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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 The authors study invertible weighted homogeneous polynomials in three variables and define their Dolgachev and Gabrielov numbers. Thereafter, they show that these numbers underlie a duality involving the Berglund-Hübsch transform of the polynomial. This gives a generalization of Arnold's strange duality between the 14 exceptional unimodal singularities. Moreover, the authors discuss additional features of this duality leading to the coincidence of certain invariants for dual invertible polynomials and new strange dualities. mirror symmetry; singularity; invertible polynomial; weighted projective line; strange duality; Dolgachev numbers; Gabrielov numbers Ebeling W., Takahashi A.: Strange duality of weighted homogeneous polynomials. Compos. Math. 147, 1413--1433 (2011) Mirror symmetry (algebro-geometric aspects), Complex surface and hypersurface singularities, Group actions on varieties or schemes (quotients) Strange duality of weighted homogeneous 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 For any finite Coxeter system \((W,S)\) we construct a certain noncommutative algebra, the so-called `bracket algebra', together with a family of commuting elements, the so-called `Dunkl elements'. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group \(W\). We prove this conjecture for classical Coxeter groups and \(I_2(m)\). We define a ``quantization'' and a multiparameter deformation of our construction and show that for Lie groups of classical type and \(G_2\), the algebra generated by Dunkl's elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in \(B_n\)-, \(D_n\)- and \(G_2\)-bracket algebras, and as an application, discover a `Pieri-type formula' in the \(B_n\)-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary \(B_n\)-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri's formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type \(A\). We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called `flat connections with constant coefficients', which describes ``a noncommutative differential geometry on a finite Coxeter group'' in the sense of S. Majid. finite Coxeter system; Dunkl elements; coinvariant algebras of Coxeter groups; multiparameter deformation; quantized bracket algebras; quantum cohomology ring; flag variety Kirillov, A., Maeno, T.: Noncommutative algebras related with Schubert calculus on Coxeter groups. Eur. J. Comb. \textbf{25}, 1301-1325 (2004). Preprint RIMS-1437, 2003 Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Reflection and Coxeter groups (group-theoretic aspects) Noncommutative algebras related with Schubert calculus on Coxeter groups | 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 Khovanov-Lauda-Rouquier algebras (or quiver Hecke algebras) were introduced independently by \textit{M. Khovanov} and \textit{A. D. Lauda} [Represent. Theory 13, 309--347 (2009; Zbl 1188.81117)] and [\textit{R. Rouquier}, ``2-Kac-Moody algebras'', preprint, \url{arXiv:0812.5023}] to construct a categorification of quantum groups associated with symmetrizable Cartan data. For a dominant integral weight \(\lambda\in P^+\), Khovanov and Lauda conjectured that the cyclotomic quotient \(R^\lambda\) of the KLR algebra \(R\) gives a categorification of the irreducible highest weight module \(V(\lambda)\), which was proved recently by \textit{S.-J. Kang} and \textit{M. Kashiwara} [Invent. Math. 190, No. 3, 699--742 (2012; Zbl 1280.17017)].
When the Cartan datum is symmetric, \textit{M. Varagnolo} and \textit{E. Vasserot} [J. Reine Angew. Math. 659, 67--100 (2011; Zbl 1229.17019)] and \textit{R. Rouquier} [Algebra Colloq. 19, No. 2, 359--410 (2012; Zbl 1247.20002)] gave a geometric realization of KLR algebra via quiver varieties and proved that the isomorphism classes of projective indecomposable modules corresponds to Kashiwara's lower global basis (or Lusztig's canonical basis).
Later, \textit{S.-J. Kang}, \textit{S.-J. Oh} and \textit{E. Park} [Int. J. Math. 23, No. 11, Paper No. 1250116, 51 p. (2012; Zbl 1283.17014)] introduced a family of KLR algebra \(R\) associated with symmetrizable Borcherds-Cartan data and showed that they provide a categorification of quantum generalized Kac-Moody algebras and their crystals.
In this paper, the authors follow the framework of the work of Varagnolo-Vasserot, they construct a geometric realization of KLR algebras associated with symmetric Borcherds-Cartan data via quivers possibly with loops. One of the main ingredients is Steinberg-type varieties arising from quivers. As an application, when \(a_{ii}\not= 0\) for any \(i\in I\), the authors prove that there exists a one-to-one correspondence between Kashiwara's lower basis (or Lusztig's canonical basis) of \(U^{-}(\mathfrak{g})\) (resp., \(V(\lambda)\)) and the set of isomorphism classes of indecomposable projective graded modules mover \(R\) (resp. the cyclotomic quotient \(R^\lambda\)). Khovanov-Lauda-Rouquier algebra; Borcherds-Cartan matrix; quiver varieties; global basis; categorification Kang, S. -J.; Kashiwara, M.; Park, E.: Geometric realization of Khovanov-lauda-Rouquier algebras associated with borcherds-Cartan data. Proc. lond. Math. soc. (3) 107, 907-931 (2013) Quantum groups (quantized enveloping algebras) and related deformations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations Geometric realization of Khovanov-Lauda-Rouquier algebras associated with Borcherds-Cartan data | 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 symplectic Grassmannian \(G_w = G_w(3,6)\) is the variety of all 3-spaces in \({\mathbb{C}}^6\) where a non-degenerate 3-form \(w\) vanishes. \(G_w\) is a smooth Fano 6-fold with \(\mathrm{Pic}(G_w) = {\mathbb{Z}}H\), and canonical class \(K_{G_w} = -4H\). The ample generator \(H\) embeds \(G_w\) in \({\mathbb{P}}^{13}\) as a smooth 6-fold of degree 16. In this paper the author studies the geometry of the general linear section \(B\) of \(G_w\) with a codimension 2 subspace \({\mathbb{P}}^{11}\) of \({\mathbb{P}}^{13}\). In Section 2 is shown that the family of lines \(F_B\) of \(B\) is a linear section of the Segre 4-fold \({\mathbb{P}}^1 \times {\mathbb{P}}^1 \times {\mathbb{P}}^1 \times {\mathbb{P}}^1\). In Section 3 is computed the Chow ring of \(B\). In Sections 1 and 4 are constructed four rank two vector bundles \(E_i\) on \(B\) such that \(E_i(1)\) give embeddings \(f_i\) of \(B\) in the Grassmannian \(G(2,6)\). One interesting property of these embeddings is that the isomorphic images of \(B\) in \(G(2,6)\) under \(f_i\), \(i = 1,\dots,4\) can be realized as congruences of lines described by \textit{E. Mezzetti} and \textit{P. De Poi} [Geom. Dedicata 131, 213--230 (2008; Zbl 1185.14042)]. Fano manifold; vector bundle; family of lines; symplectic Grassmannian Han, F, Geometry of the genus 9 Fano 4-folds, Ann. Inst. Fourier (Grenoble), 60, 1401-1434, (2010) Fano varieties, \(4\)-folds, Grassmannians, Schubert varieties, flag manifolds Geometry of the genus 9 Fano 4-folds | 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 describe a family of polynomials discovered via a particular recursion relation, which have connections to Chebyshev polynomials of the first and the second kind, and the polynomial version of Pell's equation. Many of their properties are listed in Section 3. We show that these families of polynomials in the variable \(t\) satisfy certain second-order linear differential equations that may be of interest to mathematicians in conformal field theory and number theory. We also prove that these families of polynomials in the setting of Date-Jimbo-Kashiwara-Miwa algebras when multiplied by a suitable power of \(t\) are orthogonal with respect to explicitly described kernels. Particular cases lead to new identities of elliptic integrals (see Section 5). hyperelliptic lie algebras; Krichever-Novikov algebras; universal central extensions; Date-Jimbo-Kashiwara-Miwa algebras; elliptic integrals; Pell's equation; Chebyshev polynomials Other special orthogonal polynomials and functions, Other functions coming from differential, difference and integral equations, Riemann surfaces; Weierstrass points; gap sequences, Infinite-dimensional Lie (super)algebras, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization), Geometric theory, characteristics, transformations in context of PDEs, PDEs in connection with optics and electromagnetic theory Families of orthogonal Laurent polynomials, hyperelliptic Lie algebras and elliptic integrals | 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 Start with a permutation matrix \(\pi\) and consider all matrices that can be obtained from \(\pi\) by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety \(\overline{X_\pi} \). We characterize when the ideal defining \(\overline{X_\pi} \) is toric (with respect to a \(2n-1\)-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of \(\beta \)-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of \(\beta \)-Grothendieck polynomials. Subword complexes were introduced by \textit{A. Knutson} and \textit{E. Miller} [Adv. Math. 184, No. 1, 161--176 (2004; Zbl 1069.20026)], who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations. subword complex; root polytope; matrix Schubert variety; toric variety Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies Toric matrix Schubert varieties and root polytopes (extended abstract) | 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 linear complex algebraic group, \(B\) a Borel subgroup of \(G\), and \(G/B\) the flag manifold. A Schubert variety \(X_w\) in \(G/B\) is the Zariski closure of a \(B\)-orbit in \(G/B\) and has the form \(BwB/B\), where \(w\) is a unique element in the Weyl group \(W(T,G)\), for \(T\) a maximal torus in \(B\). This paper is concerned with the Poincaré polynomial of \textit{smooth} Schubert varieties in \(G/B\). The main result is the following.
Theorem. Let \(X_w\) be a smooth Schubert variety in \(G/B\), and let \(k\) denote the largest height occurring in \(T_e(X_w)\), the Zariski tangent space of \(X_w\) at the identity coset. Then \(d_{w,k}>0\), and
\[
P_w(t)=\prod_{1\leq i\leq k}(1+t^2+\ldots+t^{2i})^{d_{w,i}}.
\]
If \(\mu\) is the partition of \(\ell(w)\) conjugate to \(\eta\) and \(i\geq 1\), then \(d_{w,i}\) is the number of times \(i\) occurs in \(\mu\).
\smallskip\noindent Here \(\ell(w)\) is the length of \(w\) in the Weyl group, \(\Phi\) is the root system determined by \(T\) and \(\Phi(w)^+=\{\alpha>0~|~ r_\alpha\leq w\}\), where \(r_\alpha\) denotes the reflection in \(\alpha\) and \(\leq\) stands for the Bruhat-Chevalley order. Moreover \(h_{w,i}=\{\alpha\in \Phi(w)^+~|~ \text{height}(\alpha)=i,~\text{for}~i>0\}\), the symbol \(\eta\) denotes the \textit{nonincreasing} partition of \(\ell(w)=|\Phi(w)^+|\) formed by the \(h_{w,i}\geq h_{w,i+1}\), and \(d_{w,i}=h_{w,i}-h_{w,i+1}\).
\noindent As a consequence of the above formula, the second Betti number of \(X_w\) is given by \(b_2(X_w)=h_{w,1}\). As the authors point out, a factorization of \(P_w(t)\) into polynomials of the form \(\mu_i(t)=1+t^2+\ldots+t^{2i}\) does not hold for smooth Schubert varieties in arbitrary flag manifolds \(G/P\).
\noindent The proof of the above theorem is obtained by generalizing to smooth Schubert varieties a cohomological proof of a formula of Kostant, Macdonald, Shapiro and Steinberg, which expresses the Poincaré polynomial of \(G/B\) as the product \(\prod_{i=1}^l(1+t^2+\ldots +t^{2m_i})\), where \(m_1,\ldots, m_l\) are the exponents of \(G\). schubert varieties Akyildiz, E; Carrell, JB, Betti numbers of smooth Schubert varieties and the remarkable formula of Kostant, Macdonald, Shapiro, and Steinberg, Mich. Math. J., 61, 543-553, (2012) Homogeneous spaces and generalizations, Homology and cohomology of homogeneous spaces of Lie groups Betti numbers of smooth Schubert varieties and the remarkable formula of Kostant, Macdonald, Shapiro, and Steinberg | 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 0741.00067.]
A classical formula, stated by Schubert in 1880, gives the number of triple contacts between curves of a \(d\)-dimensional family \(\Gamma\) and surfaces of a \((2-d)\)-dimensional family \(\Sigma\) in \(\mathbb{P}^ 3\) \((d=0,1,2)\). The formula for the case \(d=0\) has been justified by the author in a previous paper. The present paper deals with the formulas for \(d=1,2\); they are proved, when the families \(\Gamma\) and \(\Sigma\) have smooth general fibers, by a direct computation in the Chow ring \(A^*(\text{Hilb}^ 3\mathbb{P}^ 3)\). The formula for \(d=1\) contains a term which is missing in Schubert's original paper. number of triple contacts; Chow ring [R2]Rosselló, F., Triple contact formulas inP 3, inEnumerative Algebraic Geometry (Kleiman, S. L. and Thorup, A., eds.), Contemp. Math.123, pp. 223--246, Amer. Math. Soc., Providence, R. I., 1991. Enumerative problems (combinatorial problems) in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Parametrization (Chow and Hilbert schemes) Triple contact formulas in \(\mathbb{P}{}^ 3\) | 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 connected reductive group defined over the algebraic closure of a finite field, and let \(F:G \to G\) denote an isogeny of \(G\) of which some power is a (standard) Frobenius morphism. Let \(T\) be a maximal \(F\)-stable torus of \(G\), and let \(W\) denote the Weyl group of \(G\). Then to any element \(w \in W\) we can associate two Deligne-Lusztig varieties \(X(w)\) and \(Y(w)\) and a finite étale morphism \(Y(w) \to X(w)\) making \(X(w)\) a quotient of \(Y(w)\) by the action of \(T^F\), [see \textit{P. Deligne} and \textit{G. Lusztig}, Ann. Math. (2) 103, 103--161 (1976; Zbl 0336.20029)]. In [loc. cit.], Deligne and Lusztig construct a smooth compactification \(\overline{X}(w)\) of \(X(w)\); the main purpose of this paper is to give an explicit construction of the normalization \(\overline{Y}(w)\) of \(\overline{X}(w)\) in \(Y(w)\). The explicit nature of this construction allows the authors to give many properties of \(\overline{Y}(w)\), as well as to deduce a new proof of Lemme 9.11 of [loc. cit.], which is a key part of the proof of the Macdonald conjectures associating an irreducible representation of \(G^F\) to a character in general position of \(T^F\). Deligne-Lusztig varieties; compactification; normalisation; monodromy [2] Cédric Bonnafé &aRaphaël Rouquier, &Compactification des variétés de Deligne-Lusztig
nn. Inst. Fourier (Grenoble)59 (2009) no. 2, p.~621Cedram | &MR~25 | &Zbl~1167. Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Linear algebraic groups and related topics Compactification of Deligne-Lusztig 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 \textit{V. V. Batyrev} [Asterisque 218, 9--34 (1993; Zbl 0806.14041)] described the quantum cohomology ring of Fano toric varieties in terms of generators (toric divisors and formal \(q\) variable) and relations (linear relations and \(q\)-deformed monomial relations). \textit{A. Kresch} [Mich. Math. J. 48, 369--391 (2000; Zbl 1085.14519)] gave a so called quantum Giambelli formula that expresses any cohomology class in \(H^*(X,\mathbb Q)\) as a polynomial in divisor classes and formal \(q\) variables for a certain class of Fano toric varieties.
