(K\)-theory of the quintic | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We have recently constructed a bivariant analogue of the motivic Hirzebruch classes. A key idea is the construction of a suitable universal bivariant theory in the algebraic-geometric (or compact complex analytic) context, together with a corresponding ``bivariant blow-up relation'' generalizing Bittner's presentation of the Grothendieck group of varieties. Before we already introduced a corresponding universal ``oriented'' bivariant theory as an intermediate step on the way to a bivariant analogue of Levine-Morel's algebraic cobordism. Switching to the differential topological context of smooth manifolds, we similarly get a new geometric bivariant bordism theory based on the notion of a ``fiberwise bordism''. In this paper we make a survey on these theories. Schürmann, J.; Yokura, S.: Motivic bivariant characteristic classes and related topics, J. singul. 5, 124-152 (2012) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, (Equivariant) Chow groups and rings; motives Motivic bivariant characteristic classes and related topics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An invertible polynomial is a weighted homogeneous polynomial with the number of monomials coinciding with the number of variables and such that the weights of the variables and the quasi-degree are well defined. In the framework of the search for mirror symmetric orbifold Landau-Ginzburg models, P. Berglund and M. Henningson considered a pair \((f, G)\) consisting of an invertible polynomial \(f\) and an abelian group \(G\) of its symmetries together with a dual pair \((\widetilde{f}, \widetilde{G})\). We consider the so-called orbifold E-function of such a pair \((f, G)\) which is a generating function for the exponents of the monodromy action on an orbifold version of the mixed Hodge structure on the Milnor fibre of \(f\). We prove that the orbifold E-functions of Berglund-Henningson dual pairs coincide up to a sign depending on the number of variables and a simple change of variables. The proof is based on a relation between monomials (say, elements of a monomial basis of the Milnor algebra of an invertible polynomial) and elements of the whole symmetry group of the dual polynomial. mirror symmetry; singularity; mixed Hodge structure; monodromy; orbifold Ebeling, W; Gusein-Zade, SM; Takahashi, A, Orbifold E-functions of dual invertible polynomials, J. Geom. Phys., 106, 184-191, (2016) Complex surface and hypersurface singularities, Mixed Hodge theory of singular varieties (complex-analytic aspects), Mirror symmetry (algebro-geometric aspects), Group actions on varieties or schemes (quotients) Orbifold E-functions of dual invertible polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ``The paper studies three classes of Frobenius manifolds: quantum cohomology (topological sigma-models), unfolding spaces of singularities (K. Saito's theory, Landau-Ginzburg models), and the recent Barannikov-Kontsevich construction starting with the Dolbeault complex of a Calabi-Yau manifold and conjecturally producing the \(B\)-side of the mirror conjecture in arbitrary dimension. Each known construction provides the relevant Frobenius manifold with an extra structure which can be thought of as a version of ``nonlinear cohomology''. The comparison of these structures sheds some light on the general mirror problem: Establishing isomorphisms between Frobenius manifolds of different classes.
Another theme is the study of tensor products of Frobenius manifolds, corresponding respectively to the Künneth formula in quantum cohomology, direct sum of singularities in Saito's theory, and presumably, the tensor product of the differential Gerstenhaber-Batalin-Vilkovyski algebras. We extend the initial Gepner's construction of mirrors to the context of Frobenius manifold and formulate the relevant mathematical conjecture.''
The article starts with recalling the notion of a Frobenius manifold as introduced by Dubrovin. It gives an account of Gromov-Witten invariants and continues with explaining the notion of Saito's framework which axiomatizes those properties of the space of miniversal deformations of isolated singularities of functions which lead to the Frobenius structure. The direct sum of Saito's frameworks corresponds to the tensor product of the associated Frobenius manifolds. The \(A_n\) singularities are considered in detail. The remaining sections contain a self-contained account of the Barannikov-Kontsevich construction [\textit{S. Barannikov} and \textit{M. Kontsevich}, Int. Math. Res. Not. 1998, 201-215 (1998; Zbl 0914.58004)] in the axiomatic context of differential Gerstenhaber-Batalin-Vilkovyski algebras. quantum cohomology; Frobenius manifolds; Gerstenhaber-Batalin-Vilkovyski algebras; Saito theory; mirror conjecture; Künneth formula; Gromov-Witten invariants; miniversal deformations Manin Yu.: Three constructions of Frobenius manifolds: a comparitive study. Asian J. Math 3, 179--220 (1999) Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Calabi-Yau manifolds (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Other operations on complex singularities Three constructions of Frobenius manifolds: A comparative study | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(W\) denote a complex vector space of dimension \(n\). For an integer \(d>1\) let \(\text{Ver}_d:W\to\text{Sym}^d W, w\mapsto w^d\), denote the \(d\)-th Veronese map. Its image \(Z\) is called the \(d\)-th Veronese cone. The group \(\mathrm{GL}_n(\mathbb{C})\) of invertible linear transformations of \(W\) acts on \(\text{Sym}^d W\) preserving the Veronese cone \(Z\). Let \(\mathcal{D}\) be the Weyl algebra of differential operators with polynomial coefficients on the vector space \(\text{Sym}^d W\). The main results of the paper are:
(1) The description of the structure (as \(\mathrm{GL}(W)\)-representations) of the simple \(\mathrm{GL}(W)\)-equivariant holonomic \(\mathcal{D}\)-modules whose support is \(Z\). For \(d=2\) it provides a counterexample to a conjecture of Levasseur (see [\textit{T. Levasseur}, Int. Math. Res. Not. 2009, No. 3, 462--511 (2009; Zbl 1167.22006)].
(2) Let \(S=\text{Sym}(\text{Sym}^d V)\) denote the ring of polynomial functions on \(\text{Sym}^d W\). It is shown that there is a unique non-vanishing local cohomology module of \(S\) with support in \(Z\) (those for \(\text{codim}(Z)\)) (see also [\textit{A. Ogus}, Ann. Math. (2) 98, 327--365 (1973; Zbl 0308.14003)]). It is is one of the \(\mathcal{D}\)-modules in (1). Moreover, its character is described.
By the Riemann-Hilbert correspondence there are precisely \((d+1)\) simple \(\mathrm{GL}(W)\)-equivariant \(\mathcal{D}\)-modules whose support is contained in \(Z\). The main focus of the paper is the computation the character module of them. \(\mathcal{D}\)-modules; Veronese cones; local cohomology C. Raicu , 'Characters of equivariant \(\mathcal {D}\) -modules on spaces of matrices', Compositio Math., Preprint, 2015, arXiv 1507.06621. Local cohomology and commutative rings, Homogeneous spaces and generalizations, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, de Rham cohomology and algebraic geometry Characters of equivariant \(\mathcal {D}\)-modules on Veronese cones | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The odd symplectic Grassmannian \(\text{IG}:=\text{IG}(k, 2n+1)\) parametrizes \(k\) dimensional subspaces of \({\mathbb {C}}^{2n+1}\) which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on \(\text{IG}\) with two orbits, and \(\text{IG}\) is itself a smooth Schubert variety in the submaximal isotropic Grassmannian \(\text{IG}(k, 2n+2)\). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of \(\text{IG}\), i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case \(k=2\), and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Equivariant quantum cohomology of the odd symplectic Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper answers affirmatively the question in the survey article by \textit{P. A. Griffiths} [``An introduction to the theory of special divisors on algebraic curves'', Regional Conf. Ser. Math. 44 (1980; Zbl 0446.14010)]: Does the variety \(W_d^r\) of linear systems on a general curve of genus \(g\) with degree \(d\) and dimension at least \(r\) have the Brill-Noether dimension \(g - (r +1)(g - d +r)\)? Moreover the authors determine the class of this variety in the cohomology of the Jacobian and show that it is without multiple components.
The proof is by a detailed geometrical analysis of a classical degeneration idea of Castelnuovo's. It is formalized as the Castelnuovo-Severi-Kleiman conjecture (CSK): the family of \(P^k\)'s in \(P^d\) meeting the chords of a rational normal curve in \(P^d\) has the same dimension as if the chords were lines in general position, and the family has no multiple components.
The paper has three parts: (I) The reduction to CSK -- except for the absence of multiple components to \(W_d^r\); (II) The proof of CSK; (III) Proofs of the absence of multiple components.
(I) has been previously achieved by \textit{S. Kleiman} [Adv. Math. 22, 1--31 (1976; Zbl 0342.14012)]. The proof in this paper is by a geometrical argument based on duality of special divisors. (II) uses a degeneration of the chords to span an osculating flag. (III) is again by degeneration. The degenerate case is chosen so that number of intersections of two varieties as a set equals the algebraic intersection number. The techniques are those of classical algebraic geometry and Schubert calculus. variety of linear systems on a general curve; cohomology of Jacobian; Castelnuovo-Severi-Kleiman conjecture; algebraic intersection number; Schubert calculus Griffiths, P. \& Harris, J.,On the variety of special linear systems on a general algebraic curves, Duke Math. J.,47(1980), 233--272. Jacobians, Prym varieties, Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus, Enumerative problems (combinatorial problems) in algebraic geometry On the variety of special linear systems on a general algebraic curve | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In recent years the theory of Gröbner bases has found several applications in various fields of symbolic computations, in particular in applications related to combinatorics. The second author of the present book has opened the view for further applications in combinatorics [see \textit{B. Sturmfels}, ``Gröbner bases and convex polytopes'', Univ. Lect. Ser. 8 (Providence 1996; Zbl 0856.13020)].
The present book offers another application of Gröbner bases related to new algorithms for dealing with rings of differential operators. Here the Gröbner bases are reexamined from the point of geometric deformations. More precisely it provides symbolic algorithms for constructing holomorphic solutions to systems of linear partial differential equations with polynomial coefficients. Such a system is represented by a left ideal \(I\) in the Weyl algebra \(D = {\mathbb C} \langle x_1, \ldots, x_n, \partial_1, \ldots, \partial_n \rangle.\) A Gröbner deformation of the left ideal \(I\) is the initial ideal \(\text{in}_{(-w,w)}(I) \subset D\) with respect to some generic weight vector \(w = (w_1, \ldots, w_n)\) with real coordinates \(w_i.\) By methods from computational commutative algebra there is an explicit solution basis for the Gröbner deformation \(\text{in}_{(-w,w)}(I).\) The problem is to extend it to a solution basis for \(I.\) This is solved under the natural hypothesis that the given \(D\)-ideal \(I\) is regular holonomic. This is valid, for instance, for the \(D\)-ideals representing hypergeometeric integrals.
The first chapter of the book is devoted to the basic notations. The classical Gauss' hypergeometric function is expressed in the Gel'fand-Kapranov-Zelevinsky (GKZ) scheme. It contains also an introduction to holonomic systems of differential equations from the Gröbner basis point of view.
The second chapter is concerned with solving regular holonomic systems. The holomorphic solutions of \(I\) around a generic point in \({\mathbb C}^n\) form a vector space. If \(I\) is holonomic it is of finite dimension \(\text{rank} (I).\) The terms of such a holomorphic solution \(f\) are partially ordered by means of the generic weight vector \(w.\) The sum of the smallest terms, \(\text{in}_w(f),\) is a solution to the Gröbner deformation \(\text{in}_{(-w,w)} (I).\) The strategy of the solution is to find \(\text{in}_w(f)\) first and to construct \(f\) from it. This approach is justified by the inequality \(\text{rank in}_{(-w,w)} (I) \leq \text{rank} (I).\) It has to be an equality, which is true under certain circumstances. Moreover it is shown by an algorithmic approach that the equality holds whenever the system is regular holonomic. The main combinatorical arguments of the proofs grow out of the fact that the initial \(D\)-ideal \(\text{in}_{(-w,w)}(I)\) is torus-fixed for a generic \(w.\)
The chapter 3 is concerned with the GKZ-hypergeometric system \(H_A(\beta)\) associated with an integer matrix \(A\) and a complex parameter vector \(\beta,\) introduced by \textit{I. M. Gel'fand, A. V. Zelevinsky} and \textit{M. M. Kapranov} [Funct. Anal. Appl. 23, No. 2, 94-106 (1989); translation from Funkts. Anal. Prilozh. 23, No. 2, 12-26 (1989; Zbl 0787.33012)]. The matrix \(A\) represents a toric variety. If the variety lies in a projective space, then \(H_A(\beta)\) is regular holonomic, which is assumed throughout. The degree of the variety is called \(\text{vol}(A),\) since it coincides with the volume of the polytope spanned by \(A.\) By means of explicit deformations in the space of parameters the fundamental inequality \( \text{rank}(H_A(\beta)) \geq \text{vol}(A)\) is shown.
In chapter 4 the authors investigate when equality in the last estimate is true. Originally this was claimed to be true in general by Gel'fand, Zelevinsky and Kapranov [loc. cit.], which does not hold. Here the authors investigate three sufficient conditions for the equality. The first one is the Cohen-Macaulayness of a certain toric ideal \(I_A,\) the second, the so-called semi-resonance of the parameter \(\beta,\) and the third, the \(w\)-flatness of the parameter \(\beta,\) which depends on the Gröbner deformation. Moreover there is also an upper bound for the rank in general and an exact formula in the case of dimension 2.
The chapter 5 is devoted to the integration of \(D\)-modules. More precisely, hypergeometric functions arise naturally from integrals. It is the authors' goal to present algorithms for computing asymptotic expansions of these kind of integrals by the following steps: First they compute the \(D\)-ideal consisting of all operators that annihilate the integrand. Secondly they find the annihilators of the integral via the machinery of \(D\)-module theoretic integration. Thirdly they compute the Nilson series expansion of the integral. Besides of this, the chapter serves also as a more general introduction to algorithms in algebraic geometry based on \(D\)-modules. The GKZ system remains the focus example for all the general concepts and constructions in the same spirit as toric varieties serve as a ubiquitous source of examples in algebraic geometry.
In an appendix there is a description of current computer systems for \(D\)-modules and their design. This completes the algorithmic picture of the whole book with concrete samples of computations. The book is well written. The project for the book started when the authors came together to work on joint research on topics now contained in chapter 4. Then they started to develop all the necessary basic material about \(D\)-modules and linear partial differential equations not available in the literature.
The monograph requires a consequent reading in order to discover all the beauties and the surprising connections between several different branches of mathematics, coming together in the text. This book contains a number of original research results on holonomic systems and hypergeometric functions. The reviewer is sure that it will be the standard reference for computational aspects and research on \(D\)-modules in the future. It raises many open problems for future work in this area. Gröbner deformation; hypergeometric differential equation; Gröbner bases; Gröbner fan; \(D\)-modules; holonomic systems; toric variety Saito, M., Sturmfels, B., Takayama, N.: Gröbner Deformations of Hypergeometric Differential Equations, Vol.~6 of Algorithms and Computation in Mathematics. Springer, Berlin (2000) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Symbolic computation and algebraic computation, Research exposition (monographs, survey articles) pertaining to commutative algebra, Research exposition (monographs, survey articles) pertaining to computer science, Basic hypergeometric integrals and functions defined by them, Commutative rings of differential operators and their modules, Toric varieties, Newton polyhedra, Okounkov bodies, Rings of differential operators (associative algebraic aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Gröbner deformations of hypergeometric differential equations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For any triple \((i,a,\mu)\) consisting of a vertex \(i\) in a quiver \(Q\), a positive integer \(a\), and a dominant \(\operatorname{GL}_a\)-weight \(\mu \), we define a quiver current \(H^{(i,a)}_\mu\) acting on the tensor power \(\Lambda^{\!Q}\) of symmetric functions over the vertices of~\(Q\). These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple \((i=(i_1,\ldots,i_m),a=(a_1,\ldots,a_m),(\mu(1),\ldots,\mu(m)))\) of sequences of such data, we define the quiver Hall-Littlewood function \(H^{\mathbf{i},\mathbf{a}}_{\mu(\,\cdot\,)}\) as the result of acting on \(1\in\Lambda^{\!Q}\) by the corresponding sequence of quiver currents. The quiver Kostka-Shoji polynomials are the expansion coefficients of \(H^{\mathbf{i},\mathbf{a}}_{\mu(\,\cdot\,)}\) in the tensor Schur basis. These polynomials include the Kostka-Foulkes polynomials and parabolic Kostka polynomials (Jordan quiver) and the Kostka-Shoji polynomials (cyclic quiver) as special cases. We show that the quiver Kostka-Shoji polynomials are graded multiplicities in the equivariant Euler characteristic of a vector bundle (determined by \(\mu(\,\cdot\,))\) on Lusztig's convolution diagram determined by the sequences \(\mathbf{i} \)~and~\( \mathbf{a} \). For certain compositions of currents we conjecture higher cohomology vanishing of the associated vector bundle on Lusztig's convolution diagram. For quivers with no branching, we propose an explicit positive formula for the quiver Kostka-Shoji polynomials in terms of catabolizable multitableaux. We also relate our constructions to \(K\)-theoretic Hall algebras, by realizing the quiver Kostka-Shoji polynomials as natural structure constants and showing that the quiver currents provide a symmetric function lifting of the corresponding shuffle product. In the case of a cyclic quiver, we explain how the quiver currents arise in Saito's vertex representation of the quantum toroidal algebra of type \(\mathfrak{sl}_r\). quiver Hall-Littlewood functions; Kostka-Shoji polynomials Representations of quivers and partially ordered sets, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Representation theory for linear algebraic groups Quiver Hall-Littlewood functions and Kostka-Shoji polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce affine Stanley symmetric functions for the special orthogonal groups, a class of symmetric functions that model the cohomology of the affine Grassmannian, continuing the work of Lam and Lam, Schilling, and Shimozono on the special linear and symplectic groups, respectively. For the odd orthogonal groups, a Hopf-algebra isomorphism is given, identifying (co)homology Schubert classes with symmetric functions.
For the even orthogonal groups, we conjecture an approximate model of (co)homology via symmetric functions. In the process, we develop type \(B\) and type \(D\) non-commutative \(k\)-Schur functions as elements of the affine nilCoxeter algebra that model homology of the affine Grassmannian. Additionally, Pieri rules for multiplication by special Schubert classes in homology are given in both cases.
Finally, we present a type-free interpretation of Pieri factors, used in the definition of noncommutative \(k\)-Schur functions or affine Stanley symmetric functions for any classical type. affine Schubert calculus; Stanley symmetric functions; Pieri factors Pon, S.: Affine Stanley symmetric functions for classical types. J. combin. Theory ser. A 36, No. 4, 595-622 (2012) Symmetric functions and generalizations, Classical problems, Schubert calculus Affine Stanley symmetric functions for classical types | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If \((X, \widehat{X})\) is a mirror pair of Calabi-Yau manifolds, Kontsevich's homological mirror conjecture implies that \(H^\ast (X, \wedge^\ast T_X) = H^\ast (\widehat{X}, \mathbb C)\). The left hand side of this relation can be identified with the tangent space at \(X\) to the extended moduli space \(\mathcal M_{\text{compl}}\) of complex structures. This moduli space is the base of semi-infinite \(B\)-variations \(\text{VHS}^B (X)\) of Hodge structures in \(H^\ast (\widehat{X}, \mathbb C)\). On the other hand, the right hand side of the relation is the tangent space to the extended moduli space \(\mathcal M_{\text{sympl}}\) of Kähler forms on \(\widehat{X}\).
In this paper the author constructs from semi-infinite \(A\)-variations of Hodge structures over \(\mathcal M_{\text{sympl}}\) a family of solutions of the WDVV equations parametrized by isotropic increasing filtrations which are complementary to the Hodge type filtration in \(\bigoplus_{i,j} H^i (X, \wedge^j T_X)\). He also establishes canonical isomorphisms \(\text{VHS}^A (X) = \text{VHS}^B (\widehat{X})\) and \(\text{VHS}^B (X) = \text{VHS}^A (\widehat{X})\) for dual torus fibrations. Kontsevich's homological mirror conjecture; Calabi-Yau manifolds; mirror symmetry Calabi-Yau theory (complex-analytic aspects), Calabi-Yau manifolds (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects) Semi-infinite \(A\)-variations of Hodge structure over extended Kähler cone | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quiver Grassmannian is the projective variety of subrepresentations of a finite-dimensional representation of a quiver with a fixed dimension vector.
Quiver Grassmannians occur naturally in different contexts. Fomin and Zelevinsky introduced cluster algebras in 2000. Caldero and Keller used Euler characteristics of quiver Grassmannians for the categorification of acyclic cluster algebras. This was generalized to arbitrary antisymmetric cluster algebras by Derksen, Weyman and Zelevinsky. The quiver Grassmannians play a crucial role in the construction of Ringel-Hall algebras. Moreover, they arise in the study of general representations of quivers by Schofield and in the theory of local models of Shimura varieties. Motivated by this, we study the geometric properties of quiver Grassmannians, their Euler characteristics and Ringel-Hall algebras. This work is divided into three parts.
In the first part of this thesis, we study geometric properties of quiver Grassmannians. In some cases we compute the dimension of this variety, we detect smooth points and we prove semicontinuity of the rank functions and of the dimensions of homomorphism spaces. Moreover, we compare the geometry of quiver Grassmannians with the geometry of the module varieties and we develop tools to decompose the quiver Grassmannian into irreducible components.
In the following we consider some special classes of quiver representations, called string, tree and band modules. There is an important family of finite-dimensional algebras, called string algebras, such that each indecomposable module is either a string or a band module.
In the second part, for the complex field we compute the Euler characteristics of quiver Grassmannians and of quiver flag varieties in the case that the quiver representation is a direct sum of string, tree and band modules. We prove that these Euler characteristics are positive if the corresponding variety is non-empty. This generalizes some results of Cerulli Irelli.
In the third part, we consider the Ringel-Hall algebra of a string algebra. We give a complete combinatorial description of the product of an important subalgebra of the Ringel-Hall algebra.
In covering theory we resemble the results of the last two parts. Euler characteristics; coverings; quiver Grassmannians; flag varieties; representations of quivers; Ringel-Hall algebras; string algebras; band modules; string modules; tree modules Haupt, N.: Euler Characteristic and Geometric Properties of Quiver Grassmannians. Ph.D Thesis, University of Bonn (2011) Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Group actions on affine varieties, Representation theory of lattices Euler characteristics and geometric properties of quiver Grassmannians. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0716.00007.]
Let \(G\) be a semi-simple algebraic group over \(\mathbb{C}\), and \(B\) a Borel subgroup of \(G\). Then one knows that the cohomology ring \(H^*(G/B;\mathbb{C})\) is the coinvariant algebra \(A(h)/I^ W\) associated to a certain subalgebra \(h\) of the Lie algebra of \(G\) (here, \(A(h)\) is the coordinate ring of \(h\), and \(I^ W\) is the homogeneous ideal generated by the \(W\)-invariant functions \(f\) on \(h\) such that \(f(0)=0)\).
In this paper, the author proves a similar result for \(H^*(X,\mathbb{C})\), \(X\) being a Schubert variety in \(G/P\), for a parabolic subgroup \(P\supseteq B\). This paper makes a good contribution to the cohomology theory of Schubert varieties. flag manifold; cohomology theory of Schubert varieties Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Cohomology theory for linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions, Classical real and complex (co)homology in algebraic geometry \(SL_ 2\) actions and cohomology of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper provides an example of a simple highest weight module over \(\mathfrak{sl}_{12}(\mathbb{C})\) whose characteristic variety is reducible. The proof of reducibility is rather indirect, it uses the theories of \(p\)-canonical bases, \(W\)-graphs and perverse sheaves. More precisely, the paper gives two permutations \(x\) and \(y\) in \(S_{12}\) which lie in the same right Kazhdan-Lusztig cell and such that a normal slice to the Schubert variety corresponding to \(y\) along the Schubert cell corresponding to \(x\) is isomorphic to the Kashiwara-Saito singularity. This is equivalent to the reducibility of a certain characteristic variety. That characteristic varieties in other types can be reducible was already known. characteristic variety; Schubert variety; Kazhdan-Lusztig cell; highest weight module; Kashiwara-Saito singularity Williamson, Geordie, A reducible characteristic variety in type \(A\).Representations of reductive groups, Prog. Math. Phys. 312, 517-532, (2015), Birkhäuser/Springer, Cham Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Intersection homology and cohomology in algebraic topology, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grassmannians, Schubert varieties, flag manifolds A reducible characteristic variety in type \(A\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give an algebraic proof of the determinant formulas for factorial Grothendieck polynomials obtained by \textit{T. Hudson} et al. [Adv. Math. 320, 115--156 (2017; Zbl 1401.19008)] by \textit{T. Hudson} and \textit{T. Matsumura} [``Segre classes and Kempf-Laksov formula in algebraic cobordism'', Preprint, \url{arXiv:1602.05704}]. symmetric polynomials; Grothendieck polynomials; \(K\)-theory; Grassmannians; Schubert varieties Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, \(K\)-theory of schemes, Algebraic cycles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] An algebraic proof of determinant formulas of Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A theorem of the first author states that the cotangent bundle of the type \(\mathbf{A}\) Grassmannian variety can be canonically embedded as an open subset of a smooth Schubert variety in a two-step affine partial flag variety (see [\textit{V. Lakshmibai}, Transform. Groups 21, No. 2, 519--530 (2016; Zbl 1390.14148)]).
That theorem was inspired by works of Lusztig and Strickland. In particular, \textit{G. Lusztig} [J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008)] related certain orbit closures arising from the type \(\mathbf{A}\) cyclic quiver with \(h\) vertices to affine Schubert varieties. In the case \(h = 2\), Strickland [\textit{E. Strickland}, J. Algebra 75, 523--537 (1982; Zbl 0493.14030)] relates such orbit closures to conormal varieties of determinantal varieties; furthermore, any determinantal variety can be canonically realized as an open subset of a Schubert variety in the Grassmannian (see [\textit{V. Lakshmibai} and \textit{C. S. Seshadri}, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1--54 (1978; Zbl 0447.14011)]).
In this paper, the authors extend the result of [\textit{V. Lakshmibai}, Transform. Groups 21, No. 2, 519--530 (2016; Zbl 1390.14148)] to cominuscule generalized Grassmannians of arbitrary finite type (such Grassmannians occur in types \(\mathbf{A}-\mathbf{E}\)). Schubert varieties; Grassmannian; affine flag varieties Grassmannians, Schubert varieties, flag manifolds The cotangent bundle of a cominuscule Grassmanian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over \(\mathbb{C}\) and point counts over \(\mathbb{F}_q)\) to Khovanov-Rozansky homology of the associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational \(q,t\)-Catalan numbers. Via the curious Lefschetz property, this implies the \(q,t\)-symmetry and unimodality properties of rational \(q,t\)-Catalan numbers. We show that the \(q,t\)-symmetry phenomenon is a manifestation of Koszul duality for category \(\mathcal{O}\), and discuss relations with equivariant derived categories of flag varieties, and open Richardson varieties. positroid varieties; \(q,t\)-Catalan numbers; HOMFLY polynomial; Khovanov-Rozansky homology; mixed Hodge structure; equivariant cohomology; Koszul duality Grassmannians, Schubert varieties, flag manifolds, Polynomial rings and ideals; rings of integer-valued polynomials Positroids, knots, and \(q,t\)-Catalan numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(K\) be an infinite field. Let \(A\) be a Noetherian \(K\)-algebra and \(\mathbb{P}_{A}^{n}\) be the \(n\)-dimensional projective space over \(A\). In this paper, closed projective schemes are investigated. These projective schemes are investigated in the flavour of the lifting problem, stated as follows: given a closed subscheme \(Y \subset\mathbb{P}_{K}^{n-1}\), explicitly describe all closed subschemes \(W \subset \mathbb{P}_{A}^{n}\) such that \(Y\) is a general hyperplane section of \(W\), up to an extension of scalars.
The classical approach consists of considering a general hyperplane section of \(W\). The advantage of this approach is that many properties of \(W\) are preserved under general hyperplane sections. The converse approach (i.e. the lifting problem) occurs when one proceeds to the inverse problem. It consists in finding a scheme \(W\), starting from a possible hyperplane section \(Y\). Every such scheme \(W\) is called a lifting of \(Y\) and the saturated defining ideal \(I\) of \(W\) is called a lifting of the saturated defining ideal \(I'\) of \(Y\). All the liftings of \(Y\) with a given Hilbert polynomial \(p(t)\) are characterised by a parameter scheme and is described through the functor it represents. The tools which are used come from methods in Gröbner basis and marked bases theories.