In the present paper, the authors compute Gromov-Witten invariants of \(\mathbb P^d\)-bundles, \(X=\mathbb P(\oplus_{i=1}^r{\mathcal O}_{\mathbb P^1}(a_i))\) with \(\sum_{i=1}^ra_i = 2 + kr\), and give examples of quantum products with infinitely many non-trivial quantum corrections. As a main tool, the authors use the fact that Gromov-Witten invariants are invariants of the symplectic deformation class of a symplectic manifold, and combinatorial formulas by \textit{H. Spielberg} [C. R. Acad. Sci. Paris, Sér. I Math. 329, No. 8, 699--704 (1999; Zbl 1004.14014)]. Batyrev's conjecture; symplectic deformation Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli GW-invariants and quantum products with infinitely many quantum corrections | 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 extends Demazure's character formula to any Kac-Moody algebra (not necessarily symmetrizable). From this, using an argument of G. Heckman, he gets the Weyl character formula. Underlying these results, there is the identification of the characters with some Euler-Poincaré characteristic dimensions and the proof of vanishing theorems for the cohomology of semi-ample line bundles over the Schubert varieties. This machinery also allows him to prove a generalization of the Bott-Borel- Weyl theorem and Kempf's theorem, as well as some properties of the Schubert varieties. Kac-Moody algebra; Weyl character formula; Euler-Poincaré characteristic dimensions; vanishing theorems; cohomology of semi-ample line bundles; Schubert varieties; generalization of the Bott-Borel-Weyl theorem; Kempf's theorem Mathieu, Olivier, Formules de caractères pour les algèbres de Kac-Moody générales, Astérisque, 159-160, 267 pp., (1988) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Infinite-dimensional Lie (super)algebras, (Co)homology theory in algebraic geometry Character formulas for general 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 [For the entire collection see Zbl 0608.00010.]
The wedge and the vertex construction of the fundamental highest weight representations of the group \(GL_{\infty}\) are discussed, and a kind of boson-fermion correspondence is obtained. This is applied to the modified KP hierarchies yielding several definitions of these hierarchies and polynomial solutions. Geometrical interpretations of the modified KP hierarchies are given in terms of an infinite dimensional Grassmann variety and an associated flag variety. Kac-Moody algebra; wedge; vertex construction; highest weight representations; boson-fermion correspondence; KP hierarchies; polynomial solutions; Grassmann variety; flag variety Kac, V. G.; Peterson, D. H.: Lectures on infinite wedge representation and MKP hierarchy. Séminaire de math. Sup., LES presses de l'université de Montreal 102, 141-186 (1986) Partial differential equations of mathematical physics and other areas of application, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Infinite-dimensional Lie groups and their Lie algebras: general properties, Grassmannians, Schubert varieties, flag manifolds, Geometric theory, characteristics, transformations in context of PDEs Lectures on the infinite wedge-representation and the MKP hierarchy | 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 Gröbner basis for the small quantum cohomology of Grassmannian \(G_{k,n}\) is constructed and used to obtain new recurrence relations for Kostka numbers and inverse Kostka numbers. Using these relations it is shown how to determine inverse Kostka numbers which are related to the mod-\(p\) Wu formula. quantum cohomology; Gröbner basis; Grassmannian; (inverse) Kostka numbers Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Characteristic classes and numbers in differential topology Recurrence formulas for Kostka and inverse Kostka numbers via quantum cohomology of 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 review the Reshetikhin-Turaev approach for constructing noncompact knot invariants involving \(R\)-matrices associated with infinite-dimensional representations, primarily those constructed from the Faddeev quantum dilogarithm. The corresponding formulas can be obtained from modular transformations of conformal blocks as their Kontsevich-Soibelman monodromies and are presented in the form of transcendental integrals, where the main issue is working with the integration contours. We discuss possibilities for extracting more explicit and convenient expressions that can be compared with the ordinary (compact) knot polynomials coming from finite-dimensional representations of simple Lie algebras, with their limits and properties. In particular, the quantum A-polynomials and difference equations for colored Jones polynomials are the same as in the compact case, but the equations in the noncompact case are homogeneous and have a nontrivial right-hand side for ordinary Jones polynomials. Chern-Simons theory; Kontsevich-Soibelman monodromy; Wilson average; \(R\)-matrix; modular double; quantum A-polynomial Galakhov, D. and Mironov, A. and Morozov, A., {\({\mathrm SU}(2)/{\mathrm SL}(2)\)} knot invariants and {K}ontsevich--{S}oibelman monodromies, Theoretical and Mathematical Physics, 187, 2, 678-694, (2016) Yang-Mills and other gauge theories in quantum field theory, Eta-invariants, Chern-Simons invariants, Structure of families (Picard-Lefschetz, monodromy, etc.) \(\mathrm{SU}(2)/\mathrm{SL}(2)\) knot invariants and Kontsevich-Soibelman monodromies | 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 geometry of Grassmann varieties and Schubert cycles, in particular their postulation, is naturally related to the combinatorics of standard monomial theory, developed by several authors in the seventies and eighties, by means of the introduction of combinatorial structures.
\textit{C. De Concini}, \textit{D. Eisenbud} and \textit{C. Procesi} [Hodge algebras, Astérisque 91 (1982; Zbl 0509.13026)] introduced in 1981 the axioms of Hodge algebras over a partially ordered set (poset), and used them to reconstruct the study of arithmetic Cohen-Macaulayness and normality of determinantal varieties. When this theory was extended to dosets (i.e. introducing structures with combinatorial multiplicity 2), it allowed to explore the postulation of Schubert varieties \(X\subset G/P\), where \(G\) is a simply connected Chevalley group and \(P\) is a maximal subgroup of classical type. In order to extend the theory to general parabolic subgroups \(P\), De Concini proposed to use LS \textit{algebras}, i.e. multiset algebras (of higher multiplicity) having a monomial basis combinatorially described by the Lakshmibai-Seshadri paths (LS--paths), introduced for the study of representations of Kac-Moody algebras.
The book under review, based on the PhD thesis of the author, contains a development of the theory of LS algebras and their straightening relations. The author uses LS algebras to prove that Schubert varieties \(X=G/P\), for any parabolic \(P\), flatly degenerate to unions of normal toric varieties. This has consequences on the arithmetic normality of \(X\) and allows the computation of the degree of \(X\), embedded inside the projective space over the irreducible modules of \(G\).
A further generalization of LS algebras to \textit{multigraded} LS algebras (introduced through the book by means of the combinatorics of multiposets), allows the author to produce similar results on the degeneration, hence on the arithmetic Cohen-Macaulayness, of some partial flag varieties. Grassmann varieties; posets; dosets; postulation; LS paths Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) LS algebras 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 This short note studies the quantum cohomology algebra of the Grassmannian of \(l\)-dimensional planes in \(N\)-dimensional space. Take the coefficients to be \(q\)-polynomials over a cyclotomic field, the extension \(\mathbb{Q}(\zeta)\) by a primitive \(N\)-th root of \((-1)^{l+1}\). The main result is that when \(q\) is specialized to \(1\) the quantum cohomology algebra is a direct sum of its minimal prime ideals each isomorphic to \(\mathbb{Q}(\zeta)\). It follows that the rational quantum cohomology algebra with \(q=1\) is also semisimple and the eigenvalues of multiplications by Schubert classes lie in \(\mathbb{Q}(\zeta)\). Galkin, S; Golyshev, V, Quantum cohomology of Grassmannians and cyclotomic fields, Russ. Math. Surv., 61, 171, (2006) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Exterior algebra, Grassmann algebras Quantum cohomology of Grassmannians and cyclotomic fields | 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 book is the second edition of \textit{Yu. I. Manin}'s celebrated and inspiring lectures at the Université de Montréal [Quantum groups and non-commutative geometry. Montréal: Université de Montréal, Centre de Recherches Mathématiques (CRM) (1988; Zbl 0724.17006)].
This new edition has an additional chapter by Raedschelders and Van den Bergh, based on the PhD Thesis of \textit{Th. Raedschelders} [Manin's universal Hopf algebras and highest weight categories. Brussels: Vrije Universiteit Brussel (PhD Thesis) (2017)] and the authors joint works [Adv. Math. 305, 601--660 (2017; Zbl 1405.16044); J. Noncommut. Geom. 11, No. 3, 845--885 (2017; Zbl 1383.16025)]. The new chapter aims to expand upon a discussion in Manin's lectures on the possibility of ``hidden-symmetry'' in algebraic geometry. More precisely, the chapter concerns several aspects of the representation theory of certain universal bialgebras \(\underline{\mathrm{end}}(A)\) and Hopf algebras \(\underline{\mathrm{gl}}(A)\) introduced in Manin's book, where \(A\) is an algebra subject to appropriate conditions. quantum group; quantum matrix space; quadratic Hopf algebra; compact matrix pseudogroup; Yang-Baxter equation; tensor category Research exposition (monographs, survey articles) pertaining to associative rings and algebras, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Research exposition (monographs, survey articles) pertaining to group theory, Noncommutative algebraic geometry, Hopf algebras and their applications, Bialgebras, Ring-theoretic aspects of quantum groups, Yang-Baxter equations, Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Noncommutative geometry in quantum theory Quantum groups and noncommutative geometry. With a contribution by Theo Raedschelders and Michel Van den Bergh | 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 Given a two-variable invertible polynomial, we show that its category of maximally-graded matrix factorisations is quasi-equivalent to the Fukaya-Seidel category of its Berglund-Hübsch transpose. This was previously shown for Brieskorn-Pham and \(D\)-type singularities by Futaki-Ueda. The proof involves explicit construction of a tilting object on the B-side, and comparison with a specific basis of Lefschetz thimbles on the A-side. Fukaya-Seidel category; pretriangulated \(A_\infty\)-categories; derived category of singularities; Brieskorn-Pham polynomials Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Mirror symmetry (algebro-geometric aspects) Homological Berglund-Hübsch mirror symmetry for curve 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 Consider the generalized flag manifold \(G/B\) and the corresponding affine flag manifold \(\mathcal{F}\ell_{G}\). In this paper we use curve neighborhoods for Schubert varieties in \(\mathcal{F}\ell_G\) to construct certain affine Gromov -- Witten invariants of \(\mathcal{F}\ell_G\), and to obtain a family of `affine quantum Chevalley' operators \(\Lambda_0,\ldots,\Lambda_n\) indexed by the simple roots in the affine root system of \(G\). These operators act on the cohomology ring \(H^\ast(\mathcal{F}\ell_G)\) with coefficients in \(\mathbb{Z}[q_0,\ldots,q_n]\). By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for \(G= \mathrm{SL}_n(\mathbb{C})\). The first quantum ring is a deformation of the subalgebra of \(H^\ast(\mathcal{F}\ell_G)\) generated by divisors. The second ring, denoted \(QH_{aff}^\ast (G/B)\), deforms the ordinary quantum cohomology ring \(QH^\ast(G/B)\) by adding an affine quantum parameter \(q_0\). We prove that \(QH_{aff}^\ast (G/B)\) is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of \(QH_{aff}^\ast (G/B)\) by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of \(G\). Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice | 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 positive integer \(k\) and nonnegative integer \(m\), we consider the symmetric function \(G(k,m)\) defined as the sum of all monomials of degree \(m\) that contain no exponents larger than \(k-1\). We call \(G(k,m)\) a Petrie symmetric function in honor of Flinders Petrie, as the coefficients in its expansion in the Schur basis are determinants of Petrie matrices (and thus belong to \(\{-1,0,1\}\) by a classical result of \textit{M. Gordon} and \textit{E. M. Wilkinson} [Pac. J. Math. 51, 451--453 (1974; Zbl 0246.15008)]). More generally, we prove a Pieri-like rule for expanding a product of the form \(G(k,m) \cdot s_{\mu}\) in the Schur basis whenever \(\mu\) is a partition; all coefficients in this expansion belong to \(\{-1,0,1\}\). We show a further formula for \(G(k,m)\) and prove that \(G(k,1), G(k,2), G(k,3), \ldots\) form an algebraically independent generating set for the symmetric functions when \(1-k\) is invertible in the base ring. We prove a conjecture of \textit{L. Liu} and \textit{P. Polo} [``On the cohomology of line bundles over certain flag schemes. II'', Preprint, \url{arXiv:1908.08432}] about the expansion of \(G(k,2k-1)\) in the Schur basis. We then take our Pieri-like rule as an impetus to pose a different question: What other symmetric functions \(f\) have the property that each product \(fs_{\mu}\) expands in the Schur basis with all coefficients belonging to \(\{-1,0,1\}\)? We call this property MNability due to its most classical instance (besides the Pieri rules, which do not use \(-1\) coefficients) being the Murnaghan-Nakayama rule. Surprisingly, we find a number of infinite families of MNable symmetric functions besides the classical ones. symmetric functions; Petrie matrices; Murnaghan-Nakayama rule; Pieri rules; Schur functions; determinants Symmetric functions and generalizations, Classical problems, Schubert calculus The Petrie symmetric 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 An integral basis of the cohomology of a flag manifold \(G/B\) is the Schubert classes \(\mathfrak S_w\) indexed by the elements \(w\) of the Weyl group of \(G\). Hence there are, for all pairs of elements \(u,v\) in the Weyl group, integers \(c_{uv}^w\) such that