More precisely, it is proven that a subscheme \(W\) is a lifting of \(Y\) only if \(I\) belongs to a Gröbner stratum over a monomial ideal \(J\), which is a lifting of the initial ideal of \(I'\) (this is described in the Theorem 4.5). The proofs of these results are constructive and produce a method for the computation of the locally closed subschemes in a Hilbert scheme, whose disjoint union corresponds to all liftings with a given Hilbert polynomial via Gröbner strata. The main result is resumed as follows:
Theorem A. Let \(p(t)\) be the Hilbert polynomial of a lifting of \(Y\).
(1) If \(Y\) is equidimensional, the the family of the equidimensional liftings of \(Y\) with Hilbert polynnomial \(p(t)\) is parametrized by a locally closed subscheme of \(\mathrm{Hilb}^np(t)\).
(2) The family of the liftings of \(Y\) with Hilbert polynomial \(p(t)\) is parameterized by a subscheme of \(\mathrm{Hilb}^np(t)\) which can be explicitly constructed.
By part (2) of this theorem, all the liftings of \(Y\) with a given Hilbert polynomial are characterised by a parameter scheme that is obtained by gluing suitable affine open subschemes in a Hilbert scheme and is described through the functor it represents. Algorithms for computing parameter schemes for the liftings of a saturated homogeneous ideal \(I' \subset K[x]\) over a Noetherian \(K\)-algebra \(A\) in \(Hilb^n_p(t)\) by means of Gröbner strata are given.
Investigations in this topic can produce methods to obtain schemes with specific properties. For instance, the following proposition is given:
Proposition 3.5. Let \(W\) be a lifting of a scheme \(Y\). If \(Y\) is smooth on a point \(P\) then also \(W\) is smooth on \(P\).
Finally, many examples of explicit computations are provided. lifting; Gröbner basis; marked basis; equidimensionality Projective techniques in algebraic geometry, Linkage, complete intersections and determinantal ideals, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Computational aspects of higher-dimensional varieties Functors of liftings of projective schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review, the author gives a concise survey of recent developments on embeddings of orthogonal Grassmannians. After a brief review of basic definitions and notions of projective and Veronesean embeddings of point-line geometries, the author restricts himself to Grassmannians \(\mathcal B_{n,k}\) and \(\mathcal D_{n,k}\) of buildings of types \(B_n\) and \(D_n\) and seven associated embeddings. Some of the embeddings are known to be isomorphic in case the characteristic of the underlying field is not 2. Section 4 of the paper is mostly devoted to discussing the various embeddings in case of characteristic 2. Sketches of proofs are provided in order to convey the flavour of the arguments. The results of this section as well as of the following section on universality of embeddings are a summary of three papers by \textit{I. Cardinali} and \textit{A. Pasini} [J. Algebr. Comb. 38, No. 4, 863--888 (2013; Zbl 1297.14053); J. Comb. Theory, Ser. A 120, No. 6, 1328--1350 (2013; Zbl 1278.05052)] and [J. Group Theory 17, No. 4, 559--588 (2014; Zbl 1320.20041)].
The last section deals with universality of Weyl embeddings of \(\mathcal B_{n,k}\) and \(\mathcal D_{n,k}\). These are obtained from the fundamental dominant weights for the root system of types \(B_n\) and \(D_n\). \textit{A. Kasikova} and \textit{E. Shult} [J. Algebra 238, No. 1, 265--291 (2001; Zbl 0988.51001)] showed that most of these point-geometries admit universal projective embeddings. The author conjectures that the Weyl embeddings of \(\mathcal B_{n,k}\) and \(\mathcal D_{n,k}\) are universal for \(k = 2, \dots, n-1\); the ones when \(k = 1\) are known to be universal. He then considers the cases \(k = 2\) and 3 under additional assumptions and outlines a proof of universality in these situations. point-line geometry; orthogonal polar space; Grassmannian; Weyl module; Veronese variety; embedding; universal embedding Incidence structures embeddable into projective geometries, Buildings and the geometry of diagrams, Grassmannians, Schubert varieties, flag manifolds, Modular representations and characters, Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Modular Lie (super)algebras, Polar geometry, symplectic spaces, orthogonal spaces Embeddings of orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the T-equivariant generalized cohomology of flag varieties using two models, the Borel model and the moment graph model. We study the differences between the Schubert classes and the Bott-Samelson classes. After setup of the general framework we compute, for classes of Schubert varieties of complex dimension \(<4\) in rank 2 (including \(A_2\), \(B_2\), \(G_2\) and \(A_1^{(1)}\)), moment graph representatives, Pieri-Chevalley formulas and products of Schubert classes. These computations generalize the computations in equivariant \(K\)-theory for rank 2 cases which are given in [\textit{S. Griffeth} and \textit{A. Ram}, Eur. J. Comb. 25, No. 8, 1263--1283 (2004; Zbl 1076.14068)]. Ganter, N; Ram, A, Generalized Schubert calculus, J. Ramanujan Math. Soc., 28A, 149-190, (2013) Homogeneous spaces and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Classical problems, Schubert calculus Generalized Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider a \(d\)-dimensional well-formed weighted projective space \(\mathbb{P}(\overline{w})\) as a toric variety associated with a fan \(\varSigma(\overline{w})\) in \(N_{\overline{w}}\otimes\mathbb{R}\) whose 1-dimensional cones are spanned by primitive vectors \(v_0,v_1,\ldots,v_d\in N_{\overline{w}}\) generating a lattice \(N_{\overline{w}}\) and satisfying the linear relation \(\sum_iw_iv_i=0\). For any fixed dimension \(d\), there exist only finitely many weight vectors \(\overline{w}=(w_0,\ldots,w_d)\) such that \(\mathbb{P}(\overline{w})\) contains a quasi-smooth Calabi-Yau hypersurface \(X_w\) defined by a transverse weighted homogeneous polynomial \(W\) of degree \(w=\sum_{i=0}^dw_i\). Using a formula of Vafa for the orbifold Euler number \(\chi_{\mathrm{orb}}(X_w)\), we show that for any quasi-smooth Calabi-Yau hypersurface \(X_w\) the number \((-1)^{d-1}\chi_{\mathrm{orb}}(X_w)\) equals the stringy Euler number \(\chi_{\mathrm{str}}(X_{\overline{w}}^\ast)\) of Calabi-Yau compactifications \(X_{\overline{w}}^\ast\) of affine toric hypersurfaces \(Z_{\overline{w}}\) defined by non-degenerate Laurent polynomials \(f_{\overline{w}}\in\mathbb{C}[N_{\overline{w}}]\) with Newton polytope \(\mathrm{conv}(\{v_0,\ldots,v_d\})\). In the moduli space of Laurent polynomials \(f_{\overline{w}}\) there always exists a special point \(f_{\overline{w}}^0\) defining a mirror \(X_{\overline{w}}^\ast\) with a \(\mathbb{Z}\slash w\mathbb{Z}\)-symmetry group such that \(X_{\overline{w}}^\ast\) is birational to a quotient of a Fermat hypersurface via a Shioda map. mirror symmetry; Calabi-Yau hypersurfaces; toric varieties; Newton polytopes Mirror symmetry (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, \(3\)-folds, Fano varieties Mirror symmetry for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a continuation of the author's papers of the same titles, I, II and III [Adv. Stud. Pure Math. 6, 255-287 (1985; Zbl 0569.20032); J. Algebra 109, 536-548 (1987; Zbl 0625.20032); J. Fac. Sci. Univ. Tokyo, Sect. I A 34, 223-243 (1987; Zbl 0631.20028)]. The main result of this paper is the construction of a bijection between the set of two sided cells in an affine Weyl group W and the set of unipotent classes in a certain complex reductive group G, associated with W. The author shows also that the value of the \(a\)-function on a two sided cell of W (defined in I) is equal to the dimension of the variety of Borel subgroups of G containing an element of the corresponding unipotent class. The proof is based on the representation theory of affine Hecke algebras and of the algebras J introduced and studied in II and III. He also solves some conjectures which have been stated by himself [cf. Proc. Symp. Pure Math. 37, 313-317 (1980; Zbl 0453.20005)] and further, he states some new conjectures relating the algebras J with certain equivariant vector bundles. Coxeter group; two sided cells; affine Weyl group; unipotent classes; complex reductive group; variety of Borel subgroups; affine Hecke algebras; equivariant vector bundles Lusztig, G., Cells in affine Weyl groups, IV, \textit{J. Fac. Sci. Univ. Tokyo Sect. IA. Math.}, 36, 297-328, (1989) Representation theory for linear algebraic groups, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Cells in affine Weyl groups. IV | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we study the structure of the Schubert varieties in the flag manifold Fl(n) associated to Sl(n,\({\mathbb{C}})\). Two main results are obtained.
First, a combinatorial algorithm is obtained whereby one can determine the singular locus of a Schubert variety. This is summarized in the following two theorems:
Theorem I: Let \(X=X(s)\) be the Schubert variety in Fl(n) corresponding to the permutation s. Let \(A=A(s)\) be the set of all pairs (i,j) which satisfy (i) \(1\leq j<i\leq n\) and (ii) either \(\{s(1),...,s(j)\}\subset \{1,2,...,i-1\}\) or \(\{1,...,j\}\subset \{s(1),...,s(i-1)\}.\quad -\) Then X is \(non-\sin gular\quad \Leftrightarrow \quad \#A=co\dim X.\)
Theorem II: Let \(X=X(s)\) be as above and let \(Y=Y(t)\) be contained in X. For every pair (k,m), \(1\leq k,m\leq n\) let \(B(k,m)=\emptyset\) if \(\#(\{t(1),...,t(k)\}\cap \{1,...,m\})\neq \#(\{s(1),...,s(k)\}\cap \{1,.\quad..,m\})\) while if \(\#(\{t(1),...,t(k)\}\cap \{1,...,m\})=\#(\{s(1),...,s(k)\}\cap \{1,..\quad.,m\})\) let \(B(k,m)=the\) set of all pairs (i,j) which satisfy (1) i\(>m\) and \(i\not\in (\{t(1),...,t(k)\}\cap \{m+1,...,n\})\) and (2) \(j\leq k\) and t(j)\(\leq m\). Let B be the union of all such B(k,m), \(1\leq k,m\leq n\). Then X is non- singular \(along Y\quad \Leftrightarrow \quad \#B=co\dim X.\)
Secondly, a geometric description of the non-singular Schubert varieties is obtained. This result may be summarized as follows: A certain class of geometrically defined Schubert varieties may be constructed inductively by beginning with a Grassmannian and repeatedly forming a fibre bundle whose fibre is a Grassmannian and whose base space is a variety which has already been constructed during the induction. We call these repeated fibrations of Grassmannians. These Schubert varieties are all obviously non-singular from the construction. We then show that every non-singular Schubert variety in Fl(n) coincides with one of these, and consequently is a repeated fibration of Grassmannians. Schubert varieties; flag manifold; repeated fibrations of Grassmannians Ryan, Kevin M., On Schubert varieties in the flag manifold of \(\operatorname{Sl}(n, \mathbf{C})\), Math. Ann., 276, 2, 205-224, (1987), MR 870962 Grassmannians, Schubert varieties, flag manifolds On Schubert varieties in the flag manifold of Sl(n,\({\mathbb C})\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The cohomology of the affine flag variety \(\hat{F}l_G\) of a complex reductive group \(G\) is a comodule over the cohomology of the affine Grassmannian \(\mathrm{Gr}_G\). We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite Schubert classes, in (torus-equivariant) cohomology and \(K\)-theory. As an application, we deduce monomial positivity for the affine Schubert polynomials of the second author. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds On the coproduct in affine Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper proposes a generalization of the well-known triality between simply-laced Dynkin diagrams, simple Lie algebras, and Kleinian groups (with associated quotient singularities), and provides an account of the results achieved in this direction by the author and others. The exposition is exhaustive while concise, and subtle points and open questions are emphasized throughout the text.
The idea is to attach a surface singularity and a Lie algebra to a regular system of weights, which denotes a system of four integers \(W:=(a,b,c,h)\). A number of requirements is imposed upon this singularity and this algebra, in order to make this a generalization of a usual triality picture described above and also of elliptic Lie algebra theory. A singularity is defined as a zero set of a generic polynomial \(f_W\) of degree \(h\) in a weighted projective space \(P(a:b:c)\). An algebra is constructed through an intermediate step, which includes a construction of the homotopy category of graded matrix factorizations for \(f_W\), establishing a strongly exceptional collection there, and taking its associated quiver to construct a Lie algebra.
On the ``geometrical side'', investigating the singularities obtained from regular weight systems, and their vanishing cycle lattices, the author presents a notion of \(*\)-duality for regular weight systems. On one hand, it is shown to underlie the ``strange'' duality of Arnold; on the other, the author cites \textit{A. Takahashi} [Commun. Math. Phys. 205, No. 3, 571--586 (1999; Zbl 0974.14005)] where it is proven that the \(*\)-duality is equivalent to mirror symmetry for Landau-Ginzburg models. regular system of weights; simple Lie algebras; elliptic Lie algebras; homotopy category of matrix factorizations; vanishing cycles; \(*\)-duality Saito, K.: Towards a categorical construction of Lie algebras, Adv. stud. Pure math. 50, 101-175 (2008) Categories in geometry and topology, Lie algebras and Lie superalgebras, Singularities of surfaces or higher-dimensional varieties, Homology and cohomology theories in algebraic topology, Simple, semisimple, reductive (super)algebras, Derived categories, triangulated categories, Calabi-Yau manifolds (algebro-geometric aspects) Towards a categorical construction on Lie algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This interesting paper builds a bridge between two parts of noncommutative geometry: noncommutative projective geometry and quantum groups. A general theory of flag varieties for quantum groups is proposed; this amounts to considering as a quantum analog of a flag variety a well chosen graded algebra (specifically, the shape algebra) and to studing it with the methods developed by Artin, Tate, van den Bergh, Stafford and others. These considerations form Part I of the paper.
In Part II, the case of Drinfeld-Jimbo quantum groups is considered in more detail. The preceding construction is conjecturally related to geometrical objects, namely certain schemes which are unions of certain subvarieties of the classical flag variety stable under the action of the corresponding torus.
Finally, in Part III the full program is carried out completely in the case of \(\text{SL}(n)\). The proof relies on specific properties of the representation theory of this group. quantum flag varieties Ohn, C.: ''Classical'' flag varieties for quantum groups: the standard quantum \(SL(n,C)\). Adv. math. 171, No. 1, 103-138 (2002) Quantum groups (quantized enveloping algebras) and related deformations, Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry ``Classical'' flag varieties for quantum groups: the standard quantum \(\text{SL}(n,{\mathbb C})\). | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a semi-simple simply connected algebraic group (assumed, for convenience, to be defined over an algebraically closed field of characteristic 0) and let \(B\subset G\) be a Borel subgroup and \(T\subset B\) a maximal torus. Let \(W\) be the Weyl group associated to the pair \((G,T)\). For any \(w\in W\), let \(X(w)= \overline {BwB}/B \subset G/B\) be the associated Schubert variety. Also let \(V(\lambda)\) be the (finite dimensional) irreducible \(G\)-module with highest weight \(\lambda\), and (for \(w\in W)\) let \(V_w(\lambda)\) be the Demazure module defined as the smallest \(B\)-submodule of \(V(\lambda)\) containing the extremal weight vector \(wv_\lambda\) (where \(v_\lambda\) is a highest weight vector of \(V(\lambda))\). As usual, denote by \(\rho\) the half sum of the positive roots of \(G\), and the set of positive roots of \(G\) is denoted by \(R^+\).
For any \(v\leq w\in W\) (where \(\leq\) is the Bruhat-Chevalley partial order on \(W)\), \(v\in X(w)\), in particular, the identity element \(e\in X(w)\). Let \(T(w,e)\) denote the Zariski tangent space of \(X(w)\) at \(e\). Then \(T(w,e)\) can be canonically thought of as a \(T\)-stable subspace of \(u^-\) (where \(u^-\) is the span of all the negative root spaces). -- The main result of the paper under review is the following:
Fix \(w\in W\) and let \(\beta\in R^+\). Then \(-\beta\) is a weight of \(T(w,e)\) iff \(\rho-\beta\) is not a weight of \({V(\rho) \over V_w (\rho)}\).
This result is obtained by using some results of P. Littelmann on the Lakshmibai-Seshadri paths. -- For \(\beta \in R^+\), let \(F_\beta\) denote a root vector corresponding to the negative root \(-\beta\).
As an immediate consequence of her result together with a result of Polo, the author obtains that for any \(\beta \in R^+\), \(F_\beta v_\rho \in V_w(\rho)\) iff for all (not necessarily distinct) \(\beta_1, \dots, \beta_\ell \in R^+\) with \(\sum \beta_i = \beta\), \(F_{\beta_1} \cdots F_{\beta_\ell} v_\rho \in V_w(\rho)\).
As another consequence of her result, she explicitly determines the tangent space \(T(w,e)\) for any \(w\in W\) and any classical \(G\) (i.e. \(G\) of type \(A, B, C, D)\), refining her earlier works (partly obtained with Seshadri and Rajeswari). singularity; Schubert variety; Demazure module; tangent space V. Lakshmibai,Tangent spaces to Schubert varieties, Mathematical Research Letters2 (1995), 473--477. Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Tangent spaces to Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The jeu-de-taquin-based Littlewood-Richardson rule of \textit{H. Thomas} and \textit{A. Yong} [Adv. Math. 222, No. 2, 596--620 (2009; Zbl 1208.14052)] for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, \textit{A. Skovsted Buch} and \textit{M. J. Samuel} [J. Reine Angew. Math. 719, 133--171 (2016; Zbl 1431.19001)] developed a combinatorial theory of ``unique rectification targets'' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to \(K\)-theory. Separately, \textit{P.-E. Chaput} and \textit{N. Perrin} [J. Lie Theory 22, No. 1, 17--80 (2012; Zbl 1244.14036)] used the combinatorics of Proctor's ``\(d\)-complete posets'' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of \(d\)-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain \(K\)-theoretic Schubert structure constants in the Kac-Moody setting. Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Equivariant \(K\)-theory Unique rectification in \(d\)-complete posets | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review the author studies a version of the projective bundle theorem. Recall that the projective bundle theorem [\textit{I. Panin}, \(K\)-Theory 30, No. 3, 265--314 (2003; Zbl 1047.19001)] claims that \(A(\mathbb{P}^n)\) is a truncated polynomial ring over \(A(\mathrm{pt})\) with an explicit basis in terms of the powers of a Chern class. This is one of the most basic and fundamental computations for oriented cohomology theories, and allows to define higher characteristic classes and compute the cohomology of Grassmann varieties and flag varieties. There are also analogous computations for symplectically oriented cohomology theories with appropriate chosen varieties: quaternionic projective spaces \(\mathbb{H}\mathbb{P}^n\) instead of the ordinary ones and symplectic Grassmannian and flag varieties. In the present paper the author establishes analogous results for the cohomology theories with special linear orientations: these were introduced by Panin and Walter in a series of (yet) unpublished preprints, and include the universal example of a cohomology theory with a special linear orientation, namely the algebraic special linear cobordisms \(MSL\). More approachable examples include devied Witt groups in the sense of \textit{P. Balmer} [J. Pure Appl. Algebra 141, No. 2, 101--129 (1999; Zbl 0972.18006)] oriented via Koszul complexes [\textit{A. Nenashev}, J. Pure Appl. Algebra 211, No. 1, 203--221 (2007; Zbl 1140.11024)]. The precise formulation of the results is probably too technical to include in this review. The author announces some applications of the developed theory, which include the proof of the fact that Witt groups arise from the hermitian \(K\)-theory through the techniques used in the paper, as well as conceivable applications to the theory of equivariant Witt groups. special linear orientation; stable Hopf map; Euler class; Pontryagin classes; Witt groups Ananyevskiy, A., The special linear version of the projective bundle theorem, Compos. Math., 151, 461-501, (2015) Motivic cohomology; motivic homotopy theory, Witt groups of rings, \(K\)-theory of forms The special linear version of the projective bundle theorem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The present paper is a continuation of the authors' previous paper [J. Reine Angew. Math. 609, 161--213 (2007; Zbl 1157.14013); preprint \url{arXiv:math/0402143}, to appear in J.~Reine Angew. Math.]. The goal is to show that a certain multiplicity function associated to a nearby cycle sheaf on an affine flag variety is a polynomial with an explicit bound on degrees, which is expressed in term of group-theoretic data. Let \(G\) be a split connected reductive algebraic group over \(\mathbb F_p\), with an algebraic closure \(k\). Let \({\mathcal Fl}=G(k((t))/{\mathcal B}_k\) denote the affine flag variety of \(G\). This is an ind-scheme defined over \(\mathbb F_p\) and is a disjoint union of strata \({\mathcal Fl}_w\) indexed by the extended affine Weyl group \(\widetilde W\). For \(w\in \widetilde W\), let \(IC_w\) denote the intersection complexes \(j_{w\, !*}\overline{\mathbb Q}_\ell[\ell(w)]\) on the closure \(\overline {\mathcal Fl}_w\), where \(j_{w\, !*}\overline{\mathbb Q}_\ell\) is the intermediate extension of \(\overline{\mathbb Q}_\ell\) for the immersion \(j_w:{\mathcal Fl}_w\hookrightarrow {\mathcal Fl}\).
(a) When \(G\) is \(\text{GL}_n\) or \(\text{ GSp}_{2n}\), there are an ind-scheme \(M\) over \(\mathbb Z_p\) which is a deformation of the affine Grassmannian \(\text{Grass}_{{\mathbb Q}_p}\) to the affine flag variety \({\mathcal Fl}\). This is the \(p\)-adic case.
(b) In the function field case, there is an ind-scheme \(\text{FL}_X\) over a curve \(X\) and a distinguished point \(x_0\) so that the fiber of \(\text{FL}_X\) is \({\mathcal Fl}\) at \(x=x_0\) and is \(\text{Grass}_k\times G/B\) at other points.
In both cases, let \({\mathcal Q}_\mu\subset \text{ Grass}\) denote the stratum indexed by a dominant coweight \(\mu\) of \(G\), and let \(IC_\mu\) denote the intersection complexes \(j_{\mu\, !*}\overline{\mathbb Q}_\ell[\dim {\mathcal Q}_\mu]\). Write \(R\Psi_\mu\) for \(R\Psi^M(IC_\mu)\) in case (a), and for \(R\Psi^{\text{ FL}_X}(IC_\mu\boxtimes \delta)\) in case (b), where \(\delta\) is the skyscraper sheaf supported on the base point of \(G/B\). This is a \({\mathcal B}\)-equivariant perverse sheaf on \({\mathcal Fl}\). It is shown in loc.~cit.~that \(R\Psi_\mu\) is of finite length and any of its irreducible subquotients has the form \(IC_w(i)\). Thus one may define non-negative integers \(m(R\Psi_\mu, w, i)\) via the identity
\[
R\Psi_\mu^{ss}\simeq \bigoplus_{w\in \widetilde W}\bigoplus_{i\in \mathbb Z} IC_w(-i)^{m(R\Psi_\mu, w, i)},
\]
and form a multiplicity function
\[
m( R\Psi_\mu,w):=\sum_{i\in \mathbb Z} m(R\Psi_\mu, w, i)q^i\in {\mathbb Z}[q,q^{-1}].
\]
It is also shown in [loc. cit.] that \(m( R\Psi_\mu,w)\neq 0\) if and only if \(w\in \text{Adm}(\mu)\), the finite subset of \(\mu\)-admissible elements in \(\widetilde W\). The main theorem of this paper states that for any \(w\in \text{Adm}(\mu)\), the multiplicity function \(m(R\Psi_\mu,w)\) is a polynomial in \(q\) having degree at most \(\ell (\mu)-\ell(w)\), where \(\ell(w)\) denotes the length of the element \(w\). This was proved in [loc. cit.] when \(G=\text{ GL}_n\) or \(\mu\) is minuscule. The proof uses the alteration due to \textit{A. J. de Jong} [Publ. Math., Inst. Hautes Étud. Sci. 83, 51--93 (1996; Zbl 0916.14005)] to reduce to the strictly semi-stable case, and uses the calculation of \textit{M. Rapoport} and \textit{T. Zink} [Invent. Math. 68, 21--101 (1982; Zbl 0498.14010)] to bound the degree of the function.
The paper, as well as the previous one, assumes the knowledge on perverse sheaves and intersection complexes. However, it provides an excellent introduction, which explains some related contents of the previous paper, so that the reader still can reach the main points without much difficulty. affine flag varieties; nearby cycles; intersection complexes; weights; Wakimoto sheaves Görtz, U.; Haines, T.J., Bounds on weights of nearby cycles and Wakimoto sheaves on affine flag manifolds, Manuscr. Math., 120, 347-358, (2006) Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Modular and Shimura varieties, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Bounds on weights of nearby cycles and Wakimoto sheaves on affine flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The interesting paper under review aims to extend to algebraic cobordism a number of celebrated determinantal formulas intended to compute the fundamental classes of degeneracy loci of maps of vector bundles. More precisely, assume that \(X\) is a smooth complete algebraic variety endowed with some reasonable intersection theory, say \(A^*(X)\). Recall that if \(\phi\) is a map of vector bundles over \(X\), one wishes to compute the fundamental classes (provided they are defined) of degeneracy loci of the map \(\phi\). If \(A^*(X)=\mathrm{CH}(X)\), the Chow ring, then the classical celebrated Porteous' formula does the job, provided the expected dimension of the considered degeneracy locus coincides with the actual one.
It turns out that Porteous' formula is just a particular case of a celebrated determinantal formula due to Damon, Kempf and Laksov (DKL), computing the fundamental class, in the Chow ring of Grassmann bundles \(G_d(E)\), parametrizing \(d\) dimensional vector subspaces of the fibers of some vector bundle \(E\), of Schubert varieties naturally associated to filtrations of the bundle itself. If the bundle \(E\) is trivial, the aforementioned formula reduces to the classical Giambelli's formula of classical Schubert calculus.
The sequel of the story is as follows. Already more than one decade ago, Levine and Morel came up by introducing the theory of \textit{algebraic cobordism} [\textit{M. Levine} and \textit{F. Morel}, Algebraic cobordism. Berlin: Springer (2007; Zbl 1188.14015)], denoted by \(\Omega^*\). As pointed out in the paper under review, algebraic cobordism is universal among all oriented cohomology
theories, understood as families of functors. These latter include, for example, the Chow theory as well as a graded version \(K_0[\beta,\beta^{-1}]\) of the Grothendieck ring of vector bundles, a smart interpolation of the \(K\)-theory and the Chow theory. Its universality implies that formulas holding in \(\Omega^*\) suitably specialises to formulas in all other theories.
It turns out, to say it with a slogan, that the main achievement of the article under review is the lifting of the DKL formula from the Chow theory to the algebraic cobordism \(\Omega^*\). The success of the attempt is recorded and certified by the main results, a couple of very nice theorems, that in the paper are numbered 4.9 and 5.7.
In Theorem 4.9 the authors consider a Schubert variety \(X_\lambda\) in a Grassmann bundle \(G_d(E)\), perform the Damon resolution \(Y_\lambda\) and push forward its class to the algebraic cobordism ring of \(G_d(E)\). The output is a linear combination of Schur determinants evaluated at suitable lifts of Segre classes to the Grothendieck ring of vector bundles. The second main Theorem 5.7, is then concerned with the geometrical interpretation of the extended Segre class of a virtual bundle, whose evaluation is prescribed by the former theorem. The interpretation is based on an equality, taking place in the algebraic cobordism ring of \(X\), relating a suitable extension of the Segre class of a virtual bundle \(V-W\) with a certain push-forward of the Chern class of a twist of the dual of \(W\).
The paper divides itself into five sections: the climax is reached in Section 5, where the extension of the Damon-Kempf-Laksov determinantal formula is achieved. The abundant reference list is more than the potential reader may need to reconstruct all the pre-requisites one needs to fully enjoy the clean mathematics displayed in this elegant paper. Damon-Kempf-Laksov determinantal formulas in algebraic cobordism, Grassmann bundles, Schur determinants Grassmannians, Schubert varieties, flag manifolds, Bordism and cobordism theories and formal group laws in algebraic topology, Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Segre classes and Damon-Kempf-Laksov formula in algebraic cobordism | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be the odd orthogonal group \(SO(n, n + 1)\) or the symplectic group \(Sp(n,\mathbb R)\) with Lie algebra \(\mathfrak g\) given by \(\mathfrak s \mathfrak o(n, n + 1)\) or \(\mathfrak s\mathfrak p(n,\mathbb R)\), respectively. Let \(\Sigma\) denote the root system. The Grassmannians are the minimal flag manifolds \(G/P_{(k)}, 0 \leq k \leq n - 1\), where \(P_{(k)}\) is the parabolic subgroup corresponding to the maximal proper subset \((k) = \Sigma- \{a_k\}\). If \(G = SO(n, n + 1)\) then the odd orthogonal Grassmannian \(OG(n - k, 2n + 1)\) is the set of \((n-k)\)-dimensional isotropic subspaces in the vector space \(V = \mathbb R^{2n+1}\) equipped with a nondegenerate symmetric bilinear form. If \(G = Sp(n,\mathbb R)\) then the isotropic Grassmannian \(IG(n - k, 2n)\) is the set of \((n - k)\)-dimensional isotropic subspaces in the symplectic vector space \(V = \mathbb R^{2n}\).