\[
\mathfrak S_u\mathfrak S_v =\sum_wc_{uv}^w \mathfrak S_w,
\]
where the summation is over the elements in the Weyl group. A Pieri type formula is a formula that describes the structure constants \(c_{uv}^w\) when \(\mathfrak S _v\) is a special Schubert class pulled back from the projection \(G/B\to G/P\), where \(P\) is a maximal parabolic subgroup. When \(G=\text{Gl}_n(\mathbb{C})\) the classical Pieri formula gives such a description. For other \(G\) there are formulas by \textit{H. Hiller} and \textit{B. Boe} [Adv. Math. 62, 49-67 (1986; Zbl 0611.14036)] and by \textit{P. Pragacz} and \textit{J. Ratajski} [J. Reine Angew. Math. 476, 143-189 (1996; Zbl 0847.14029); C. R. Acad. Sci., Paris, Sér. I 317, 1035-1040 (1993; Zbl 0812.14034); Manuscr. Math. 79, 127-151 (1993; Zbl 0789.14041)]. When \(G=\text{Gl}_n(\mathbb{C})\) an interpretation of the structure constants \(c_{uv}^w\) in Pieri's formula, in terms of chains in the Bruhat order, was conjectured by \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, 257-269 (1993; Zbl 0803.05054)] and given algebraic, geometric and combinatorial proofs by \textit{A. Postnikov} [Prog. Math. 172, 371-383 (1999; Zbl 0944.14019)], \textit{F. Sottile} [Ann. Inst. Fourier 46, 89-110 (1996; Zbl 0837.14041)], and \textit{M. Kogan} and \textit{A. Kumar} [Proc. Am. Math. Soc. 130, 2525-2534 (2002; Zbl 1001.05121)], respectively. The main result of the present article are analogous Pieri type formulas when \(G\) is \(\text{Sp}_{2n}(\mathbb{C})\) and \(\text{SO}_{2n+1}(\mathbb{C})\). One of the techniques used is to explicitly determine triple intersections of Schubert varieties. \textit{F. Sottile} has used this technique with success earlier [see, e.g., Colloq. Math. 82, 49-63 (1999; Zbl 0977.14023)], and shows that the coefficients in the Pieri type formulas are the intersection number of a linear space with a collection of quadrics, and thus are either \(0\) or a power of \(2\). special Schubert classes; Schubert varieties; Bruhat order; Pieri type formulas; Weyl groups; parabolic groups; isotropic flag manifolds; cohomology \beginbarticle \bauthor\binitsN. \bsnmBergeron and \bauthor\binitsF. \bsnmSottile, \batitleA Pieri-type formula for isotropic flag manifolds, \bjtitleTrans. Amer. Math. Soc. \bvolume354 (\byear2002), no. \bissue7, page 2659-\blpage2705 \bcomment(electronic). \endbarticle \OrigBibText ----, A Pieri-type formula for isotropic flag manifolds , Trans. Amer. Math. Soc. 354 (2002), no. 7, 2659-2705 (electronic). \endOrigBibText \bptokstructpyb \endbibitem Classical groups (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Symmetric functions and generalizations, Combinatorics of partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] A Pieri-type formula for isotropic 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 authors consider cases of the quantum differential equation (QDE) which is an ordinary linear differential equation with a regular singular point at \(0\) and an irregular singular point at \(\infty\). At the latter the differences between the formal (symbolic) solution and the actual solution are estimated in sectors by the Stokes data. The classical Stokes matrices for the QDE of \(\mathbb{P}^n\) are computed using multisummation and the `monodromy identity'. The authors recover the results of \textit{D. Guzzetti} [Commun. Math. Phys. 207, No. 2, 341--383 (1999; Zbl 0976.53094)] that confirm Dubrovin's conjecture for projective spaces. The same method yields explicit formulas for the Stokes matrices of quantum differential equations of smooth Fano hypersurfaces in \(\mathbb{P}^n\) and for weighted projective spaces. Stokes matrices; quantum cohomology; monodromy identity; quantum differential equations; Fano variety Morales, J.A.C., van der Put, M.: Stokes matrices for the quantum differential equations of some Fano varieties. arXiv:1211.5266 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Fano varieties, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Formal solutions and transform techniques for ordinary differential equations in the complex domain Stokes matrices for the quantum differential equations of some Fano 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 well-known generalized flag manifolds model important geometric situations, such as all the compact Kähler manifolds or the coadjoint orbits of compact semisimple Lie groups. They are quotients \(G/P\) of a complex semisimple Lie group \(G\) by a parabolic subgroup \(P\). The paper under review is concerned with the quantization of the sub-class of (irreducible) compact Hermitian symmetric spaces, within the class of Kähler manifolds.
In particular, the quantization considered here is a subalgebra \(A\) of the quantized function algebra \(\mathbb C[G]_q\). In order to quantize the Kähler metric as well, one is naturally lead to Connes' spectral triples. In particular, one needs to formulate a Dolbeault-Dirac operator as a (non-trivial) equivariant K-homology class and study its spectrum. A spectral triple as such, and a Dolbeault-Dirac operator \(D\) were given by \textit{U. Krähmer} in [Lett. Math. Phys. 67, No. 1, 49--59 (2004; Zbl 1054.58005)], however the formula of \(D\) given there does not allow the computation of its spectrum, or a rigorous proof that it defines an appropriate K-homology class.
The authors construct an element \(D = \overline{\partial} + \overline{\partial}^{\ast}\) in \(U_q(\mathfrak{g}) \otimes Cl_q\), referred to as the Dolbeault-Dirac operator. Here \(U_q(\mathfrak{g})\) is the compact real form of the quantized universal enveloping algebra and \(Cl_q\) is a quantized Clifford algebra. Their construction is quite algebraic and uses the theory of the braided symmetric and exterior algebras of \textit{A. Berenstein} and \textit{S. Zwicknagl} [Trans. Am. Math. Soc. 360, No. 7, 3429--3472 (2008; Zbl 1220.17004); Adv. Math. 220, No. 1, 1--58 (2009; Zbl 1174.17019)]. The main result of the paper is that \(\overline{\partial}\) squares to zero, so that \(D^2 = \overline{\partial}\overline{\partial}^* + \overline{\partial}^* \overline{\partial}\). As the authors state in the introduction, this is a first effort towards a quantum version of Parthasarathy's formula, which in the classical case allows the computation of the spectral properties of the Dolbeault-Dirac operator.
We would like to point out that a thorough geometric insight to the motivation and the ideas of the authors is given in the last section of the paper. quantization; compact Hermitian symmetric spaces; Dolbeault-Dirac operator Kr''ahmer, U., and M. Tucker-Simmons, On the Dolbeault-Dirac Operator of Quantized Symmetric Spaces, Trans. of the London Math. Soc. 2 (2015), 33--56. Noncommutative geometry (à la Connes), Grassmannians, Schubert varieties, flag manifolds, Quadratic and Koszul algebras, Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Harmonic analysis on homogeneous spaces, Spin and Spin\({}^c\) geometry, Geometry of quantum groups On the Dolbeault-Dirac operator of quantized symmetric 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 Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. That is, let \(G\) be a complex semisimple group and consider the affine Grassmannian \(\mathrm{Gr}_G = G((t^{-1}))/G[t]\) of \(G\). There are the spherical Schubert cells \(\mathrm{Gr}^\lambda := G[t]t^\lambda\) in the affine Grassmannian for a dominant coweight \(\lambda\) of \(G\). Let \(\mathcal{W}_\mu := G_1[[t^{-1}]]t^\mu\) be transverse orbits for a dominant coweight \(\mu\). One can then construct and study affine Grassmannian slices, which are their intersections \(\overline{\mathcal{W}}^\lambda_\mu := \mathcal{W}_\mu \cap \overline{\mathrm{Gr}^\lambda}\).
\textit{A. Braverman} et al. [Adv. Theor. Math. Phys. 23, No. 1, 75--166 (2019; Zbl 1479.81044)] generalized these spaces \(\overline{\mathcal{W}}_\mu^\lambda\) and \(\mathcal{W}_\mu\) to the case where \(\mu\) is not dominant. When \(G\) is simply laced, then \(\overline{\mathcal{W}}^\lambda_\mu\) is the Coulomb branch of a quiver gauge theory, whose Higgs branch is a Nakajima quiver variety. This was extended to the non-simply-laced case, to describe \(\overline{\mathcal{W}}^\lambda_\mu\) as the Coulomb branch of a quiver gauge theory with symmetrizers.
The authors prove that neighboring generalized affine Grassmannian slices are related by Hamiltonian reduction by the action of the additive group (Theorem 1.1, page 2): letting \(\lambda\) to be a dominant coweight, \(\mu\) a coweight, and \(\alpha_i^\vee\) any simple coroot, there is a Poisson isomorphism \(\overline{\mathcal{W}}^\lambda_\mu /\!\!/_1 \mathbb{G}_a \cong \overline{\mathcal{W}}^\lambda_{\mu + \alpha_i^\vee}\). This results in Theorem 1.3 (page 3) for shifted Yangians \(Y_{\mu}\), which are used to quantize the spaces \(\mathcal{W}_{\mu}\): upon applying associated graded, the comultiplication map, \( Y_{\mu_1 + \mu_2} \rightarrow Y_{\mu_1} \otimes Y_{\mu_2}\) becomes the multiplication map \(\mathbb{C}[\mathcal{W}_{\mu_1 + \mu_2}] \rightarrow \mathbb{C}[\mathcal{W}_{\mu_1}] \otimes \mathbb{C}[\mathcal{W}_{\mu_2}]\). They also prove Theorem 1.4 (page 3): comultiplication gives an isomorphism \(Y_\mu[(E_i^{(1)})^{-1}] \cong Y_{-\alpha_i^\vee}^0 \otimes Y_{\mu + \alpha_i^\vee}\), obtaining \(Y_{\mu + \alpha_i^\vee}\) as the quantum Hamiltonian reduction of \(Y_\mu\). Here, \(E_i^{(1)}\) is inverted since it is the quantum moment map for the \(\mathbb{G}_a\)-action. The algebra \(Y_{-\alpha_i^\vee}^0\) is isomorphic to the ring of differential operators on \(\mathbb{C}^\times\), a quantum version of the isomorphism \( \overline{\mathcal{W}}^0_{-\alpha_i^\vee} \cong T^* \mathbb{C}^\times\).
Finally, the authors also prove a similar version for their quantization algebras known as truncated shifted Yangians for \(G=SL_n\) unconditionally and for general \(G\), with certain conditions on the reducedness conjecture (Theorem 1.5, page 4): comultiplication gives an isomorphism \(Y_\mu^{\lambda}[(E_i^{(1)})^{-1}] \cong Y_{-\alpha_i}^0 \otimes Y_{\mu + \alpha_i^\vee}^{\lambda}\), resulting in \(Y_{\mu + \alpha_i^\vee}^{\lambda}\) as the quantum Hamiltonian reduction of \(Y_\mu^{\lambda}\). affine Grassmannian slices; shifted Yangians; Hamiltonian reduction; Coulomb branches Grassmannians, Schubert varieties, flag manifolds Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians | 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 method for computing the three-point genus zero Gromov--Witten invariants of the flag manifold \(G/B\). For this one should define the quantum multiplication, i.e. choose the isomorphism between vector spaces \(QH^*(G/B)\) and \(H^*(G/B)\otimes \mathbb C[q_1,\ldots,q_r]\). This isomorphism is called an (abstract) quantum evaluation map and should satisfy some conditions, in particular, the integrability condition (vanishing of some particular differential 2-form). The existence of a basis that gives this map is equivalent to the existence of a particular \(D\)-module, which is called a quantization of \(QH^*(G/B)\). In general, such quantization gives an evaluation map that does not satisfy the integrability condition. To correct this, one should choose the particular operator. After a natural choice of a \(D\)-module that provides the quantization one gets the usual quantum product operation (Theorem \(1.5\)).
The authors prove that in fact the correcting operator is polynomial (Proposition 2.2). This means that the system of partial differential equations on it given by the integrability condition can be solved ``by quadrature''. Finally, the authors consider the case of \(G= \text{GL}_n \mathbb C\) for \(n=2,3,4\) as an example and get the particular formulas for quantum evaluation maps. A. Amarzaya and M. Guest, Gromov-Witten invariats of flag manifolds, via \(D\)-modules , J. London Math. Soc. (2) 72 (2005), 121--136. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Gromov-Witten invariants of flag manifolds, via \(D\)-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 (classical, small quantum, equivariant) cohomology ring of the grassmannian \(G(k,n)\) is generated by certain derivations operating on an exterior algebra of a free module of rank \(n\) ( Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree \(n\) into the product of two monic polynomials, one of degree \(k\). L. Gatto, T. Santiago, Schubert Calculus on a Grassmann Algebra. math.AG/0702759, Canad. Math. Bull., to appear. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Schubert calculus on a Grassmann 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 An element \([\Phi] \in Gr_n \left(\mathcal{H}_+, \mathbf{F}\right)\) of the Grassmannian of \(n\)-dimensional subspaces of the Hardy space \(\mathcal{H}_+ = H^2\), extended over the field \(\mathbf{F} = \mathbf{C} (x_{1},\dots, x_{n})\), may be associated to any polynomial basis \(\phi = \{\phi_0, \phi_1,\dots\}\) for \(\mathbf{C}(x)\). The Plücker coordinates \(S_{\lambda, n}^{\phi}(x_1, \ldots, x_n)\; \text{of}\; [\Phi]\), labeled by partitions \(\lambda\), provide an analog of Jacobi's bi-alternant formula, defining a generalization of Schur polynomials. Applying the recursion relations satisfied by the polynomial system \(\phi\) to the analog \(\{h_i^{(0)} \}\) of the complete symmetric functions generates a doubly infinite matrix \(h_i^{(j)}\) of symmetric polynomials that determine an element \([H] \in \mathrm{Gr}_n(\mathcal{H}_+, \mathbf{F})\). This is shown to coincide with \([\Phi]\), implying a set of generalized Jacobi identities, extending a result obtained by \textit{A. N. Sergeev} and \textit{A. P. Veselov} [Mosc. Math. J. 14, No. 1, 161--168 (2014; Zbl 1297.05244)] for the case of orthogonal polynomials. The symmetric polynomials \(S_{\lambda, n}^{\phi}(x_1, \ldots, x_n)\) are shown to be KP (Kadomtsev-Petviashvili) \(\tau\)-functions in terms of the power sums \([x]\) of the \(x_{a}\)'s, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums \(\sum_{\lambda} S_{\lambda, n}^{\phi}([x]) S_{\lambda, n}^{\theta}(\mathbf{t})\) associated to any pair of polynomial bases \((\phi, \theta)\), which are shown to be 2D Toda lattice \(\tau\)-functions. A number of applications are given, including classical group character expansions, matrix model partition functions, and generators for random processes.{
\copyright 2018 American Institute of Physics} Harnad, J.; Lee, E., Symmetric polynomials, generalized Jacobi-trudi identities and \textit{ {\(\tau\)}}-functions, J. Math. Phys., 59, 091411, (2018) Symmetric functions and generalizations, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Polynomials and rational functions of one complex variable, Hardy spaces, Grassmannians, Schubert varieties, flag manifolds, Schur and \(q\)-Schur algebras, KdV equations (Korteweg-de Vries equations) Symmetric polynomials, generalized Jacobi-Trudi identities and \(\tau\)-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 Let \(F\) denote the Frobenius endomorphism on a projective variety \(X\) over a field of positive characteristic. Then \(X\) is called Frobenius split if the map \(\mathcal O_ X\to F_* \mathcal O_X\) splits. The main result in this paper says that Schubert varieties in \(G/B\), \(G\) a connected reductive group with Borel subgroup \(B\), are all Frobenius split. As a consequence the authors obtain the following vanishing theorem: If \({\mathcal L}\) is an ample line bundle on a Schubert variety \(X\), then \(H^i(X,\mathcal L) = 0\) for \(i>0\). In the case \(X=G/B\) this is a special case of Kempf's vanishing theorem. For certain classes of Schubert varieties it was established by \textit{V. Lakshmibai}, \textit{C. Musili} and \textit{C. Seshadri}, see e.g. [Bull. Am. Math. Soc., New. Ser. 1, 432--435 (1979; Zbl 0466.14020)].
The second author has followed up on these results by proving -- in joint work with \textit{S. Ramanan} -- that Schubert varieties are normal [Invent. Math. 79, 217--224 (1985; Zbl 0553.14023)] and Cohen-Macaulay [Invent. Math. 80, 283--294 (1985; Zbl 0541.14039)]. The normality, a more general vanishing theorem and, as a consequence, Demazure's character formula were also proved by the reviewer [Invent. Math. 79, 611--618 (1985; Zbl 0591.14036)]. Generalizations to the Kac-Moody case have been obtained by \textit{O. Mathieu} [C. R. Acad. Sci., Paris, Ser. I 303, 391--394 (1986; Zbl 0602.17008)] and \textit{S. Kumar} [''Demazure character formula in arbitrary Kac-Moody setting'', preprint (Tata Inst. 1986); Invent. Math. 89, 395--423 (1987; Zbl 0635.14023)]. line bundles on homogeneous spaces; Frobenius endomorphism; Schubert varieties; Frobenius split; normality; Demazure's character formula Brion, M.: Variétés sphériques, Notes de la session de la S. M. F. Opérations hamiltoniennes et opérations de groupes algébriques, Grenoble (1997). http://www-fourier.ujf-grenoble.fr/~mbrion/spheriques.ps Grassmannians, Schubert varieties, flag manifolds, Finite ground fields in algebraic geometry, Cohomology theory for linear algebraic groups Frobenius splitting and cohomology vanishing for 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 Fix an ideal \(I\) in a polynomial ring and consider its zero scheme \(X\) inside a coordinatized vector space. Each term order yields a Gröbner basis for \(I\), or geometrically, a Gröbner degeneration of \(X\) into a possibly nonreduced union of coordinate subspaces. The authors investigate an intermediate degeneration obtained by taking the limit of \(X\) under rescaling just one axis at a time. The limit \(X'\) breaks into two collections of pieces: a projection part and a cone part. In cases where \(X'\) is reduced, quantitative information such as multidegrees and Hilbert series of the original variety \(X\) can be derived from the parts of this geometric vertex decomposition of \(X\) and combined later. Under suitable hypotheses, repeating the degeneration-decomposition procedure for each coordinate axis in turn eventually yields the Gröbner degeneration, but with extra inductive information. When the limit \(X''\) of this sequence is defined by a squarefree monomial ideal, the inductive procedure corresponds exactly to the usual notion of vertex decomposition for simplicial complexes.
The main example of this article is the family of vexillary matrix Schubert varieties, whose ideals are also known as (one-sided) ladder determinantal ideals. Using a diagonal term order to specify the Gröbner degeneration, the authors show that vexillary matrix Schubert varieties have geometric vertex decompositions into simpler varieties of the same type. Using the combinatorics of [\textit{S. Fomin} and \textit{A. N. Kirillov}, Discrete Math. 153, No. 1--3, 123--143 (1996; Zbl 0852.05078)], a new formula for the numerators of their multigraded Hilbert series, the double Grothendieck polynomials, is obtained. The formula is given in terms of so-called flagged set-valued tableaux. This combinatorial notion unifies work [\textit{M. L. Wachs}, J. Comb. Theory, Ser. A 40, 276--289 (1985; Zbl 0579.05001)] on flagged tableaux, and [\textit{A. S. Buch}, Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)] on set-valued tableaux.
The article focuses on diagonal term orders, giving results complementary to those of [\textit{A. Knutson} and \textit{E. Miller}, Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)], where it was shown that the generating minors form a Gröbner basis for any antidiagonal term order and any matrix Schubert variety. It is proved that under a diagonal term order, the only matrix Schubert varieties for which generating minors form Gröbner bases are the vexillary ones. degeneration; Gröbner bases; simplicial complexes; Stanley-Reisner schemes; matrix Schubert varieties; ladder determinantal ideals; diagonal term orders A. Knutson, E. Miller, and A. Yong, \textit{Gröbner geometry of vertex decompositions and of flagged tableaux}, J. Reine Angew. Math., 630 (2009), pp. 1--31. Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Gröbner geometry of vertex decompositions and of flagged tableaux | 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 two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a \(K\)-theoretic deformation of the quasi-key basis and also a lift of the \(K\)-analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the \(K\)-analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)].
The second new basis is the kaon basis, a \(K\)-theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis.
Throughout, we explore how the relationships among these \(K\)-analogues mirror the relationships among their cohomological counterparts. We make several ``alternating sum'' conjectures that are suggestive of Euler characteristic calculations. Demazure character; Demazure atom; Lascoux polynomial; Lascoux atom; Grothendieck polynomial; quasi-Lascoux polynomial; kaon Symmetric functions and generalizations, Classical problems, Schubert calculus, Hopf algebras and their applications, Connections of Hopf algebras with combinatorics, Grassmannians, Schubert varieties, flag manifolds Polynomials from combinatorial \(K\)-theory | 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 Following \textit{A. Braverman} et al. [Commun. Math. Phys. 308, No. 2, 457--478 (2011; Zbl 1247.81169)], we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite \(W\)-algebra of type \(A\). This is a finite analog of the AGT conjecture [\textit{L. F. Alday} et al., Lett. Math. Phys. 91, No. 2, 167--197 (2010; Zbl 1185.81111)] on 4-dimensional supersymmetric Yang-Mills theory with surface operators. Our new observation is that the \(\mathbb C^\ast \)-fixed point set of a handsaw quiver variety is isomorphic to a graded quiver variety of type \(A\), which was introduced by the author in connection with the representation theory of a quantum affine algebra. As an application, simple modules of the \(W\)-algebra are described in terms of \(IC\) sheaves of graded quiver varieties of type \(A\), which were known to be related to Kazhdan-Lusztig polynomials. This gives a new proof of a conjecture by \textit{J. Brundan} and \textit{A. Kleshchev} [Mem. Am. Math. Soc. 918, 107 p. (2008; Zbl 1169.17009), Conj. 7.17] on composition multiplicities on Verma modules, which was proved by \textit{I. Losev} [``On the structure of the category \(\mathcal O\) for \(W\)-algebras.'' \url{http://arxiv.org/abs/0812.1584}], in a wider context, by a different method. quiver variety; shifted Yangian; finite \(W\)-algebra; quantum affine algebra; Kazhdan-Lusztig polynomial H. Nakajima, \textit{Handsaw quiver varieties and finite W-algebras}, arXiv:1107.5073 [INSPIRE]. Quantum groups (quantized enveloping algebras) and related deformations, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Handsaw quiver varieties and finite \(W\)-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 \textit{V. V. Deodhar} [Invent. Math. 79, 499--511 (1985; Zbl 0563.14023)] introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan-Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context of soliton solutions to the KP equation, as shown by Kodama and the second author. Deodhar components \(\mathcal R_D\) of the Grassmannian are in bijection with certain tableaux \(D\) called Go-diagrams, and each component is isomorphic to \((\mathbb K^*)^a\times(\mathbb K)^b\) for some nonnegative integers \(a\) and \(b\). {
} Our main result is an explicit parametrization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram \(D\) we construct a weighted network \(N_D\) and its weight matrix \(W_D\), whose entries enumerate directed paths in \(N_D\). By letting the weights in the network vary over \(\mathbb K\) or \(\mathbb K^*\) as appropriate, one gets a parametrization of the Deodhar component \(\mathcal R_D\). One application of such a parametrization is that one may immediately determine which Plücker coordinates are vanishing and nonvanishing, by using the Lindström-Gessel-Viennot lemma. We also give a (minimal) characterization of each Deodhar component in terms of Plücker coordinates. A main tool for us is the work of \textit{R. J. Marsh} and \textit{K. Rietsch} [Represent. Theory 8, 212--242 (2004; Zbl 1053.14057)] on Deodhar components in the flag variety. Grassmannian; network; total positivity; Deodhar decomposition Talaska, K; Williams, L, Network parametrizations for the Grassmannian, Algebra Number Theory, 7, 2275-2311, (2013) Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over the reals, the complexes, the quaternions, Combinatorial aspects of representation theory Network parametrizations 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 The authors study the cohomology ring of the Grassmannian \(G\) of iso\-tro\-pic \(n\)-subspaces of a complex \(2m\)-dimensional vector space, endowed with a non-degenerate orthogonal form (here \(1\leq n<m\)). A multiplication formula ``of Pieri type'' for the cohomology ring of \(G\) is given. The paper is a continuation of an earlier one by the authors [\textit{P.~Pragacz} and \textit{J.~Ratajski}, J. Reine Angew. Math. {476}, 143--189 (1996; Zbl 0847.14029)], where the symplectic and odd orthogonal Pieri-type formulas were given. For group-theoretic treatment of a Pieri-type formula in the case of maximal isotropic subspaces (\(n=m\)) see [\textit{H.~Duan} and \textit{P.~Pragacz}, ``Divided differences of type D and the Grassmannian of complex structures'', in: Quantum Groups and Algebraic Combinatorics, N.~Jing (ed.), World Sci. (2003), to appear].
The method of finding and proving this type of formulas is based on the study of an iterated Leibniz-type formula for divided differences and related deformations of some ``distinguished'' reduced decompositions of elements of the Weyl group. A summary of this method may be found in section~6 of the paper [\textit{P.~Pragacz}, Banach Cent. Publ. {36}, 125--177 (1996; Zbl 0851.05094)]. cohomology ring; Grassmannians; Schubert classes; isotropic subspaces; Pieri-type formulas Pragacz, P.; Ratajski, J.: A Pieri-type formula for even orthogonal grassmannians. Fund. math. 178, 49-96 (2003) Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Grassmannians, Schubert varieties, flag manifolds, Étale and other Grothendieck topologies and (co)homologies, Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] A Pieri-type formula for even 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 [For the entire collection see Zbl 0741.00064.]