The Schubert cells provide a cellular structure of these Grassmannians via Bruhat decomposition. In addition, the minimal representatives \(\mathcal{W}^{(k)}_n\) of the Weyl group modulo the subgroup generated by reflections in \((k)\) parametrize such Schubert cells. In the paper they use a permutation model to identify cells with the set of signed \(k\)-Grassmannians permutations. They associate with each permutation \(w \in \mathcal{W}^{(k)}_n\) a double partition \((\alpha, \lambda)\) for which they have the corresponding half-shifted Young diagram.
The cellular homology appears after computing the boundary map coefficients. In this paper, they provide an explicit formula for the coefficients of those real Grassmannians, generalizing the second author's results in [Adv. Geom. 16, No. 3, 361--379 (2016; Zbl 1414.57018)] for the Lagrangian and maximal isotropic Grassmannians. This contributes to the study of the topology of real flag manifolds in comparison with the complex ones which have torsion-free homology groups.
Authors' abstract: ``We obtain a combinatorial expression for the boundary map coefficients of real isotropic and odd orthogonal Grassmannians. It provides a natural generalization of the known formulas for Lagrangian and maximal isotropic Grassmannians. The results derive from the classification of Schubert cells into four types of covering pairs when identified with signed k-Grassmannian permutations. Our formulas show that the coefficients depend on the changed positions for each permutation pair type. We apply this to obtain an orientability criterion and compute the first and second homology groups for these Grassmannians. Furthermore, we exhibit an apparent symmetry of the boundary map coefficients.'' real isotropic and odd orthogonal Grassmannians; Schubert calculus; boundary map coefficients Grassmannians, Schubert varieties, flag manifolds, Homology and cohomology of homogeneous spaces of Lie groups, Permutations, words, matrices, Combinatorial aspects of groups and algebras Integral homology of real isotropic and odd orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The following theorem is proved. If the Jacobian of a pair of polynomials \(P(x,y)\) an \(Q(x,y)\) is equal to 1, \(n\) is an arbitrary positive integer, and the parameters \(p_1,p_2,\dots,p_n\) take any values, with the restriction \(p_2\neq 0\), then, after the substitution \(x\mapsto xy^n+p_n y^{n-1}+\dots p_2y+p_1\), \(y\mapsto 1/y\), at least one of the functions \(P(x,y)\) or \(Q(x,y)\) is no longer polynomial (in the new variables). This is a generalization of the similar result obtained by A. G. Vitushkin for \(n\leq 3\) without the restriction \(p_2\neq 0\). In the paper there is also described a geometric construction having a direct application to the Jacobian conjecture. exactness of holomorphic 2-forms Jacobian problem On linear chains of blow-ups related to the Jacobian conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\underline a=(0<a_1<a_2< \cdots <a_k<n)\) be a sequence of integers and consider the general flag variety \(\mathbb{F}(\underline a, \mathbb{C}^n)\). Its homology is generated by the Schubert varieties. The Pieri formula gives a description of the cup product when one of the factors is a Chern class of a tautological bundle.
The (small) quantum cohomology of \(\mathbb{F}(\underline a, \mathbb{C}^n)\) is studied. The main approach to compute the quantum products is a quantum Pieri formula from \textit{I. Ciocan-Fontanine} [Duke Math. J. 98, No.3, 485--524 (1999; Zbl 0969.14039)]. This formula is reproven here by a geometrical argument which reduces it to the usual Pieri formula. The algorithm for computing a general quantum product is thus to use a quantum Giambelli formula (which expresses a quantum cohomology class as a polynomial in the Chern classes of tautological bundles) and then applying the quantum Pieri formula. quantum cohomology; flag manifold; quantum Pieri formula; quantum Giambelli formula Buch A.S.: Quantum cohomology of partial flag manifolds. Trans. Am. Math. Soc. 357, 443--458 (2005) (electronic) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum cohomology of partial flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simply connected complex semisimple Lie group of rank \(r\) with a fixed Borel subgroup \(B\) and a maximal torus \(H\subset B\). Let \(W=\text{Norm}_G(H)/H\) be the Weyl group of \(G\). The generalized flag manifold \(G/B\) can be decomposed into the disjoint union of Schubert cells \(X^\circ_w=(BwB)/B\), for \(w\in W\). To any weight \(\gamma\) that is \(W\)-conjugate to some fundamental weight of \(G\), one can associate a generalized Plücker coordinate \(p_\gamma\) on \(G/B\). In the case of type \(A_{n-1}\) (i.e., \(G=SL_n)\), the \(p_\gamma\) are the usual Plücker coordinates on the flag manifold. The closure of a Schubert cell \(X^\circ_w\) is the Schubert variety \(X_w\), an irreducible projective subvariety of \(G/B\) that can be described as the set of common zeroes of some collection of generalized Plücker coordinates \(p_\gamma\). It is also known that every Schubert cell \(X^\circ_w\) can be defined by specifying vanishing and/or non-vanishing of some collection of Plücker coordinates. The main two problems studied in this paper are the following.
(1) Describe a given Schubert cell by as small as possible number of equations of the form \(p_\gamma=0\) and inequalities of the form \(p_\gamma\neq 0\).
(2) Suppose a point \(x\in G/B\) is unknown to us, but we have access to an oracle that answers questions of the form: ``\(p_\gamma(x)=0\), true or false?'' How many such questions are needed to determine the Schubert cell \(x\) is in?
The number of equations of the form \(p_\gamma=0\) needed to define a Schubert variety is generally much larger than its codimension. We show that for a certain Schubert variety \(X_w\) in the flag manifold of type \(A_{n-1}\), one needs exponentially many such equations to define it, even though \(\text{codim}(X_w) \leq\dim (G/B)={n\choose 2}\). Given this kind of ``complexity'' of Schubert varieties, it may appear surprising that for the types \(A_r,B_r,C_r\), and \(G_2\), we provide a description of an arbitrary Schubert cell \(X^\circ_w\) that only uses \(\text{codim} (X_w)\) equations of the form \(p_\gamma=0\) and at most \(r\) inequalities of the form \(p_\gamma\neq 0\). For the type \(D\), a description of Schubert cells is slightly more complicated. Our main result regarding (2) is an algorithm that recognizes a Schubert cell \(X^\circ_w\) containing an element \(x\). For the types \(A_r,B_r,C_r\), and \(G_2\), our algorithm ends up examining precisely the same Pücker coordinates of \(x\) that appear in the previous result. In the case of type \(A_{n-1}\), recognizing a cell requires testing the vanishing of at most \({n\choose 2}\) Plücker coordinates. Finally, we discuss the problem of presenting a subset of Plücker coordinates whose vanishing pattern determines which cell a point is in. Schubert variety; flag manifold; Plücker coordinate; Bruhat cell; vanishing pattern S. Fomin and A. Zelevinsky, ''Recognizing Schubert cells,'' preprint math. CO/9807079, July 1998. Grassmannians, Schubert varieties, flag manifolds Recognizing Schubert cells. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let X denote the flag variety associated to a semi-simple complex group G. For an element w of the Weyl group of G let \(L_ w\) denote the irreducible D-module with regular singularities on X supported on the Schubert variety \(\bar X_ w\). The following conjecture is due to \textit{Kazhdan-Lusztig}: ``For \(G=SL_ n\), the singular support of \(L_ w\) is an irreducible subvariety of \(T^*X''.\)
The authors treat the case of Grassmannians (degenerate flag varieties for \(SL_ n)\) and prove the following theorem: ``Let \(G_{k,\ell}\) denote the Grassmannian of k-planes in \(k+\ell\) dimensional complex vector space. Let \(\bar X_{\lambda}\) be a Schubert variety in \(G_{k,\ell}\). Let \(L_{\lambda}\) denote the irreducible holonomic D- module with regular singularities on \(G_{k,\ell}\) supported on \(\bar X_{\lambda}\). Then the singular support of \(L_{\lambda}\) is an irreducible subvariety of \(T^*G_{k,\ell}''.\)
The theorem is reduced to study of Zelevinsky's resolutions using the formula of Brylinski for calculation of multiplicities of components of the characteristic cycle of a D-module in terms of vanishing cycles. Then, the proof is based on a detailed analysis of the geometry of the resolutions of singularities of Schubert varieties in Grassmannians introduced by Zelevinsky. flag variety; Grassmannians; multiplicities of components of the characteristic cycle of a D-module; vanishing cycles; resolutions of singularities; Schubert varieties P. Bressler, M. Finkelberg, and V. Lunts, Vanishing cycles on Grassmannians, Duke Math. J. 61 (1990), no. 3, 763 -- 777. Grassmannians, Schubert varieties, flag manifolds, Algebraic cycles, Global theory and resolution of singularities (algebro-geometric aspects), Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Semisimple Lie groups and their representations Vanishing cycles on Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum cohomology ring of an algebraic variety is a certain deformation of its cohomology ring. Roughly speaking, in such a deformation two subvarieties are considering intersecting if they are connected by one or more rational curves. The virtual numbers of such curves, counted by Gromov-Witten invariants, appear as coefficients in the decomposition of the quantum cohomology cup product. These ideas feature prominently in mathematical physics, especially in string theory, where Gromov-Witten invariants appear as quantum corrections to classical notions of geometry, hence the name quantum cohomology.
In this paper the authors study the equivariant quantum cohomology ring of hypertoric varieties. Hypertoric varieties are a version of toric varieties, which can be obtained as an hyperKähler quotient. Well known examples are crepant resolutions of \(A_n\) singularities. One can study such varieties using combinatorial techniques from toric geometry, and as a result their geometry is captured by certain hyperplane arrangements. The paper contains two results. The first one is an explicit presentation of the equivariant cohomology ring of such varieties, given in terms of generators and relations. This result follows from an explicit formula for the quantum multiplication by a divisor.
The second result concerns a mirror formula for the quantum connection on such varieties. The quantum connection depends on the equivariant parameters, and for fixed equivariant parameters is a deformation of an ordinary connection, which involves the quantum product. This results identifies such a quantum connection on a hypertoric variety with the Gauss-Manin connection of a certain mirror family of complex manifolds (with a local system). Such a family is defined in term of the toric data and hyperplanes of the original variety. The proof follows by identifying a certain quantum differential equation for the quantum connection, with a Picard-Fuchs equation for the periods of a specific cohomology class of the mirror family.
The results of the paper are quite explicit and can be useful for people working on hypertoric varieties. The authors have also put some effort in making the paper self-contained, and the relevant concepts of hypertoric geometry and quantum cohomology are reviewed, albeit rather concisely. quantum cohomology; hypertoric variety; symplectic resolution; mirror symmetry McBreen, M.; Shenfeld, D., Quantum cohomology of hypertoric varieties, Lett. Math. Phys., 103, 11, 1273-1291, (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Quantum cohomology of hypertoric varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear spaces. It is intended as a guide for readers with a combinatorial bent to understand and appreciate the geometric and topological aspects of Schubert calculus, and conversely for geometric-minded readers to gain familiarity with the relevant combinatorial tools in this area. We lead the reader through a tour of three variations on a theme: Grassmannians, flag varieties, and orthogonal Grassmannians. The orthogonal Grassmannian, unlike the ordinary Grassmannian and the flag variety, has not yet been addressed very often in textbooks, so this presentation may be helpful as an introduction to type B Schubert calculus. This work is adapted from the author's lecture notes for a graduate workshop during the Equivariant Combinatorics Workshop at the Center for Mathematics Research, Montreal, June 12--16, 2017. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Variations on a theme of Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A derived version of Grothendieck's Quot scheme is constructed. Let \(X\) be a projective scheme over a field \(\mathbb{K}\) and \({\mathcal F}\) a fixed coherent sheaf on \(X\). We take a \(h'\in\mathbb{Q}[t]\) and put \(h= h^{{\mathcal F}}- h'\) in which \(h^{{\mathcal F}}\) is the Hilbert polynomial of \({\mathcal F}\). Informally, the Quot scheme can be thought of as a Grassmannian of subsheaves in \({\mathcal F}\); its closed points are in 1:1 correspondence with \(\text{Sub}_h({\mathcal F})= \{{\mathcal K}\subset{\mathcal F}: h^{{\mathcal F}}= h\}\). In the same situation, the authors construct a dg-manifold \(\text{RSub}_h({\mathcal F})\) as a graded version of the derived Grassmannian. Quot schemes; Grassmannian Ciocan-Fontanine, Ionuţ; Kapranov, Mikhail, Derived Quot schemes, Ann. Sci. École Norm. Sup. (4), 34, 3, 403-440, (2001) Homogeneous spaces and generalizations, Derived categories, triangulated categories Derived Quot schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper contains a computation of the K-group of Grassmannians and flag varieties over an arbitrary Noetherian base scheme, and the K-group of forms of Grassmannians and flag varieties associated to a sheaf of Azumaya algebras. An extension to objects over \({\mathbb{Z}}\) of main points in the proof of the characteristic zero case is used in the computations. This include a weaker form of the Cauchy formula for \(Gl_ n\)- representations and the Bott theorem over \({\mathbb{Z}}\) in the appropriate Grothendieck group. K-group of Grassmannians; flag varieties; Azumaya algebras Marc Levine, V. Srinivas, and Jerzy Weyman, \?-theory of twisted Grassmannians, \?-Theory 3 (1989), no. 2, 99 -- 121. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, \(K\)-theory of schemes K-theory of twisted Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce the notion of a cominuscule point in a Schubert variety in a generalized flag variety for a semisimple group. We derive formulas expressing the Hilbert series and multiplicity of a Schubert variety at a cominuscule point in terms of the restrictions of classes in torus-equivariant \(K\)-theory and cohomology to that point, generalizing previously known formulas for flag varieties of cominuscule type. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. The formulas for Schubert varieties are special cases of more general formulas valid at generalized cominuscule points of schemes with torus actions. flag variety; Schubert variety; cominuscule; minuscule; equivariant; \(K\)-theory Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Equivariant \(K\)-theory Cominuscule points and Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A family of deformations \(\mathcal W_f\) of the Witt algebra \(\mathcal W\) parametrized by the space \(\varepsilon\) of even polynomials with vanishing constant terms is defined. The existence of an isomorphism \(\widehat{\mathcal W}_f \simeq \widehat{\mathcal W}\) , where \( \widehat{ } \) refers to suitable completions of \(\mathcal W\), is proved. A relation between \(\mathcal W_f\) and Krichever-Novikov algebras of genus 0 and 1 is given.