In his fundamental paper in Invent. Math. 83, 333-382 (1986; Zbl 0621.35097), the author characterizes the Jacobian varieties among all principally polarized abelian varieties (p.p.a.v.) with the help of the Kadomtsev-Petviashvili (differential)-equation hierarchy (KP-hierarchy). In the article under review he extends these techniques with the goal to characterize Prym varieties among the p.p.a.v.; Prym varieties are varieties which can be defined for pairs \((C,\iota)\) where \(C\) is an irreducible algebraic curve and \(\iota\) is an involution of the curve \(C\). The KP hierarchy is replaced by the one-component BKP hierarchy (Kadomtsev-Petviashvili of \(B_ \infty\) type), resp. two-component BKP hierarchy. As in the Jacobian case the technique is to assign to certain geometric data (curve, involution, torsion-free sheaf, local trivialization,...) a point in a suitable infinite dimensional Grassmannian via the (Burchnall-Chaundy-)Krichever construction. Now solutions of the above hierarchies can be connected with this Grassmannian, via the assignment of a ``wave function'' to every point of the Grassmannian. For the two component BKP the construction is the extension of the above Krichever construction generalized to Riemann surfaces with two points removed. The author gives characterizations of the Prym varieties in terms of the one or two component BKP hierarchies if certain additional conditions for the pairs \((C,\iota)\) are fulfilled. This additional conditions are conditions on the singularities of \(C\), the fixed points of \(\iota\), and the branching numbers. \(KP\)-equation; Krichever map; Schottky-problem; Kadomtsev-Petviashvili of \(B_ \infty\) type; principally polarized abelian varieties; Prym varieties; p.p.a.v.; BKP hierarchy; infinite dimensional Grassmannian Shiota, T., Prym varieties and soliton equations, in: Infinite-dimensional Lie algebras and groups, Luminy-Marseille, 1988, 407-448, Adv. Ser. Math. Phys., 7, World Sci. Publishing, Teaneck, NJ, 1989. i i i i i Geometric Bäcklund Transformations 519 Theta functions and curves; Schottky problem, Abelian varieties and schemes, KdV equations (Korteweg-de Vries equations) Prym varieties and soliton 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 As a generalization of Brieskorn polynomials in \({\mathbb{C}}^ m\) weighted homogeneous polynomials having an isolated singularity are defined and considered. The method of complete intersections used for calculations of invariants calculated long ago, are discussed. A side result for the hypersurface case is obtained by determining same related homology groups. hypersurface singularity; Brieskorn polynomials; weighted homogeneous polynomials; isolated singularity; complete intersections; invariants H.A. Hamm , Invariants of weighted homogeneous singularities , Journées Complexes, 1985 (Nancy, 1985) 6-13. Local complex singularities, Singularities of surfaces or higher-dimensional varieties Invariants of weighted homogeneous 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 manuscript under review requires a minimal amount of knowledge of algebraic or complex geometry. Indeed, one can replace ``schemes'' with ``varieties'' or ``complex manifolds'' throughout the paper. Furthermore, the author provides many examples and exercises to support the process of understanding the main ideas and concepts. The theory of Donaldson-Thomas (DT) invariants associates integers to moduli spaces of stable sheaves on a compact Calabi-Yau \(3\)-fold. K. Behrend realized that these same numbers, originally written as integrals over algebraic cycles or characteristic classes, can also be obtained by an integral over a constructible function (which is called a Behrend function), with respect to the measure given by the Euler characteristic. This new perspective extends DT theory to non-compact moduli spaces, as well as ``motivic nature'' of this new invariant.
This manuscript aims to give a thorough discussion on DT theory in the case of quivers with potential. This results in omitting the role of orientation data as well as derived algebraic geometry arising in DT theory. However, many important ideas and concepts are still evident in the case of quiver representations that this paper is an excellent starting point to learn about DT theory. The paper begins with a notion of classifying objects, which form an abelian category. The author then goes into the concept of Artin and moduli stacks, in particular, of quotient stacks (Section \(2\), Example \(2.20\)), where a thorough discussion is given in Section \(2\).
We then study quiver moduli spaces and stacks in Section \(3\). Furthermore, the path algebra of a quiver is categorical, noncommutative analogue of a polynomial algebra in ordinary commutative algebra, and although we are introduced to quotients of the path category of a quiver by some ideal of relations, results in Section \(6.1\) only depend on the linear category. That is, a potential \(W\) is an element of the vector space \(\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]\), where \([\mathbb{C}Q,\mathbb{C}Q]\) is the \(\mathbb{C}\)-linear span of all commutators (a potential can also be thought of as the \(0\)-th Hochschild homology of the \(\mathbb{C}\)-linear category \(\mathbb{C}Q\), or \(\mathbb{C}\)-linear combination of equivalence classes of cycles in a quiver \(Q\) where two cycles are equivalent if one can be transformed into the other by a cyclic permutation).
A constructible function is a function \(a:X(\mathbb{C})\rightarrow \mathbb{Z}\) on the set of closed points of a scheme \(X/\mathbb{C}\) of finite-type with only finitely-many values on each connected component of \(X\), and such that the level sets of \(a\) are the closed points of locally closed subsets in each connected component of \(X\). Introduced in Section \(4\), they can be pulled back and multiplied, and using fiberwise integrals with respect to the Euler characteristic, they can be pushed forward. Moreover, every locally closed subscheme determines a constructible (characteristic) function.
Motivic theory (see Section \(4.1\) and \(4.2\)) is a generalization of constructible functions, in that one associates to every scheme \(X\) an abelian group \(R(X)\) of functions on \(X\) which can be pulled back, pushed forward, and multiplied. In addition, there is a ``characteristic function'' in \(R(X)\), where \(X\) is a locally closed subscheme such that the characteristic function of a disjoint union is the sum of the characteristic function of its summands.
Vanishing cycles for schemes and quotient stacks are introduced in Section \(5\), which form an additional structure on stacky motivic theories formalizing the properties of ordinary classical vanishing cycles. For example, the Behrend function is a good example of a vanishing cycle on the theory of constructible functions. In Section \(6\), DT functions and invariants are introduced, where many examples are provided in Section \(6.2\) to illustrate the theory. We end this section in discussing Ringel-Hall algebras (Section \(6.3\)), an integration map (Section \(6.4\)), and a wall-crossing identity (Section \(6.5\)), which are some of the fundamental tools used in DT theory. Donaldson-Thomas theory; moduli stacks; constructible functions; quivers with potential; vanishing cycles; quotient stacks; Ringel-Hall algebra; wall-crossing; Grothendieck group of varieties Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Stacks and moduli problems, Representations of quivers and partially ordered sets, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Intersection homology and cohomology in algebraic topology An introduction to (motivic) Donaldson-Thomas theory | 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 \(p(x)=p(x_1,\dots,x_g)\) be a noncommutative polynomial in \(g\) variables with real coefficients. Such a polynomial \(p\) is called symmetric if \(p^T=p\), where the involution \(^T\) is first defined on monomials (words in the letters \(x_1,\dots,x_g\)) by sending a word to a word with the same letters written in the reverse order, e.g., \((x_jx_l)^T=x_lx_j\), and then naturally extended to polynomials (which are linear combinations of words). A noncommutative polynomial \(p\) can be evaluated on \(g\)-tuples \(X=(X_1,\dots,X_g)\) of real symmetric \(n\times n\) matrices for any \(n\), so that \(p(X)\) is a symmetric matrix if \(p\) is symmetric. The positivity domain \(\mathcal{D}_p^n\) of \(p\) in dimension \(n\) is the closure of the component of \(0\) of the set of \(g\)-tuples \(X\) of symmetric \(n\times n\) matrices such that \(p(X)\) is positive definite. The positivity domain \(\mathcal{D}_p\) of \(p\) is a disjoint union of the domains \(\mathcal{D}_p^n\), \(n=1,2,\dots\).
The main result of the present paper is that if the symmetric noncommutative polynomial \(p\) satisfies some natural conditions (\(p\) is decreasing near the boundary of \(\mathcal{D}_p\), \(p\) is a minimum degree defining polynomial for \(\mathcal{D}_p\), \(p\) has the generic full rank boundary property), which are defined and discussed in detail in the paper, and \(\mathcal{D}_p\) is convex, then \(p\) has degree four or less. In addition, a finer structure of \(p\) is discussed in detail. linear matrix inequalities; convex sets of matrices; noncommutative semialgebraic geometry H. Dym, W. Helton, and S. McCullough, ''Irreducible noncommutative defining polynomials for convex sets have degree four or less,'' Indiana Univ. Math. J., vol. 56, iss. 3, pp. 1189-1231, 2007. Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), Miscellaneous inequalities involving matrices, Semialgebraic sets and related spaces, Linear operator inequalities, Dilations, extensions, compressions of linear operators, Convex sets and cones of operators, Convexity of real functions of several variables, generalizations Irreducible noncommutative defining polynomials for convex sets have degree four or less | 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 arose as a companion to the article ``Mirror manifolds in higher dimension'' by \textit{B. R. Greene}, \textit{D. R. Morrison} and \textit{M. R. Plesser} [Commun. Math. Phys. 173, No. 3, 559-597 (1995)] which used mirror symmetry and what is now called quantum cohomology to predict the number of rational curves of low degree satisfying appropriate Schubert conditions on Calabi-Yau hypersurfaces in projective spaces of arbitrary dimension. In the paper discussed here, the predictions for lines and conics are verified up to dimension 10. Verifications of other predictions of mirror symmetry are done for some weighted projective spaces. -- Computations are done using the Maple package ``Schubert: a Maple package for intersection theory'' developed by the author and \textit{S. A. Strømme}.
Finally, a low degree consequence of quantum cohomology is shown to follow easily from Pieri's formula. Calabi-Yau manifolds; Maple; mirror symmetry; quantum cohomology; number of rational curves; Calabi-Yau hypersurfaces S. Katz, Rational curves on Calabi-Yau manifolds: Verifying predictions of Mirror Symmetry, Algebraic Geometry , Marcel-Dekker, New York, 1994. CMP 95:04 Computational aspects of algebraic curves, Enumerative problems (combinatorial problems) in algebraic geometry, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Calabi-Yau manifolds (algebro-geometric aspects) Rational curves on Calabi-Yau manifolds: Verifying predictions of mirror symmetry | 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, \(B\) its Borel subgroup and \(T\) a maximal torus of \(G.\) The authors study equivariant \(K\)-theory of a generalized flag variety \(K_{T}(G/B).\) They construct a model of \(K_{T}(G/B)\) as a subalgebra of a braided Hopf algebra known as the Nichols-Woronowicz algebra [\textit{N. Andruskiewitsch, H.-J. Schneider}, in: New directions in Hopf algebras. Math. Sci. Res. Inst. Publ. 43, 1--68 (2002; Zbl 1011.16025); \textit{Y. Bazlov}, J. Algebra 297, No. 2, 372--399 (2006; Zbl 1101.16027)] \({\mathcal B}(V_{W})\) associated to the Yetter-Drinfeld module \(V_{W}\) over the Weyl group \(W.\)
The model is based on the multiplication formula developed by \textit{C.Lenart} and \textit{A. Postnikov} [Affine Weyl groups in \(k\)-theory and representation theory, preprint 2005, \url{arXiv:math/0309207}] using the so-called alcove paths. The authors define the multiplicative Dunkl elements in \({\mathcal B}(V_{W})\) and show that the commutative algebra generated by them is isomorphic to the \(K\)-ring \(K(G/B)\) (Thm. 4.2). This result is then extended to the equivariant case. The authors remark that their result might be viewed as realization a of the idea of
\textit{S. Fomin} and \textit{A. Kirillov} [in: Advances in geometry. Prog. Math. 172, 147--182 (1999; Zbl 0940.05070)] who suggested embedding of various basic algebras such as cohomology or \(K\)-theory into commutative parts of non-commutative Hopf algebras. Nichols-Woronowicz algebra; equivariant \(K\)-theory; Dunkl elements Lenart, C., Maeno, T.: Alcove path and Nichols-Woronowicz model of the equivariant \(K\)-theory of generalized flag varieties. Int. Math. Res. Not. (2006). 10.1155/IMRN/2006/78356 Grassmannians, Schubert varieties, flag manifolds, Equivariant \(K\)-theory Alcove path and Nichols-Woronowicz model of the equivariant \(K\)-theory of generalized 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 Let \(G\) be a simply-connected, semi-simple algebraic group over an algebraic closed field, let \(T\) be a maximal torus of \(G\) and let \(B\) be a Borel subgroup of \(G\) containing \(T\). The authors are interested to determine what Schubert varieties of a projective homogeneous variety \(G/P\) contain a semistable point with respect to the action of \(T\) and to a fixed ample line bundle \(\mathcal{L}\). In this work there are two main results. In the first part of this article the authors restrict themselves to the case where \(G\) is a simple group of type \(B\), \(C\) or \(D\) and \(P\) is a maximal parabolic subgroup. With these hypotheses they classify the minimal elements \(w\in W/W_{P}\) such that \(X(w)_{T}^{ss}(\mathcal{L})\neq\emptyset\). Here \(X(w)=\overline{BwP/P}\) is the Schubert variety associated to \(w\). The authors affirm that, when \(G\) exceptional or \(P\) non-maximal, the same problem is more complicated.
In the second part of this work, the authors classify the Coxeter elements \(\tau\) of \(W\) such that there is a non-trivial line bundle \(\mathcal{L}\) on \(G/B\) with \(X(\tau)_{T}^{ss}(\mathcal{L})\neq\emptyset\). An element of \(W\) is a Coxeter element if it can be written as a product of distinct simple reflections. The authors are interested to such elements for the following reason: a Schubert variety \(X(w)\) contains a (rank \(G\))-dimensional \(T\)-orbit if and only if \(w\geq \tau\) for some Coxeter element \(\tau\). In this part, they do not make special assumption on \(G\).
Let \(\chi\) be the \(B\)-character associated to a fixed globally generated line bundle \(\mathcal{L}\). Supposing that \(\chi\) belongs to the root lattice, the authors prove that \(X(w)_{T}^{ss}(\mathcal{L})\neq\emptyset\) if and only if \(w\chi\leq0\). This fact allows them to prove their main theorems in a combinatorial way.