The article supplies the proofs of some results used by the author in [Lett. Math. Phys. 46, No. 2, 121--129 (1998; Zbl 0932.17024)]. DOI: 10.1142/S0129055X99000118 Lie algebras of vector fields and related (super) algebras, Relationships between algebraic curves and physics, Virasoro and related algebras, Differentials on Riemann surfaces, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Formal and analytic rigidity of the Witt algebra. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The article under review is very similar to, but in many regards much stronger than, [\textit{A. Krishna} et al., Algebr. Geom. Topol. 15, No. 2, 823--861 (2015; Zbl 1323.14016)]. The author establishes the existence of a lax monoidal ``homotopy semitopologization'' functor and studies some of its properties. In establishing the first goal, he constructs a lax monoidal fibrant replacement functor on motivic symmetric spectra, which is of independent interest.
The studied properties are manifold. For example, similar to the above reference, it is shown that applying the functor to spectra representing motivic cohomology or algebraic K theory yields the standard semitopological versions; and applying it to algebraic cobordism yields a semitopological version of algebraic cobordism which has the expected properties. Some of these properties can only be formulated knowing that the semitopologization functor is lax monoidal, and so are completely new. motivic homotopy theory; semi-stable symmetric motivic spectra; semi-topological cohomology theory Motivic cohomology; motivic homotopy theory, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Motivic strict ring spectra representing semi-topological cohomology theories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review continues the study of the authors about singular quiver Grassmannians, providing desingularizations of irreducible components of arbitrary quiver Grassmannians over Dynkin quivers.
Given a Dynkin quiver \(\mathcal Q\), being a directed graph whose unoriented underlying graph is a Dynkin diagram, a representation \(M\) of \(\mathcal Q\), and a dimension vector \(e\), the quiver Grassmannian \(Gr_e(M)\) is the variety of subrepresentations of \(M\) of dimension vector \(e\). When \(M\) has good homological properties, \(Gr_e(M)\) is smooth. However, the general analysis needs a desingularization of the quiver Grassmannian.
The construction of a desingularization lies on the definition of an algebra \(B_{\mathcal Q}\) for the quiver \(\mathcal Q\) which has global dimension at most two, such that the original module category \(\mathrm{mod\,}k\mathcal Q\) embeds into the subcategory \(\mathrm{mod\,}B_{\mathcal Q}\) of objects of projective and injective dimensions at most one, where all non-trivial extensions in \(\mathrm{mod\,}k\mathcal Q\) vanish after the embedding. This is to avoid the natural embedding \(M\mapsto\Hom(-,M)\) of \(\mathrm{mod\,}k\mathcal Q\) into \((\mathrm{mod\,}k\mathcal Q)^{op}\) which give projective functors of dimension two, in general.
A quiver \(\widehat{\mathcal Q}\) is constructed from \(B_{\mathcal Q}\) and, for every representation \(M\) over \(\mathcal Q\), a representation \(\widehat M\) over \(\widehat{\mathcal Q}\) arises. This, together with a fully faithful functor \(\Lambda\) from the category of representations of \(\mathcal Q\) to the one in \(B_{\mathcal Q}\) with good homological properties, completes the ingredients of the construction.
Then, it is constructed the desingularization map \(\pi_{[N]}\colon Gr_{\dim\widehat N}(\widehat M)\to Gr_e(M)\), where \(\widehat N\) is a representation over \(\widehat{\mathcal Q}\) coming from an isomorphism class of representations \([N]\) over \(\mathcal Q\) of dimension vector \(e\). The variety \(Gr_{\dim\widehat N}(\widehat M)\) is smooth with irreducible equidimensional connected components, and the fibers of the map can be described in terms of a quiver Grassmannian over \(\widehat{\mathcal Q}\) itself, once we know the irreducible components of the singular \(Gr_e(M)\). The article finishes by showing some particular examples of the construction. quiver Grassmannians; desingularizations; Dynkin quivers; Auslander algebras; flag varieties; Auslander-Reiten theory; quiver representations; categories of representations; irreducible components Cerulli Irelli, G., Feigin, E., Reineke, M.: Desingularization of quiver Grassmannians for Dynkin quivers. Adv. Math. \textbf{245}, 182-207 (2013) Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Singularities of surfaces or higher-dimensional varieties Desingularization of quiver Grassmannians for Dynkin quivers. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{M. Gillespie} et al. [``A crystal-like structure on shifted tableaux'', Preprint, \url{arXiv:1706.09969}] introduced coplactic operators on shifted semistandard tableaux, which form a crystal-like structure that recovers the combinatorics of type B Schubert calculus. In the paper under review, the authors provide local axioms that uniquely characterize this crystal-like structure on shifted tableaux. This characterization resembles that developed by \textit{J. R. Stembridge} [Trans. Am. Math. Soc. 355, No. 12, 4807--4823 (2003; Zbl 1047.17007)] for type A tableau crystals, and provides a tool for the study of Schur Q-positive expansions in symmetric function theory. local axioms; crystal; coplactic operators; shifted tableaux Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus Axioms for shifted tableau crystals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Givental's work on equivariant Gromov-Witten invariants has established the relationship between quantum cohomology and hypergeometric series [see \textit{A. B. Givental'}, Int. Math. Res. Not. 1996, No. 13, 613-663 (1996; Zbl 0881.55006)]. Consider a toral action on a compact Kähler manifold \(X\) with finitely many fixed points \(x_w\). The solutions of the differential equations arising from quantum cohomology are related to equivariant correlators \(Z_w\) associated with the \(x_w\). The correlators \(Z_w\) are the hypergeometric series associated with equivariant quantum cohomology, and can be uniquely determined by linear recursion relations. The main result in the paper under review is the explicit determination of the recursion relations for the \(Z_w\) on flag spaces \(X=G/B\). Here \(G\) is the simply connected algebraic group associated with a finite root system \(R\). Hence \(X\) is a homogeneous space for the action of the maximal torus of \(G\). The set of fixed points is in this case finite and parametrized by the Weyl group of \(R\). A simple explicit formula for the \(Z_w\) is presented in the case \(G=SL(3)\). Givental' recursion relations; hypergeometric functions; quantum cohomology; equivariant Gromov-Witten invariants; flag spaces Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Other hypergeometric functions and integrals in several variables, Quantization in field theory; cohomological methods, Grassmannians, Schubert varieties, flag manifolds On hypergeometric functions connected with quantum cohomology of flag spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials I cite the abstract of the paper: ``The theory of algebraic varieties gives an algebraic interpretation of differential geometry, thus of our physical world. To treat, among other physical properties, the theory of entanglement, we need to generalize the space parametrizing the objects of physics. We do this by introducing noncommutative varieties''.
The paper under review (3 pages long) contains many mistakes, I'm unsure if these are the mistakes of the author or the publisher. The author defines a ``matrix polynomial algebra'' \(R\), the non-commutative spectrum of a ``matrix polynomial algebra'' \(R\) and the non-commutative structure sheaf of \(R\). The points of \(\mathrm{Spec}(R)\) are the set of simple left \(R\)-modules of \(R\). The author gives some elementary examples and makes many claims without supplying any proofs. In the second chapter of the paper ``The dynamics in non-commutative affine varieties'' the author introduce the 1-radical of a matrix polynomial algebra \(R\) and the tangent space. He gives some examples and also calculates the tangent space of a free matrix polynomial algebra. The author ends the paper with a short section called ``Entanglement: Blowing up a singularity'' where the author studies a non-commutative version of the \(x^4+y^3\) singularity. The reviewer wants to encourage the author to write a more detailed paper on the subject where he includes all definitions and proofs -- a self-contained paper. Send it to a journal where the refereeing is more thorough. noncommutative algebra; differential geometry; matrix polynomial algebras Noncommutative algebraic geometry, Physics Physical interpretation of noncommutative algebraic varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Q\) be a Dynkin quiver of type \(\mathbb{A},\;\mathbb{D},\) or \(\mathbb{E}.\) Associated to \(Q\) is the path algebra \(\mathbb{C}Q,\) from which one can construct the repetitive algebra \(\widehat{\mathbb{C}Q}:=\bigoplus _{n\in\mathbb{Z}}\mathbb{C}Q\oplus\bigoplus_{n\in\mathbb{Z}}D\left( \mathbb{C}Q\right) ,\) where \(D\left( -\right) \) indicates \(\mathbb{C} \)-linear duality. Generalizing an earlier result of [\textit{D. Hernandez} and \textit{B. Leclerc}, ``Quantum Grothendieck rings and derived Hall algebras'', \url{arXiv:1109.0862}], the authors show that all of the orbit closures of \(\widehat{\mathbb{C}Q}\) are isomorphic to a certain collection of varieties \(\mathfrak{M}_{0}^{\cdot}\left( V,W\right) .\)
Let \(\mathbf{d}\) be a finite dimension vector\(.\) One can construct a certain quiver \(\widehat{\Gamma}\) which contains the repetition quiver \(\widehat{Q}\) as a subquiver. Together with \(\mathbf{d},\) \(\widehat{\Gamma}\) can be used to construct certain complex vector spaces \(V\) and \(W\), and an affine algebraic group \(G_{V}\) which acts on the rep\(_{\mathbf{d}}\left( \mathbb{C} \widehat{\Gamma}/R\right) \) for \(R\) a certain ideal. The affine quotient of rep\(_{\mathbf{d}}\left( \mathbb{C}\widehat{\Gamma}/R\right) \) by \(G_{V}\) is denoted \(\mathfrak{M}_{0}^{\cdot}\left( V,W\right) .\) We define \(\mathfrak{M}_{0}^{\cdot}\left( W\right) \) to the the union of such \(\mathfrak{M}_{0}^{\cdot}\left( V,W\right) \) over all \(V\). We also have an open subset of \(\mathfrak{M}_{0}^{\cdot\text{reg}}\left( V,W\right) \) of \(\mathfrak{M}_{0}^{\cdot}\left( V,W\right) \;\)which parameterizes the closed free \(G_{V}\)-orbits, and \(\mathfrak{M}_{0}^{\cdot}\left( V\right) \) is the disjoint union (over all \(V\)) of these subsets. It can be shown that \(\mathfrak{M}_{0}^{\cdot\text{reg}}\left( V,W\right) \) is nonempty if and only if there is a stable representation of \(\mathbb{C}\widehat{\Gamma}/R\) whose underlying graded vector space is \(\left( V,W\right) \) and \(\left( V,W\right) \) is dominant.
The variety of \(\mathbf{d}\)-dimensional representations of \(\widehat {\mathbb{C}Q}\) is shown to be isomorphic to \(\mathfrak{M}_{0}^{\cdot}\left( W^{\mathbf{d}}\right) \),via a map \(\psi,\) and this isomorphism is \(G_{W} \)-equivariant.
The main result of this paper is as follows. Let \(\mathbf{d}\) be a finite dimension vector. supported on \(\psi\left( \text{proj }\widehat{\mathbb{C} Q}\right) .\) Then there is a bijection between (i) the set of isomorphism classes of representations of \(\widehat{\mathbb{C}Q}\) with dimension vector \(\mathbf{d};\) (2)\ the set of dominant pairs \(\left( V,W^{\mathbf{d}}\right) ;\) and (3) the set of non-empty \(\mathfrak{M}_{0}^{\cdot\text{reg}}\left( V,W^{\mathbf{d}}\right) .\) This allows for a description of the space \(V\) in the dominant pair \(\left( V,W\right) \) associated to a representation of \(\widehat{\mathbb{C}Q}.\) Nakajima variety; repetitive algebra; quantum loop algebra; quiver; orbit closure; intersection cohomology Leclerc, B., Plamondon, P.: Nakajima varieties and repetitive algebras. arXiv:1208:3910[math.QA]. To appear in Publ. RIMS, Kyoto Geometric invariant theory, Representations of quivers and partially ordered sets, Quantum groups (quantized enveloping algebras) and related deformations Nakajima varieties and repetitive algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider two different subjects: the \(q\)-deformed universal characters \(\widetilde S_{[\lambda,\mu]}(t,\hat t;x,\hat x)\) and the \(q\)-deformed universal character hierarchy. The former are an extension of \(q\)-deformed Schur polynomials, and the latter can be regarded as a generalization of the \(q\)-deformed KP hierarchy. We investigate solutions of the \(q\)-deformed universal character hierarchy and find that the solution can be expressed by the boson-fermion correspondence. We also study a two-component integrable system of \(q\)-difference equations satisfied by the two-component universal character. \(q\)-deformation; universal character; \(q\)-deformed universal character hierarchy; boson-fermion correspondence; lattice \(q\)-deformed universal character hierarchy Particle exchange symmetries in quantum theory (general), Formal methods and deformations in algebraic geometry, Ordinary representations and characters, Schur and \(q\)-Schur algebras, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Difference equations, scaling (\(q\)-differences) \(q\)-universal characters and an extension of the lattice \(q\)-universal characters | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The representation theory of affine Lie algebras can be encoded by semi-infinite flag varieties. These can also be interpreted as spaces of rational maps, and so play a major role in the computation of the quantum \(K\)-theory of flag varieties. \textit{A. Braverman} and \textit{M. Finkelberg} [J. Am. Math. Soc. 27, No. 4, 1147--1168 (2014; Zbl 1367.17011); Math. Ann. 359, No. 1--2, 45--59 (2014; Zbl 1367.17010); J. Lond. Math. Soc., II. Ser. 96, No. 2, 309--325 (2017; Zbl 1384.17009)] used this last interpretation to give fundamental properties, such as normality, rationality of the singularities, an analogue of the Borel-Weil theorem, the computation of quantum \(J\) functions (in the line of Givental-Lee) and its connection with \(q\)-Whittaker functions.
This article has two main objectives. The first is to extend a cohomology formula for line bundles given by Braverman and Finkelberg to include a special class of twisted sheaves. The other main objective is to generalize the other results to all Schubert varieties giving more applications to representation theory. This last extension provides a natural realization of specializations of non-symmetric Macdonald polynomials together with some difference equations characterizing them. This generalizes the link of non-symmetric Macdonald polynomials to representation theory of algebras currently defined by \textit{A. Braverman} and \textit{M. Finkelberg} [Zbl 1367.17010], \textit{C. Lenart} et al. [Int. Math. Res. Not. 2015, No. 7, 1848--1901 (2015; Zbl 1394.05143); ``A uniform model for Kirillov-Reshetikhin crystals. II: Alcove model, path model, and \(P=X\)'', Int. Math. Res. Not. 2017, No. 14, 4259--4319 (2017; \url{doi:10.1093/imrn/rnw129}); Transform. Groups 22, No. 4, 1041--1079 (2017; Zbl 1428.05325)], \textit{I. Cherednik} and \textit{D. Orr} [Math. Z. 279, No. 3--4, 879--938 (2015; Zbl 1372.20009)], \textit{S. Naito} et al. [Trans. Am. Math. Soc. 370, No. 4, 2739--2783 (2018; Zbl 1432.17015)], and \textit{E. Feigin} and \textit{I. Makedonskyi} [Sel. Math., New Ser. 23, No. 4, 2863--2897 (2017; Zbl 1407.17028)].
Let \(G\) be a simply connected simple algebraic group and let \(W\) be its Weyl group with set of simple reflections \(\{s_i\}.\) Let \(\Lambda\) be the weight lattice, \(\Lambda_+\) the set of dominant weights, let \(Q^\vee\) be the co-root lattice of \(G\). Consider the space \(\mathcal Q\) of rational maps from \(\mathbb P^1\) to \(G/B\) with subspace \(\mathcal Q(w)\) defined as the closure of the set of rational maps whose value at \(0\) is in a Schubert variety corresponding to \(w\in W\). This space has a natural line bundle \(\mathcal O(\lambda)\) for each \(\lambda\in\Lambda\).
Associated to \(G\), the current algebra is defined as \(\mathfrak g[z]=\text{Lie}G\otimes_{\mathbb C}\mathbb C[z]\), and \(G\) comes with the associated Iwahori subalgebra \(\mathcal J\). For each \(\lambda\in\Lambda_+\), \(\mathfrak g[z]\) has a representation \(W(\lambda)\) called a global Weyl module. \textit{M. Kashiwara} [Publ. Res. Inst. Math. Sci. 41, No. 1, 223--250 (2005; Zbl 1147.17306)] defined its Demazure submodule \(W(\lambda)_w\) for \(w\in W\) as the cyclic \(\mathcal J\)-module generated by a vector with weight \(w\lambda\in\Lambda\). As these last modules are graded, this defines their character \(\text{ch}W(\lambda)_w\in\mathbb C(\!(q)\!)[\Lambda].\)
The first main theorem is stated more or less verbatim, as it is stated in the category of ind-schemes:
Theorem A. For each \(\lambda\in\Lambda\) and \(w\in W\); {\parindent=0.8cm \begin{itemize}\item[1.] The ind-scheme \(\mathcal Q(w)\) is normal, and projectively normal; \item[2.] There is an isomorphism of \(\mathcal J\)-modules \(H^i(\mathcal Q(w),\mathcal O_{\mathcal Q(w)}(\lambda))^\ast=\begin{cases} W(\lambda)_w,\;(i=0,\lambda\in\Lambda_+)\\\{0\}\text{ otherwise }\end{cases};\) \item[3.] For each \(i\in\mathsf{I}\) so that \(s_iw>w\), \(\text{ch}W(\lambda)_{s_i w}=D_i(\text{ch} W(\lambda)_w)\), where \(D_i\) is a Demazure operator acting on \(\mathbb C (\!(q)\!)[\Lambda]\); \item[4.] There exists a Demazure operator \(D_{t\beta}\) for each \(\beta\in\mathcal Q^\vee\) such that \(\langle\beta,w\alpha\rangle>0\) for every positive root \(\alpha\). This operator is mutually commutative, and \(D_{t\beta}(\text{ch}W(\lambda)_w)=q^{-\langle\beta,w\lambda\rangle}\cdot\text{ch}W(\lambda)_w\).
\end{itemize}} The author remarks that (2) to (4) above are regarded as semi-infinite analogues of the Demazure character formula due to Demazure-Joseph-Kumar in the ordinary setting.
We state the second main result (more or less) verbatim:
Theorem B. For each \(w\in W\) and \(\lambda\in\Lambda_+\), the module \(W(\lambda)_w\) admits a free action of a certain polynomial ring and its specialization to \(\mathbb C\) gives the Feigin-Makedonskyi module \(W_{w\lambda}\). In particular, \(\Gamma(\text{FI}_G^\frac{\infty}{2})(w),\mathcal O_{\text{FI}_G^\frac{\infty}{2}(w)}(\lambda))^\ast\cong W_{w\lambda},\) where \(\text{FI}_G^\frac{\infty}{2}(w)\) is a variant of \(Q(w)\).
The third theorem is a result of the comparison of Cerednik-Orr's recursive formula for non-symmetric Macdonald polynomials specialized to \(t=\infty\) with the authors construction, verbatim:
Theorem C. For each \(\lambda\in\Lambda_+\) and \(w\in W\), there exists an \((\mathbf{I}\rtimes\mathbb G_m)\)-equivariant sheaf \(\mathcal E_w(\lambda)\) such that \(\text{ch}H^0(Q(w),\mathcal E_w(\lambda))^\ast=(\prod_{i\in I}\prod_{k=1}^{\langle\alpha_i^\vee,\lambda_w\rangle}\frac{1}{1-q^k})\cdot E^\dagger_{-w\lambda}(q^{-1},\infty),\) where \(\lambda_w\) is a dominant weight determined by \(\lambda\) and \(w\), and \(E^\dagger_{-w\lambda}(q,t)\) is the bar-conjugate of a non-symmetric Macdonald polynomial, in addition, \(H(Q(w),\mathcal E_w(\lambda))=\{0\}\) for \(i>0\).
To prove the results, an analogue of the Kodaira vanishing theorem is presented. This leads to the fourth main result of the article, which gives relations between different specializations of non-symmetric Macdonald polynomials.
The article starts off with two sections on the preliminary material regarding current algebra representations and semi-infinte flag varieties. This involves many very complicated definitions, and includes a lot of high-end results which the author proves because of lack of references. Then the necessary observation that the semi-infinite flag variety is projectively normal is proved, so that the proof of Theorem A can be given by algebraic manipulations.
The last sections contain the proofs of the remaining main results. These results are simple observations from the author's point of view, but it would take a lot of labour to understand them fully, at least for this reviewer. We understand that the theory of ind-schemes, limiting to derived schemes and a touch of deformation theory are important ingredients, and they are put together in a concise and detailed manner to prove these very important results. semi-infinite flag varieties; Schubert varieties; affine Lie algebras; quantum \(K\)-theory; Borel-Weil theorem; current algebra; non-symmetric Macdonald polynomials; Weyl module Kato, S., Demazure character formula for semi-infinite flag manifolds Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Frobenius induction, Burnside and representation rings, Representation theory of associative rings and algebras, Character groups and dual objects Demazure character formula for semi-infinite flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the Grassmannian \(\mathcal G_K(k,n)\) over any field \(K\), some flags \(F_i\), \(i=1,\dots,m\), and consider the corresponding sets of conditions \(\Omega(F_i)\) on the intersection of \(k\)-planes with the elements of each \(F_i\). This datum defines a Schubert problem \(\omega=\{\Omega(F_i)\}\). The author studies the case where the expected set of solutions is finite. The problem \(\omega\) is enumerative over \(K\) when one finds the expected number \(\deg(\{\Omega(F_i)\})\) of solutions, formed by distinct \(k\)-planes, all of them defined over \(K\).
The author proves that any Schubert problem is enumerative over \(\mathbb R\), as well as over algebraically closed field of any characteristic. Moreover, he finds that Schubert problems over finite fields are enumerative in a set of positive density. One tool for proving the results is an extension of the Bertini-Kleiman smoothness condition. The author proves that the map from the set of solutions in the universal Grassmannian, to the product of \(m\) copies of the flag variety, is generically smooth.
The author finally studies the Galois group of an enumerative problem, for the Grassmannians of lines and planes. Schubert cycles R. Vakil, Schubert induction. Ann. Math. 164, 489--512 (2006) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Enumerative problems (combinatorial problems) in algebraic geometry Schubert induction | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{T. Springer} introduced [in Invent. Math. 44, 279-293 (1978; Zbl 0376.17002)] a construction of representations of Weyl groups: let G be a complex semi-simple connected Lie group, and A a nilpotent element in Lie G; on the set \({\mathcal B}^ A\) of Borel subgroups of G containing exp CA, the centralizer of A acts, and this gives an action of its fundamental group C(A) on the homology \(H({\mathcal B}^ A,{\mathbb{Q}})\); this action commutes with an action of the Weyl group W defined using a Cartan subgroup of G, and the simple components of the action of W on the top homology modules for all A's give all the simple representations of W. On the other hand, \textit{A. Joseph} [in J. Algebra 88, 238-278 (1984; Zbl 0539.17006)] gives another construction of the representations of W: fix a nilpotent subalgebra with normalizer a Borel subgroup B in G, to each irreducible component of its intersection with the orbit of A Joseph defines a polynomial of the dual of the Cartan algebra of Lie B, and the \({\mathbb{Q}}\)-span of these polynomials is a W-module. This article shows that this latter representation corresponds to the former for the invariants of C(A) in \(H_{top}({\mathcal B}^ A,{\mathbb{Q}})\). The appendix contains a proof of the identification of the Kazhdan-Lusztig representation [\textit{D. Kazhdan} and \textit{G. Lusztig}, Adv. Math. 38, 222-228 (1980; Zbl 0458.20035)] of W with the Springer representation. This article is clearly written, and has already had interesting applications on the characteristic cycles of holonomic systems on flag manifolds. representations of Weyl groups; complex semi-simple connected Lie group; nilpotent element; Borel subgroups; Cartan subgroup; top homology modules; simple representations; Cartan algebra; holonomic systems on flag manifolds Morrison, S.E.: A Diagrammatic Category for the Representation Theory of \(U_q(sl(n))\). ProQuest LLC, Ann Arbor, MI (2007). Thesis (Ph.D.)-University of California, Berkeley Representation theory for linear algebraic groups, Lie algebras of linear algebraic groups, Group varieties, Cohomology theory for linear algebraic groups, Analysis on real and complex Lie groups On Joseph's construction of Weyl group representations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a very well written paper which deals with an interesting problem, namely, the characterization of the so-called double fibrations. It is known that the language of double fibrations is convenient for describing a family of submanifolds and the construction of the dual family of submanifolds. It has also deep applications in integral geometry.
Here the author introduces the notion of admissible double fibration in the sense that there exists a collection of densities for which the integration operator admits a local inversion formula ({\S} 1). A necessary condition for admissibility is given in terms of the degree of a certain covering map ({\S} 2). The author states the conjecture that the condition \(d(\Gamma)=1\) is also sufficient for the admissibility of n-parameter families of k-dimensional planes in \({\mathbb{C}}{\mathbb{P}}^ n\) ({\S} 3) and, in the remaining part of the paper verifies that the conjecture is true for linear complexes and also for n-parameter families of (n-2)-dimensional subspaces of \({\mathbb{C}}{\mathbb{P}}^ n\) ({\S} 7). dual family of submanifolds; integral geometry; admissible double fibration; covering map A. B. Goncharov, Admissible families of \(k\)-dimensional submanifolds , Dokl. Akad. SSSR 300 (1988), no. 3, 535-539. Structure of families (Picard-Lefschetz, monodromy, etc.), Analytic subsets and submanifolds, Integral geometry Admissible families of k-dimensional submanifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0717.00009.]
The purpose of the paper under review is to generalize the results of \textit{J. L. Brylinski} and the author [Invent. Math. 64, 387-410 (1981; Zbl 0473.22009)] and \textit{A. Beilinson} and \textit{J. Bernstein} [C. R. Acad. Sci., Paris, Sér. I 292, 15-18 (1981; Zbl 0476.14019)] to the symmetrizable Kac-Moody algebra case by using the infinite dimensional flag varieties, established earlier by the author. The main result is that the generalization of the Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody algebras is true. The proof is not much different from the one of Brylinski-Kashiwara. However, since the flag variety is infinite dimensional in general, special care is needed.
Let \(X=G/B_ -\) be the flag variety of a symmetrizable Kac-Moody algebra \({\mathfrak g}\) and \({\mathcal D}_ X\) be the sheaf of differential operators on X. For an integrable dominant weight \(\lambda\), let \({\mathcal O}(\lambda)\) be the corresponding line bundle over X and let \({\mathcal D}_{\lambda}={\mathcal O}(\lambda)\otimes {\mathcal D}_ X\otimes {\mathcal O}(- \lambda)\) be the twisted ring of differential operators on X. For an admissible \({\mathfrak g}\)-module M, the functor from the category of admissible \({\mathcal D}_{\lambda}\)-modules \({\mathcal M}\mapsto Hom_{{\mathfrak g}}(M,\Gamma (X,{\mathcal M}))\) is representable in the category of admissible \({\mathcal D}_{\lambda}\)-modules. Denote by \({\mathcal D}_{\lambda}{\hat \otimes}M\) the admissible \({\mathcal D}_{\lambda}\)- module that represents the functor above. If M is a (\({\mathfrak g},B)\)- module, then \({\mathcal D}_{\lambda}\otimes M\) is a B-equivariant \({\mathcal D}_{\lambda}\)-module. For an admissible B-equivariant \({\mathcal D}_{\lambda}\)-module \({\mathcal M}\), set \(\tilde H^ n(X,{\mathcal M})=\oplus_{\mu \in P}\lim_{\overset\leftarrow U}H^ n(U,{\mathcal M})_{\mu},\) where U ranges over quasi-compact B-stable open subsets. The main theorem says that, for an admissible B-equivariant \({\mathcal D}_{\lambda}\)-module \({\mathcal M}\), (1) \(\tilde H^ n(X,{\mathcal M})=0\) for any \(n\neq 0\). (2) \({\mathcal D}_{\lambda}\otimes {\tilde \Gamma}(X,{\mathcal M})\to {\mathcal M}\) is an isomorphism. D-module; infinite dimensional flag varieties; generalization of the Kazhdan-Lusztig conjecture; symmetrizable Kac-Moody algebras; twisted ring of differential operators Kashiwara, Masaki, Kazhdan-Lusztig conjecture for a symmetrizable Kac-Moody Lie algebra.The Grothendieck Festschrift, Vol.\ II, Progr. Math. 87, 407-433, (1990), Birkhäuser Boston, Boston, MA Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Kazhdan-Lusztig conjecture for a symmetrizable Kac-Moody Lie algebra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author studies the behavior of the (genus zero) Gromov-Witten invariants of an arbitrary smooth projective variety \(X\) under blow-ups of some points. This task is motivated by several reasons, as the Gromov-Witten invariants of a blow-up variety \(\widetilde X\) might provide both interesting examples in general Gromov-Witten theory and explicit solutions to particular enumerative problems on the original variety \(X\) itself.