Remark that in the Proposition 3 (and only in that Proposition) the authors use a definition of \(\omega\geq0\) different from the usual one. They say that a weight \(\omega\) is greater or equal to 0 if it can written as a positive linear combination of simple roots. In particular, they do not assume that the coefficients are integral. semistable points; line bundle; Coxeter element; Schubert varieties Kannan, S. S.; Pattanayak, S. K., Torus quotients of homogeneous spaces-minimal dimensional Schubert varieties admitting semi-stable points, \textit{Proc. Indian Acad. Sci. (Math. Sci.)}, 119, 4, 469-485, (2009) Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds Torus quotients of homogeneous spaces --- minimal dimensional Schubert varieties admitting semi-stable points | 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 is well known that classical Schubert varieties in the Grassmannian of \(d\)-planes in a certain ambient vector space parametrise subspaces of dimension \(d\) satisfying certain incidence conditions as e.g. with flag varieties. Except for the case of Grassmannians the Schubert varieties are singular and for this class of varieties there are the so called Chern-Schwartz-MacPherson (CSM) classes introduced independently by \textit{M. H. Schwartz} [C. R. Acad.~Sci., Paris 260, 3262--3264 (1965; Zbl 0139.16901)] and \textit{R. D. MacPherson} [Ann. Math. (2) 100, 423--432 (1974; Zbl 0311.14001)]. The authors deal with the computation of CSM classes for singular Schubert varieties in terms of the Schubert cells which are determined by certain coefficients. The computation of these coefficients is stated as their main theorem, namely theorem 1.1 of this paper.
The paper is organized as follows. In section two, they recall material concerning classical Schubert varieties. They recall in subsections 2.1 and 2.2 the basic notions of partitions, diagrams, Schubert varieties, the ordering \( \leq \) associated to the partitions. Their objective as already mentioned is to express the CSM class of a Schubert cell as a combination of Schubert classes in terms of coefficients \( \gamma_{\alpha, \beta} \) for all \( \beta \leq \alpha \). In subsection 2.3 they associate to a Schubert variety \(S(\alpha)\) a so called Bott-Samelson variety \( V( \alpha)\) and a map \( \pi_{\alpha}: V(\alpha) \rightarrow S(\alpha)\) which is a birational isomorphism. These varieties are realized as a tower of projective bundles over a point in subsection 2.4 and are therefore non-singular. They verify that the complement of \( S(\alpha)^0 \) in \( V( \alpha)\) is a divisor with simple normal crossings in proposition 2.10. In proposition 2.12 the push-forward of \(\pi_{\alpha}\) is computed.
In section three, they state the formula \( c_{SM}( S(\alpha))^0 = \sum_{ \alpha \leq \beta} \gamma_{\alpha, \beta} [S(\beta)] \). They compute the coefficients \(\gamma_{\alpha, \beta}\) in theorem 3.4 and in theorem 3.6. As a consequence of theorem 3.4 they obtain theorem 3.8 giving determinantal expressions for these coefficients. In theorem 3.10 they present the coefficients \(\gamma_{\alpha, \beta}\) as coefficients of the series expansions of a rational function which is their main theorem, already stated as theorem 1.1 of the introduction of the paper. Other expressions for the coefficients \(\gamma_{\alpha, \beta}\) are given as e.g. in corollary 3.11 which is for the case of a one-row diagram.
In section four, they consider the positivity of the coefficients of the CSM classes proving positivity of CSM classes only for the case of two-row diagrams using lemma 4.4 and corollary 3.4. In this section they also prove proposition 4.1 and proposition 4.2 as examples of formulas that imply that some of the coefficients in the CSM class of a Schubert cell are determined by the CSM class of Schubert cells for smaller diagrams. The authors conclude with a formula for \(\gamma_{\alpha, \beta}\) in terms of non-intersecting lattice paths for 2-row diagrams in theorem 4.5. intersection theory; characteristic classes; Grassmanians; Schubert varieties; flag manifolds; classical problems; Schubert calculus Aluffi, P.; Mihalcea, L. C., \textit{Chern classes of Schubert cells and varieties}, J. Algebraic Geom., 18, 63-100, (2009) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Chern classes of Schubert cells and 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 From the author's abstract: We show that the small quantum product of the generalized flag manifold \(G/B\) is a product operation on \(H^*(G/B)\otimes \mathbb R[q_1,\dots, q_l]\) uniquely determined by the fact that it is a deformation of the cup product on \(H^*(G/B)\), it is commutative, associative, graded with respect to \(\deg(q_i)=4\), it satisfies a certain relation (of degree two), and the corresponding Dubrovin connection is flat. We deduce that it is again the flatness of the Dubrovin connection which characterizes essentially the solutions of the ``quantum Giambelli problem'' for \(G/B\). This result gives new proofs of the quantum Chevalley formula (proved by Peterson and Fulton-Woodward), and of Fomin, Gelfand and Postnikov's description of the quantization map for \(Fl_n\). quantum cohomology; quantum multiplication; Schubert variety; quantum Chevalley formula Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds A characterization of the quantum cohomology ring of \(G/B\) 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 The present paper is an extended version of the paper [the author, C. R., Math., Acad. Sci. Paris 355, No. 1, 40--44 (2017; Zbl 1357.34138)]. The main results of the paper are the following.
1. The author shows ``that a classical problem of geometry, set up and solved by Pierre-Ossian Bonnet in 1867, yields as a by-product of a new, isomonodromic, very symmetric second order matrix Lax pair of a codimension-two \(\mathrm P_{\mathrm{VI}}\), which is easily extrapolated to the generic \(\mathrm P_{\mathrm{VI}}\).
2. The author gives a ``rigorous derivation of a nice property of \(\mathrm P_{\mathrm{VI}}\), unveiled by Suleimanov and known as the ``quantum correspondence''.
3. The author matches ''the completeness property of \(\mathrm P_{\mathrm{VI}}\) and the completeness property of the Gauss-Codazzi equations by building a solution of the Gauss-Codazzi equations in terms of the full \(\mathrm P_{\mathrm{VI}}\).'' Painlevé equation; Lax pair; Bonnet surface; quantum correspondence Conte, R., Generalized Bonnet surfaces and Lax pairs of <?CDATA \(P_{VI}\) ?>, J. Math. Phys., 58, (2017) Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Structure of families (Picard-Lefschetz, monodromy, etc.) Generalized Bonnet surfaces and Lax pairs of \(\mathrm{P}_{\mathrm{VI}}\) | 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 chapter is devoted to a discussion of Gromov-Witten-Welschinger (GWW) classes and their applications. In particular, Horava's definition of quantum cohomology of real algebraic varieties is revisited by using GWW classes and is introduced as a differential graded operad. In light of this definition, we speculate about mirror symmetry for real varieties. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Real algebraic and real-analytic geometry, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Towards quantum cohomology of real 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 main object of the paper under review is a smooth, projective, geometrically integral variety \(X\) defined over a number field \(k\). The author is interested in comparing various obstructions to the Hasse principle and weak approximation. She subdivides all known obstructions into two big classes: those related to the Brauer group of \(X\) and to descent on torsors under a linear algebraic group \(G\). One of her goals is to reconcile the Brauer-type and descent-type languages used to describe the obstructions. Towards this end, she gives an overview of the known results relating these two types of obstructions and also presents some new ones. In particular, such comparison theorems are proved for the case where \(G\) is unipotent or solvable. At the end, she produces a nice diagram showing all known interrelations between different types of obstructions. rational points; Brauer-Manin obstruction; Étale-Brauer obstruction; torsors; linear algebraic groups Balestrieri, F., Obstruction sets and extensions of groups, Acta arith., 173, 151-181, (2016) Rational points, Linear algebraic groups over global fields and their integers, Varieties over global fields Obstruction sets and extensions of groups | 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} [Algebra Number Theory 3, No. 2, 121--148 (2009; Zbl 1229.05285)] 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 R. 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. unique rectification target; jeu de taquin; \(d\)-complete poset; Schubert calculus; Kac-Moody group Rahul Ilango, Oliver Pechenik, Michael Zlatin, Unique rectification in \(d\)-complete posets: towards the \(K\)-theory of Kac-Moody flag varieties, preprint 2018, 34 pages, arXiv:1805.02287. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorics of partially ordered sets, Grassmannians, Schubert varieties, flag manifolds Unique rectification in \(d\)-complete posets: towards the \(K\)-theory of Kac-Moody 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 A unitableau \((p_{ij})\) is a tabular arrangement of positive integers \(p_{ij}\) taken from some set \(\{\) 1,2,...,p\(\}\), with strictly increasing rows. Unitableau is said to be standard (strongly standard) if its row lengths are nonincreasing and its columns are (strictly) increasing also. Strongly standard unitableaux were introduced by A. Young (1901) in his work on invariant theory. Thereafter he used them for describing the irreducible representations of the symmetric group; Young was led to this theme while investigating how Gordan-Capelli series, occurring in the classical invariant theory of forms, can be derived from an identity involving Young tableaux. Since then, Young tableaux have played an important role in quantum mechanics (H. Weyl, 1930-1940), they have occurred in the theory of elementary particles, in many combinatorial and computer science problems. W. Hodge (1947) used Young tableaux to study Schubert varieties of flag manifolds.
This monograph is a detailed and carefully commented research account, resulting from the author's interest (1982-1985) in the structure of Schubert varieties. Having investigated certain determinantal ideals, he was led to the problem of enumerating Young tableaux and especially some related, more general objects - multitableaux.
To give idea of the results in the book, some definitions are needed. A tableau with q sides of the same shape but with possibly different entries, is called a multitableau (or simply, a tableau) of width q; in the special case \(q=2\) it is called bitableau. The tableau of width q having only one row, is called a multivector of width q. A tableau is said to be bounded by \(m,m=(m_ 1,...,m_ q)\in {\mathbb{N}}^ q\), if for any \(j\in \{1,...,q\}\) all the entries on the j-th side are \(\leq m_ j\). A few words more about these bitableaux. Let \(X=(x_{ij})\) be an \(m_ 1\times m_ 2\)-matrix with its elements being indeterminates over a field K. A bitableau bounded by \(m=(m_ 1,m_ 2)\) is a sequence of bivectors bounded by m, and each such bivector indicates some minor of X. Taking the product of all these minors for a given bitableau, we get a monomial in the minors of X; this monomial is called standard if the given tableau is standard. The Straightening Formula [see \textit{J. Désarménien}, \textit{J. P. Kung} and \textit{G.-C. Rota}, Adv. Math. 27, 63-92 (1978; Zbl 0373.05010)] says, that the standard monomials of X form a K-basis of the algebra K[X] of polynomials in \(x_{ij}\). Some more definitions. The number of entries (rows) on each side of the tableau is called its area (depth), the length of the tableau is its largest row length. A standard tableau of width q is said to be predominated by a multivector a of width q if a is bounded by m and the tableau, is again standard.
Chapter 1 contains comments about new and complicated notation, some preliminary remarks, and a systematic and extensive treatment of binomial coefficients. Chapter 2 gives formulas for counting the sets stab(q,T) and \(mon(2,T)\), where \(stab(q,T)\) is the set of all standard tableaux of width q and area V, which are bounded by m and predominated by the multivector a of width q and length p, and \(mon(q,T)\) is the corresponding set of monomials. These formulas are examples of determinantal polynomials in binomial coefficients. Chapter 3 contains a certain universal identity, satisfied by minors of X. Chapter 4 gives several applications of these results: enumerative proofs of the Straightening Formula and of certain generalizations of the Second Fundamental Theorem of invariant theory, computations of Hilbert functions of determinantal ideals in K[X].
All chapters, though parts of the whole, are self-contained. They begin with an informal discussion, contain a summary, and motivation and hints about further use of the main results, as well as comments on underlying principles. Some useful illustrations and mental experiments are given for better understanding of basic points of reasoning. Proofs are divided into a sequence of independent statements, all these steps are numbered and their interrelations are indicated. The new symbol-codes, regardless of being quite puzzling locally, are suitably in the whole and form a complete system.
This book will be a valuable reference for research and applications of enumerating multitableaux, it may be used also as a text in the present- day-combinatorics for graduate courses. Young tableaux; Schubert varieties; determinantal ideals; bitableau; binomial coefficients; invariant theory; multitableaux S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux, Marcel Dekker, New York, 1988. Research exposition (monographs, survey articles) pertaining to combinatorics, Exact enumeration problems, generating functions, Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Factorials, binomial coefficients, combinatorial functions, Polynomial rings and ideals; rings of integer-valued polynomials Enumerative combinatorics of Young tableaux | 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 considers the generalization of the Pfaffian-Grassmanian double mirrors to higher dimensions. The original example is due to \textit{E. A. Rødland} [Compos. Math. 122, No. 2, 135--149 (2000; Zbl 0974.14026)],
and its generalization to higher dimensions is
suggested by \textit{A. Kuznetsov} [in: Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13--21, 2014. Vol. II: Invited lectures. Seoul: KM Kyung Moon Sa. 635--660 (2014; Zbl 1373.18009)].
Let \(V\) be an \(n\)-dimensional complex vector space for an odd \(n\geq 5\). Let \(W\subset \Lambda^2V^{\vee}\) be a generic \(n\)-dimensional space of skew
forms on \(V\). Then the two families of Calabi-Yau varieties \(\{X_W\}\) and \(\{Y_W\}\) associated to \(W\) are defined as follows. \(X_W\) is a subvariety of
the Grassmannian \(\mathrm{G}(2,V)\) of dimension \(2\) subspaces \(T_2\subset V\), which is defined as the
locus of \(T_2\in \mathrm{G}(2,V)\) with \(w|_{T_2}=0\) for all \(w\in W\). \(Y_W\) is a
subvariety of the Pfaffian variety \(\mathrm{Pf}(V)\subset\mathbb{P}\Lambda^2 V\) of
skew forms on \(V\) whose rank is less than \(n-1\), and defined as the
intersection of \(\mathrm{Pf}(V)\) with \(\mathbb{P}W\subset\mathbb{P}\lambda^2 V\).
When \(n\geq 11\), \(Y_W\) is singular, so one ought to consider
the stringy Hodge numbers. The first result of this paper is the
following
Theorem 1: For any odd \(n\geq 5\), Then there is
the equality of the Hodge numbers \(h^{p,q}(X_W)=h^{p,q}_{st}(Y_W)\).
Viewing the Pfaffian-Grassmannian correspondence as a special case
of a more general correspondence between Calabi-Yau complete
intersections \(X_W\) and \(Y_W\) in dual Pfaffian varieties \(\mathrm{Pf}(2k, V^{\vee})\)
and \(\mathrm{Pf}(n-1-2k, V)\) for a vector space \(V\) of odd dimension \(n\),
the following result is obtained. This is the main result of this
paper.
Theorem 2: For any odd \(n\geq 5\), the varieties \(X_W\) and
\(Y_W\) have well-defined
stringy Hodge numbers. Moreover, \(h^{p,q}_{rt}(X_W)=h^{p,q}_{st}(Y_W).\)
To establish Theorem 1, first
a remarkably simple formula for the stringy \(E\)-function of
the Pfaffian variety \(\mathrm{Pf}(\mathbb{C}^{2r+1})\) of skew forms of rank at
most \(2r-2\) in \(\mathbb{P}\Lambda^2(\mathbb{C}^{2r+1})\) is obtained for any
\(r\geq 2\). Then the statement on the equality of Hodge numbers is
reduced to the equality of the \(E\)-functions: \(E(X_W;u,v)=E_{st}(Y_W;u,v)\).
Proof of Theorem 2 is more technical, but the main idea is the
same as for Theorem 1, namely, to establish the equality of the
stringy \(E\)-functions.
Finally the analogous discussions are carried out for even
dimensional case. However, it is remarked that
\(X_W\) and \(Y_W\) have different dimensions
and the notion of double mirrors ought to be generalized and
the definition of stringy \(E\)-functions needs to be modified. double mirror; stringy \(E\)-functions; Calabi-Yau varieties Mirror symmetry (algebro-geometric aspects), Homogeneous spaces and generalizations, Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Calabi-Yau manifolds (algebro-geometric aspects) Stringy \(E\)-functions of Pfaffian-Grassmannian double mirrors | 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 study the relation between the scaling 1-hook property of the coloured Alexander polynomial $\mathcal {A}_R^{\mathcal{K}}\left( q \right)$ and the KP hierarchy. The Alexander polynomial arises as a special case of the HOMFLY polynomial $\mathcal {H}_R^{\mathcal{K}}\left(q,a \right)$ of the knot $\mathcal{K}$ coloured with representation $R$ [\textit{E. Witten}, Commun. Math. Phys. 121, No. 3, 351--399 (1989; Zbl 0667.57005); \textit{ D. Bar-Natan}, J. Knot Theory Ramifications 4, No. 4, 503--547 (1995; Zbl 0861.57009)]:
\[{\mathcal {H}}_R^{\mathcal {K}}\left( {q,a} \right) = \frac{1} {Z}\int {DA{e^{ - \frac{i} {\hbar }{S_{CS}}\left[ A \right]}}{W_R}\left( {K,A} \right)}, \]
where the Wilson loop is
\[{W_R}\left( {K,A} \right) = {\text{t}}{{\text{r}}_R}P \exp \left( {\oint {A_\mu ^a\left( {\text{x}} \right){T^a}d{{\text{x}}^\mu }} } \right),\]
and the Chern-Simons action is
\[{S_{CS}}\left[ A \right] = \frac{\kappa } {{4\pi }}\int\limits_M {\text{Tr} \left( {A \wedge dA + \frac{2} {3}A \wedge A \wedge A} \right)}, \]
with $q = {e^\hbar },a = N\hbar $ and $\hbar = \frac{{2\pi i}} {{\kappa + N}}.$
The limiting case $\hbar \to 0,N \to \infty $ such that $N\hbar$ remains fixed, i.e., $q=1$, of the HOMFLY polynomials gives the special polynomials ${\mathcal {H}}_R^{\mathcal {K}}\left( {q,a} \right) = \sigma _R^{\mathcal {K}}\left( a \right)$, whose $R$ dependence makes them expressible in the form $\sigma _R^{\mathcal {K}}\left( a \right) = {\left( {\sigma _{\left[ 1 \right]}^{\mathcal {K}}\left( a \right)} \right)^{\left| R \right|}}$ for the Young diagram $R = \left\{ {{R_i}} \right\},{R_1} \geqslant {R_2} \geqslant \ldots \geqslant {R_{l\left( R \right)}},\left| R \right|: = \sum\nolimits_i {{R_i}} $. This provides the construction of a KP $\tau$-function (see for example [\textit{P. Dunin-Barkowski} et al., J. High Energy Phys. 2013, No. 3, Paper No. 021, 85 p. (2013; Zbl 1342.57004)]).
\par The dual limit as $a \to 1$ of the HOMFLY polynomials, i.e., ${\mathcal {H}}_R^{\mathcal {K}}\left( {q,1} \right)$, for the fundamental representation $R$ gives the Alexander polynomial, the coloured version of which exhibits a dual property with respect to $R$, viz., ${\mathcal {A}}_R^{\mathcal {K}}\left( q \right) = {\mathcal {A}}_{\left[ 1 \right]}^{\mathcal {K}}\left( {{q^{\left| R \right|}}} \right)$, which holds only for the representations corresponding to 1-hook Young diagrams ${R = \left[ {r,{1^L}} \right]}.$
\par In this paper the authors study this property perturbatively and claim that while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials induce the equations of the KP hierarchy.
\par The main result of the paper is stated in Section 5, where, by considering the generating function of the KP hierarchy, replacing the Hirota operators with the Casimir eigenvalues and symmetrizing the identity, the authors find that Hirota KP bilinear equations are satisfied if and only if
\({\mathcal {A}}_R^{\mathcal {K}}\left( q \right) = {\mathcal {A}}_{\left[ 1 \right]}^{\mathcal {K}}\left( {{q^{\left| R \right|}}} \right)\).
The authors give only the first half of the proof of this result. \par The paper explores interesting
connections between the KP hierarchy and the coloured Alexander polynomials. Chern-Simons theory; knot invariant; Kontsevich integral; Vassiliev invariants; Hirota bilinear identities; KP hierarchy; Young diagrams; Gromov-Witten theory; Schur polynomial Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Knots and links in the 3-sphere, Invariants of knots and \(3\)-manifolds, Knots and links (in high dimensions) [For the low-dimensional case, see 57M25], Eta-invariants, Chern-Simons invariants, Casimir effect in quantum field theory, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Coloured Alexander polynomials and KP hierarchy | 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 localization theorem for affine \(\mathcal{W} \)-algebras in the spirit of Beilinson-Bernstein and Kashiwara-Tanisaki. More precisely, for any non-critical regular weight \(\lambda \), we identify \(\lambda \)-monodromic Whittaker \(D\)-modules on the enhanced affine flag variety with a full subcategory of Category \(\mathcal{O}\) for the \(\mathcal{W} \)-algebra.
To identify the essential image of our functor, we provide a new realization of Category \(\mathcal{O}\) for affine \(\mathcal{W} \)-algebras using Iwahori-Whittaker modules for the corresponding Kac-Moody algebra. Using these methods, we also obtain a new proof of Arakawa's character formulae for simple positive energy representations of the \(\mathcal{W} \)-algebra. \( \mathcal{W} \)-algebra; localization; highest weight module; character formula Localization for affine \(\mathcal{W} \)-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 I construct a correspondence between the Schubert cycles on the variety of complete flags in \(\mathbb{C}^n\) and some faces of the Gelfand-Zetlin polytope associated with the irreducible representation of \(SL_n (\mathbb{C})\) with a strictly dominant highest weight. The construction is motivated by the geometric presentation of Schubert cells using Demazure modules due to \textit{I. N. Bernstein, I. M. Gelfand} and \textit{S. I. Gelfand} [Russ. Math. Surveys 28, No. 3, 1--26 (1973; Zbl 0286.57025)]. The correspondence between the Schubert cycles and faces is then used to interpret the classical Chevalley formula in Schubert calculus in terms of the Gelfand-Zetlin polytopes. The whole picture resembles the picture for toric varieties and their polytopes. V. Kiritchenko, \textit{Gelfand-Zetlin polytopes and flag varieties}, Int. Math. Res. Not. IMRN (2010), no. 13, 2512-2531. Grassmannians, Schubert varieties, flag manifolds, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Gelfand-Zetlin polytopes and 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 We discuss some connections between the closure \(\overline F\) of a Steinberg fiber in the wonderful compactification of an adjoint group and the affine Deligne-Lusztig varieties \(X_w(1)\) in the affine flag variety. Among other things, we describe the emptiness/nonemptiness pattern of \(X_w(1)\) if the translation part of \(w\) is quasi-regular. As a by-product, we give a new proof of the explicit description of \(\overline F\), first obtained by the author [in Adv. Math. 203, No. 1, 109-131 (2006; Zbl 1102.20032)]. affine Deligne-Lusztig varieties; unipotent varieties; Steinberg fibers; group compactifications; connected simple algebraic groups; wonderful compactification X. He, ''Closure of Steinberg fibers and affine Deligne-Lusztig varieties,'' Int. Math. Res. Not., vol. 2011, iss. 14, pp. 3237-3260, 2011. Linear algebraic groups over arbitrary fields, Classical groups (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Compactifications; symmetric and spherical varieties, Representation theory for linear algebraic groups Closure of Steinberg fibers and affine Deligne-Lusztig 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 \(\pi: Y\to X\) be the Hesse pencil of elliptic curves, given as the projection onto the first factor of the surface in \(\mathbb{P}^ 1\times \mathbb{P}^ 2\) with equation \(u_ 0 (x_ 0^ 3+ x_ 1^ 3+ x_ 2^ 3)- u_ 1 x_ 0 x_ 1 x_ 2 =0\). Let \(W= Y\times_ X Y\) and let \(\widetilde {W}\) be the desingularization of \(W\) obtained by blowing up its double points. The author shows that for \(F\) a quadratic extension of \(\mathbb{Q}\) such that the sign in the functional equation of \(L_ F (H^ 3 (\widetilde {W}, s))\) is positive, the group \(\text{CH}^ 2 (\widetilde{W}_ F)_{CM}:= \ker [\text{CH}^ 2 (\widetilde {W}_ F)_{\hom}\to \text{CH}^ 2 (\widetilde {W}\times_ X \eta_ F)]\) has rank 0. Here \(\eta_ F\) is the generic point of \(X_ F\). The hypothesis about \(F\) is satisfied for real quadratic fields unramified at 3. The author also verifies that \(L_ F (H^ 3 (\widetilde {W},2))\neq 0\) for \(\text{disc}(F)= \pm1\bmod 3\) and \(d<250\), \(d\neq 172\). So for such \(d\), one has that the rank of \(\text{CH}^ 2 (\widetilde{W}_ F)_{CM}\) equals \(\text{ord}_{s=2} L_ F(H^ 3 (\widetilde{W}),s)\), a fact which supports a conjecture of Bloch and Beilinson. complex multiplication; Chow groups; \(L\)-function; Bloch-Beilinson conjecture; real quadratic fields C. Schoen, Complex multiplication cycles and a conjecture of Beĭlinson and Bloch, Trans. Amer. Math. Soc. 339 (1993), no. 1, 87-115. Parametrization (Chow and Hilbert schemes), Quadratic extensions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Algebraic cycles, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Modular and Shimura varieties, Holomorphic modular forms of integral weight, Forms of half-integer weight; nonholomorphic modular forms Complex multiplication cycles and a conjecture of Beilinson and Bloch | 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 complex semisimple simply connected linear algebraic group. The main result of this note is to give several equivalent criteria for the {it untwistedness} of the {it twisted cubes} introduced by Grossberg and Karshon. In certain cases arising from representation theory, Grossberg and Karshon obtained a Demazure-type character formula for irreducible \(G\)-representations as a sum over lattice points (counted with sign according to a density function) of these twisted cubes. A twisted cube is untwisted when it is a ``true'' (i.e., closed, convex) polytope; in this case, Grossberg and Karshon's character formula becomes a purely {it positive} formula with no multiplicities, i.e., each lattice point appears precisely once in the formula, with coefficient \(+1\). One of our equivalent conditions for untwistedness is that a certain divisor on the special fiber of a toric degeneration of a Bott-Samelson variety, as constructed by Pasquier, is basepoint-free. We also show that the strict positivity of some of the defining constants for the twisted cube, together with convexity (of its support), is enough to guarantee untwistedness. Finally, in the special case when the twisted cube arises from the representation-theoretic data of \(\lambda\) an integral weight and \(\underline{w}\) a choice of word decomposition of a Weyl group element, we give two simple necessary conditions for untwistedness which is stated in terms of \(\lambda\) and \(\underline{w}\). twisted cubes; Demazure character formula; Bott-Samelson variety; toric variety; toric divisor M. Harada and J. J. Yang, Grossberg-Karshon twisted cubes and basepoint-free divisors , J. Korean Math. Soc. 52 (2015), no. 4, 853-868. Toric varieties, Newton polyhedra, Okounkov bodies, Representation theory for linear algebraic groups, Divisors, linear systems, invertible sheaves Grossberg-Karshon twisted cubes and basepoint-free divisors | 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 theory of DAHA-Jones polynomials is extended from torus knots to iterated torus knot, for any reduced root systems and weights. This is inspired by \textit{P. Samuelson}'s construction for the \(\mathfrak{sl}_2\) case [``Iterated torus knots and double affine Hecke algebras'', Preprint, \url{arXiv:1408.0483}]. The paper proves polynomiality, duality and other properties, and computes several examples. They conjecture that these polynomials specialize to Khovanov-Rozansky polynomials, which was since proven by \textit{H. Morton} and \textit{P. Samuelson} [Duke Math. J. 166, No. 5, 801--854 (2017; Zbl 1369.16034)].