The first part of the author's work exhibits an efficient algorithm to compute the Gromov-Witten invariants of a blow-up \(\widetilde X\) in terms of those of a smooth variety \(X\). This is followed by an analysis of the enumerative significance of the Gromov-Witten invariants of the blow-up \(\widetilde X\) mainly with a view toward counting certain irreducible curves on \(X\) not contained in the exceptional divisor of the blow-up. In the second part of the paper, the author specializes his general results to blow-ups of a projective space \(\mathbb{P}^r\). It turns out that many Gromov-Witten invariants of these blow-ups can be interpreted as the number of rational curves in \(\mathbb{P}^r\) having prescribed global multiplicities in the blow-up points. This is enhanced by various concrete numerical examples given at the end, including an efficient method to compute multiple cover contribution factors for finite covers of infinitesimally rigid smooth rational curves on a Calabi-Yau threefold.
This work is part of the author's Ph.D. thesis written at the University of Hannover, Germany, with \textit{K. Hulek} as academic advisor [Gromov-Witten and degeneration invariants: Computation and enumerative significance (Hannover: Universität Hannover) (1998; Zbl 0902.14021)]. quantum cohomology; enumerative geometry; birational maps; rational curves; coverings Gathmann, A., \textit{Gromov-Witten invariants of blow-ups}, J. Algebraic Geom., 10, 399-432, (2001) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Ramification problems in algebraic geometry, Rational and birational maps, Enumerative problems (combinatorial problems) in algebraic geometry Gromov-Witten invariants of blow-ups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Starting from the meromorphic functions on the complex plane with poles at \(0\) and \(\infty\), and the meromorphic functions doubly-periodic under the lattice \(L=\langle 1,\tau \rangle\) with poles at most at the points lying in the orbit of \(0\) and \(1/2\) under \(L\), a nonassociative algebra is constructed. This is done with the help of a certain differential-difference operator \(T\), which depends on two parameters \(x\) and \(y\). The algebra structure is calculated in terms of certain exhibited basis elements. Subalgebras are studied. In general, the subalgebras obtained will not be Lie algebras. Certain subalgebras that are Lie algebras, are classified. Special cases are considered. Depending on the parameters, the Witt algebra of meromorphic vector fields on \(\mathbb P^1(\mathbb C)\) holomorphic outside \(0\) and \(\infty\), and the Krichever-Novikov algebra on the torus \(\mathbb C/L\), consisting or meromorphic vector fields holomorphic outside \({0}\bmod L\) and \({1/2}\bmod L\) show up again. (They were the basic starting objects). The results are obtained via explicit calculations.
Two small formal points should be mentioned. Firstly, the systematic use of the misnomer ``Virasoro-De-Witt'' algebra to denote the Witt algebra (or equivalently the Virasoro algebra without central extension). Secondly, that the list of references is not in accordance with the referencing scheme in the article. Witt algebra; Virasoro algebra; elliptic functions; nonassociative algebras; Lie algebras Lie algebras of vector fields and related (super) algebras, Virasoro and related algebras, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Elliptic curves On discretized generalizations of Krichever-Novikov algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author shows how the techniques of modified psi classes developed by \textit{T. Graber, J. Kock} and \textit{R. Pandharipande} [Am. J. Math. 124, 611--647 (2002; Zbl 1055.14056)] can also be used to solve the characteristic number problem for cuspidal rational curves in \( \mathbb{P}^2\) or \(\mathbb{P}^1 \times \mathbb{P}^1\) (as well as that of a single triple contact to a given curve). A slight generalization of the tangency quantum potential is given in section 1.5, namely incorporating invariants corresponding to top products, one factor of which is a square of a modified psi class or a certain codimension 2-boundary class defined in section 2. Via certain topological recursions (introduced in equation (3) of the paper as the quantum tangency potential), these new potentials are related to the usual tangency quantum potential. The locus of cuspidal curves and the locus of curves with a triple contact are now described in terms of these classes as shown in proposition 3.4, and thus the corresponding generating functions can be expressed in terms of the slightly enriched potentials (as given in equations (11), (12), (13) of the paper). In this way, one obtains the characteristic numbers for cuspidal plane curves in proposition 3.9 and for irreducible cuspidal curves in \( \mathbb{P}^1 \times \mathbb{P}^1\) in proposition 4.6. In proposition 5.3 the author calculates the closure of the locus of irreducible inmersions having a triple contact to \(V\) at a mark \(p_1\) and concludes by giving two examples of calculation: the characteristic numbers of rational curves having a triple contact with a general curve in \(\mathbb{P}^2\) and for rational curves in \(\mathbb{P}^1 \times \mathbb{P}^1 \) with one triple contact to a \((1,1)\) curve (examples 5.5 and 5.7). enumerative geometry; characteristic numbers; cuspidal curves; quantum cohomology Kock, J, Characteristic number of rational curves with cusp or prescribed triple contact, Math. Scand., 92, 223-245, (2003) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Singularities of curves, local rings, Rational and unirational varieties Characteristic numbers of rational curves with cusp or prescribed triple contact | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For the domain \(\mathcal{P}=1+x\mathbb{Q}(w)_0[[x]]\), where \(\mathbb{Q}(w)_0=\mathbb{Q}(w)\cap\mathbb{Q}[[w]]\), and a function \(f(w,x)\in\mathcal{P} \), we consider the Zinger operator
\[\mathbf{M} f(w,x)=\biggl(1+\frac xw\frac{\partial}{\partial x}\biggr)\frac{f(w,x)}{f(0,x)}\]
and define \(I_p(x)=\mathbf{M}^p(f(w,x))\mid_{w=0} \). In this article, we study a class of periodic functions under the iterations of \(\mathbf{M}\) and show that \(I_p\) have interesting properties. A typical element of this class is constructed from the holomorphic solution of a differential equation with maximal unipotent monodromy. For this solution we define a kind of deformation (Zinger deformation) as a member of \(\mathcal{P} \). This deformation is a natural generalization of what Zinger did for the hypergeometric function
\[\mathcal{F}(x)=\sum_{d=0}^\infty\biggl(\frac{(nd)!}{(d!)^n}\biggr)x^d.\]
Finally for a family of Calabi-Yau manifolds, we consider the associated Picard-Fuchs equation. Then under the mirror symmetry hypothesis, we show that the Yukawa couplings can be interpreted as these new functions \(I_p\). Zinger functions; Yukawa couplings; maximal unipotent monodromy; Calabi-Yau equations; mirror symmetry Period matrices, variation of Hodge structure; degenerations, Calabi-Yau manifolds (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Applications of deformations of analytic structures to the sciences, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Structure of families (Picard-Lefschetz, monodromy, etc.) Zinger functions and Yukawa couplings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For positive integers \(p, q\), let \(f_{p,q}(x,y)=x^p(xy(x+2)\cdots(x+q)-1)\). The author calls them generalized Broughton polynomials after the work of \textit{S. A. Broughton} [Invent. Math. 92, 217--241 (1988; Zbl 0658.32005)], which investigates the topology of the polynomial function \(f(x,y)=x(xy-1)\). Let \(C_0:f_{p,q}(x,y)/x^{p}=0\), \(C_1:f_{p,q}(x,y)=1\), and let \(M=\mathbb{C}^2-(C_0\cup C_1)\). The main result of this paper determines the structures of the resonance varieties \(\mathcal{R}_k(M)\) and the characteristic varieties \(\mathcal{V}_k(M)\). In particular he shows that strictly positive dimensional components of \(\mathcal{V}_1(M)\) consist of \(p-1\) one-dimensional translated tori. Broughton polynomial; characteristic varieties; resonance varieties; translated components Pencils, nets, webs in algebraic geometry, Plane and space curves, Singularities of curves, local rings, Relations with arrangements of hyperplanes Broughton polynomials and characteristic varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathfrak g\) be a simple complex (finite dimensional) Lie algebra, and let \(R\) be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed. Lie algebras of the form \({\mathfrak g} \otimes_\mathbb{C} R\) are considered; these generalize Kac-Moody loop algebras since for a curve of genus zero with two punctures \(R \simeq \mathbb{C}[t,t^{-1}]\). The universal central extension of \({\mathfrak g} \otimes R\) is analogous to an untwisted affine Kac-Moody algebra. By Kassel's theorem the kernel of the universal central extension is linearly isomorphic to the Kähler differentials of \(R\) modulo exact differentials. The dimension of the kernel for any \(R\) is determined first. Restricting to hyperelliptic curves with 2, 3, or 4 special points removed, a basis for the kernel is determined. Restricting further to an elliptic curve with punctures at two points (of orders one and two in the group law) we explicitly determine the cocycles which give the commutation relations for the universal central extension. The results involve Pollaczek polynomials, which are a genus-one generalization of ultraspherical (Gegenbauer) polynomials. Krichever-Novikov algebras; universal central extension; untwisted affine Kac-Moody algebra; Kassel's theorem; Kähler differentials; hyperelliptic curves; elliptic curve with punctures at two points; cocycles; Pollaczek polynomials M. R. Bremner, ''Universal central extensions of elliptic affine Lie algebras'',J. Math. Phys. 35, 6685--6692 (1994). Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Elliptic curves, Differentials on Riemann surfaces Universal central extensions of elliptic affine Lie algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Finding examples of compact hyperkähler manifolds is a difficult problem in algebraic geometry. In this paper, the authors propose a new conjectural method to associate to each Lie algebra of type \(G_2, D_4, F_4, E_6, E_7, E_8\) a family of polarized hyperkähler fourfolds.
It is known that to each simple complex Lie algebra one can associate a projective homogeneous variety \(X_{\textrm{ad}}\), called the adjoint variety; this adjoint variety is covered by a family of special subvarieties, called Legendrian cycles.
The authors conjecture that for the considered Lie algebras, it is possible to construct a projective variety \(X\) such that
\begin{itemize}
\item \(\dim X = \dim X_{\textrm{ad}}\),
\item \(X\) is also covered by Legendrian cycles,
\item there is a suitable cycle space \(P\) for these Legendrian cycles,
\item for a generic hyperplane section \(X_H\) of \(X\), the subspace \(P_H\subset P\) parametrizing the Legendrian cycles contained in \(X_H\) is birational to a hyperkähaler fourfold.
\end{itemize}
The conjecture is proven and \(P_H\) is explicitly described for the Lie algebras \(G_2, D_4, F_4, E_6\); the cases \(E_7, E_8\) stay open. Fano varieties; hyperkähler varieties; Legendrian varieties; Tits-Freudenthal square \(K3\) surfaces and Enriques surfaces, Fano varieties, Grassmannians, Schubert varieties, flag manifolds Hyperkähler manifolds from the Tits -- Freudenthal magic square | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author computes the number of points, in the Galois field \(GF(q)\) with \(q\) elements, of the Grassmann variety of \(d\)-planes in projective \(n\)-space, and all its Schubert subvarieties, all defined over \(GF(q)\). He observes in the case of Grassmann manifolds that the formula he obtains is the same as the one given by Sylvester for Gauss polynomials. number of points of Grassmann variety; Gauss polynomials; Sylvester's formula; Galois geometries; Galois field; Schubert subvarieties Grassmannians, Schubert varieties, flag manifolds, Combinatorial identities, bijective combinatorics, Enumerative problems (combinatorial problems) in algebraic geometry, Finite ground fields in algebraic geometry A geometric interpretation of an equality by Sylvester | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a continuation of part I [Adv. Stud. Pure Math. 6, 255-287 (1985; Zbl 0569.20032)]. The main theme of the previous paper was the study of a numerical function \(w\mapsto a(w)\) on a Coxeter group W which in the case of Weyl groups is closely related to the Gelfand-Kirillov dimension of certain modules over the corresponding enveloping algebra. There, the function was defined purely in terms of multiplication in the Hecke algebra and several properties of it were proved for Weyl groups.
In this paper, the author proves them for a larger class of Coxeter groups including the affine Weyl groups. One of the main themes of the paper is provided by certain distinguished involutions of W, one in the left cell. \textit{A. Joseph} has shown that for each left cell of a Weyl group, the function \(y\mapsto \ell(y)-2\delta(y)\) \((\ell=\) length, \(\delta=\) degree of the polynomial \(P_{e,w})\) reaches its minimum at a unique element of that left cell [cf. J. Algebra 73, 295-326 (1981; Zbl 0482.17002); \textit{D. Kazhdan} and the author, Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. The author shows that this minimal value is \(a(y)\) for any y in the left cell, from this Joseph's conjecture follows easily and also gives the analogous result for affine Weyl groups. An important role in the author's proof is played by a ring J which may be regarded as an asymptotic version of the Hecke algebra H and he proves the comparison theorem between H and J. Gelfand-Kirillov dimension; enveloping algebra; Hecke algebra; Coxeter groups; affine Weyl groups; involutions; left cell Lusztig, G., Cells in affine Weyl groups, II, \textit{J. Algebra}, 109, 536-548, (1987) Representation theory for linear algebraic groups, Universal enveloping (super)algebras, Homological dimension in associative algebras, Associated Lie structures for groups, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Cells in affine Weyl groups. II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules. equivariant D-modules; Kostka polynomials; Poisson-de Rham homology; W-algebras; Springer fibers; nilpotent cone; Harish-Chandra homomorphism; Grothendieck-Springer resolution Bellamy, G., Schedler, T.: Kostka polynomials from nilpotent cones and Springer fiber cohomology. arXiv:1509.02520 (2015) Poisson algebras, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grassmannians, Schubert varieties, flag manifolds Filtrations on Springer fiber cohomology and Kostka polynomials | 0 |
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