The same authors have since extended the DAHA-Jones polynomials to iterated torus links [\textit{A. Beliakova} (ed.) and \textit{A. D. Lauda} (ed.), Categorification in geometry, topology, and physics. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1362.81007)]. double affine Hecke algebra; Jones polynomials; HOMFLY-PT polynomial; Khovanov-Rozansky homology; iterated torus knot; cabling; MacDonald polynomial; plane curve singularity; generalized Jacobian; Betti numbers; Puiseux expansion Cherednik, I.; Danilenko, I., DAHA and iterated torus knots, Algebr. Geom. Topol., 16, 843-898, (2016) Singularities of curves, local rings, Knots and links in the 3-sphere, Hecke algebras and their representations, Braid groups; Artin groups, Compact Riemann surfaces and uniformization, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Lie algebras of linear algebraic groups, Singular homology and cohomology theory DAHA and iterated torus knots | 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 \(f\in K[x_0,\dots,x_n]\), \(n\geq 1\), be a polynomial over an algebraically closed field of characteristic zero and let \(H_f\) be its Hessian determinant. In 1851 Hesse claimed that if \(f\) is homogeneous then \(H_f=0\) if and only if after a suitable invertible linear change of variables \(T:(y_0,\dots,y_n)\to (x_0,ldots,x_n)\) the composition \(f\circ T\) depends on a smaller number of variables, i.e. \(f\circ T\in K[y_0,\dots,y_{n-1}]\). Gordan and Noether in 1876 showed that it is not true in general (precisely the implication \(\Rightarrow\) fails). They proved that the equivalence holds for \(n\leq 3\) and they gave counter-examples for \(n\geq 4\).
The paper analyses and modernizes the Gordan and Noether results. In particular the author gives a complete classification of homogeneous polynomials for which \(H_f=0\) in the case \(n=4\). For non-homogeneous case see the recent paper by \textit{M. de Bondt} and \textit{A. van den Essen} [J. Algebra 282, 195--204 (2004; Zbl 1060.14089)]. Hessian determinant; polynomial; homogeneous polynomial Lossen, C, When does the Hessian determinant vanish identically? (on Gordan and Nöther's proof of hesse's claim), Bull. Braz. Math. Soc. (N.S.), 35, 71-82, (2004) Real and complex fields, Hypersurfaces and algebraic geometry, Jacobian problem When does the Hessian determinant vanish identically? (On Gordan and Noether's proof of Hesse's claim) | 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 \(V_{\mathbb{R}}\) be a real vector space with an irreducible action of a finite reflection group \(W\). The author constructs a one-parameter family of semialgebraic polyhedra \(K_W(\lambda)\) in \(V_\mathbb{R}\) which are dual to the Weyl chamber decomposition of \(V_\mathbb{R}\). The key of such a construction is a theorem on linearization of the tube domain in \((V/\!/W)_\mathbb{R}\) over the simplicial cone.
As an application, the author obtains two geometric descriptions of generators for\break \(\pi_1 ((V/\!/W)_\mathbb{C}^{\text{reg}})\), satisfying the Artin braid relations. K. Saito, ''Polyhedra dual to the Weyl chamber decomposition: a précis,'' Publ. Res. Inst. Math. Sci., vol. 40, iss. 4, pp. 1337-1384, 2004. Semialgebraic sets and related spaces, Reflection and Coxeter groups (group-theoretic aspects) Polyhedra dual to the Weyl chamber decomposition: a précis | 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 review properties of generalized Macdonald functions arising from the AGT correspondence. In particular, we explain a coincidence between generalized Macdonald functions and singular vectors of a certain algebra \(\mathcal{A}(N)\) obtained using the level-\((N, 0)\) representation (horizontal representation) of the Ding-Iohara-Miki algebra. Moreover, we give a factored formula for the Kac determinant of \(\mathcal{A}(N)\), which proves the conjecture that the Poincaré-Birkhoff-Witt-type vectors of the algebra \(\mathcal{A}(N)\) form a basis in its representation space. AGT correspondence; Macdonald symmetric function; Ding-Iohara-Miki algebra; singular vector Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Formal methods and deformations in algebraic geometry, Yang-Mills and other gauge theories in quantum field theory, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Singular vectors of the Ding-Iohara-Miki 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 Let \(A\) be a finite alphabet (independent variables), \({\mathcal R}(A)\) the ring of rational functions on \(A\), \({\mathcal G}(A)\) the symmetric group of \(A\), and \(\mathcal E\) the group algebra of \({\mathcal G}(A)\) on the field \({\mathcal R}(A)\). The elements of \(\mathcal E\) are linear operators on \({\mathcal R}(A)\), and \(\mathcal E\) is an \({\mathcal R}(A)\)-module the ``canonical'' basis of which is formed by the permutations of \({\mathcal G}(A)\). Here, several other classical bases of \(\mathcal E\) consisting of symmetrizing operators are considered, namely Newton's divided differences, the convex symmetrizers, and the concave symmetrizers. The authors deal with the problem to determine the matrices of change of such bases explicitly. Their main result is that the elements of these transformation matrices are just specializations of Schubert or Grothendieck polynomials. An important tool is the fact that all but one specialization of the maximal (twofold) Schubert polynomial vanish. (Unfortunately, there are misprints in some formulas.) Moreover, the paper contains instructive remarks on references to algebraic geometry regarding the interpretation of the matrices (cohomology and \(K\)-theory rings of flag manifolds, Schubert varieties). group algebra; permutations; symmetrizing operators; Newton's divided differences; Grothendieck polynomials; Schubert polynomial Lascoux, Alain; Schützenberger, Marcel-Paul, Décompositions dans l'algèbre des différences divisées, Discrete Math., 99, 1-3, 165-179, (1992) Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Orthogonal polynomials [See also 33C45, 33C50, 33D45], Grassmannians, Schubert varieties, flag manifolds, \(K\)-theory and homology; cyclic homology and cohomology, Computations of higher \(K\)-theory of rings Décompositions dans l'algèbre des différences divisées. (Decompositions in the algebra of divided differences) | 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 studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs an explicit paving of all Steinberg varieties in Lie type A in terms of semistandard tableaux. As a result, we obtain an elementary proof of a theorem of Steinberg and Shimomura that the well-known Kostka numbers count the maximal-dimensional irreducible components of Steinberg varieties. The second application proves an open conjecture for certain parabolic Hessenberg varieties in Lie type A by showing that their Betti numbers equal those of a specific union of Schubert varieties. The third application proves that the irreducible components of parabolic Hessenberg varieties are in bijection with the irreducible components of the Steinberg variety. All three of these applications extend our geometric understanding of the three varieties at the heart of this paper, a full understanding of which is unknown even for Springer varieties, despite over forty years' worth of work. Springer fibers; Steinberg varieties; Hessenberg varieties; standard tableaux; affine pavings Classical real and complex (co)homology in algebraic geometry, Classical groups (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Hessenberg varieties of parabolic type | 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 apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-\(A\) affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a \(k\)-Schur function. Consequently, we prove that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for \(\mathbb C^n\) enumerate the highest weight elements under these operators. flag Gromov-Witten invariants; Littlewood-Richardson coefficients; crystal graphs; Specht modules J. Morse, A. Schilling, Crystal operators and flag Gromov-Witten invariants, preprint. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Quantum groups (quantized enveloping algebras) and related deformations Flag Gromov-Witten invariants via 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 Let \(X\) be the full flag variety of a connected simple complex algebraic group \(G\) and let \(X_w\subset X\) be a Schubert variety (associated to a Weyl group element \(w\)). If \(G\) is of type \(A_n\), \textit{V. Lakshmibai} and \textit{G. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45-52 (1990; Zbl 0714.14033)] have shown that \(X_w\) is rationally smooth iff \(w\) avoids certain length-4 patterns. Further, \textit{V. Gasharov} has shown that (again in the \(A_n\)-case) \(X_w\) is rationally smooth iff the Poincaré polynomial \(p_w(t):= \sum_{v\leq w}t^{\ell(v)}\) (where \((\ell(v)\) is the length of \(v\)) factors into polynomials of the form \((1+t+ t^2+\cdots+ t^r)\). Now, Billey, in the paper under review, generalizes both of these results for classical groups \(G\) of type \(B\) and \(C\). As a corollary, she obtains that \(X_w\) (for \(G\) of type \(A\) and \(B\)) is rationally smooth iff the poset \([e,w]:= \{v\leq w\}\) has the Peck property (i.e., is rank symmetric, rank unimodal, and \(k\)-Sperner for all \(k\)). She also obtains a characterization for smooth \(X_w\) (for \(G\) of type \(B,C\) and \(D\)) in terms of pattern avoidance. full flag variety; Schubert variety; pattern avoidance Billey, S, \textit{pattern avoidance and rational smoothness of Schubert varieties}, Adv. in Math., 139, 141-156, (1998) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Pattern avoidance and rational smoothness 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 Let \(\mathfrak{g}\) be a complex reductive Lie algebra, \(\mathfrak{h}\) be its Cartan subalgebra, and \(W\) be the Weyl group. Then, by Chevalley's theorem on polynomial \(W\)-invariants, the quotient \(W \backslash \mathfrak{h}\) is an affine space. Moreover, it carries a natural stratification \(\mathcal{S}^{(0)}\). The paper under review studies the category \(\operatorname{Perv}(W \backslash \mathfrak{h})\) of perverse sheaves on the quotient which are smooth with respect to \(\mathcal{S}^{(0)}\), and gives an equivalence to the category of objects of mixed functoriality with respect to a natural cell decomposition of \(W \backslash \mathfrak{h}\) refining \(\mathcal{S}^{(0)}\), called mixed Bruhat sheaves in this paper. The sections \S\S1--6 are devoted to this study.
The paper also proposes the viewpoint that the category \(\operatorname{Perv}(W \backslash \mathfrak{h})\) provides a conceptual encoding of ``the algebra of parabolic induction'', i.e., the entire package of results related to principal series representations and those obtained by parabolic induction in all the brunches of representation theory. Several illustrations of this viewpoint are given in \S 7 and \S 8. This part seems to have various open problems and relations to topics of algebraic and geometric representation theory. perverse sheaves; configuration spaces; parabolic induction; Hopf algebras Sheaves in algebraic geometry, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Structure theory for Lie algebras and superalgebras, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Representation theory for linear algebraic groups Parabolic induction and perverse sheaves on \(W \backslash \mathfrak{h} \) | 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 lecture is based on a former paper of the authors [J. Geom. Phys. 19, No. 3, 287-313 (1996; Zbl 0965.81096)]. \(b-c\) systems with integer spin on algebraic curves given by Weierstrass polynomials are described by an operator formalism. To that end Riemannian surfaces are considered and represented as an \(n\)-fold branching covering of the Riemannian sphere where \(n\) is the degree of the Weierstrass polynomial. From this an \(n\)-fold splitting of the Hilbert space into Fock spaces is deduced, such that in each Fock space only modes with the same monodromy characteristics propagate. The correlation functions are calculated. integer spin; Weierstrass polynomials; Riemannian surfaces; correlation functions Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Coverings of curves, fundamental group, Compact Riemann surfaces and uniformization Creation and annihilation operators for \(b-c\) systems on general algebraic curves | 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 \(\dot{G}\) be a connected reductive algebraic group over a field \(\mathbf{k}=\overline{\mathbf{k}}\), with simply-connected derived subgroup, and let \(\dot{B}\subseteq \dot{G}\) be a Borel subgroup. Let \(\widetilde{\mathcal{N}}=T^*(\dot{G}/\dot{B})\) be the cotangent bundle of the flag variety \(\dot{G}/\dot{B}\). Consider the derived category \(D^{\text{b}}\text{Coh}^{\dot{G}\times \mathbb{G}_m}(\widetilde{\mathcal{N}})\) of \((\dot{G}\times \mathbb{G}_m)\)-equivariant coherent sheaves on \(\widetilde{\mathcal{N}}\). \textit{R. Bezrukavnikov} [Invent. Math. 166, No. 2, 327--357 (2006; Zbl 1123.17002)] has introduced the exotic \(t\)-structure on this category on the cotangent bundle \(T^*(G/B)\) of its flag variety. It plays an important role in Bezrukavnikov-Mirković's proof of Lusztig's conjectures on modular representations of Lie algebras, in the proof of the Mirković-Vilonen conjecture on torsion on the affine Grassmannian, in derived equivalences for local geometric Langlands duality in characteristic zero and in positive characteristic, in the cohomology of tilting modules for Lusztig's quantum groups and algebraic groups, to name a few.
The exotic \(t\)-structure is defined using the \(t\)-structure of a certain graded, exceptional set of objects in \(D^{\text{b}}\text{Coh}^{\dot{G}\times \mathbb{G}_m}(\widetilde{\mathcal{N}})\). That is, if \(\mathfrak{C}\) is a \(\mathbf{k}\)-linear triangulated category of finite type (the \(\mathbf{k}\)-vector space \(\text{Hom}^{\bullet}(X,Y)\) is finite-dimensional for any objects \(X, Y\in \mathfrak{C}\)), an ordered subset \(\nabla=\{ \nabla^i: i\in I\}\) of \(\text{ob}(\mathfrak{C})\) is exceptional if \(\text{Hom}^{\bullet}(\nabla^i,\nabla^j)=0\) for \(i<j\), \(\text{Hom}^n(\nabla^i,\nabla^i)=0\) for \(n\not=0\), and \(\text{End}(\nabla^i)=\mathbf{k}\).
Now for each \(i\in I\), define \(\mathfrak{C}_{<i}\) as the full triangulated subcategory generated by \(\nabla^j\), where \(j<i\). Another subset \(\Delta=\{ \Delta_i:i\in I \}\) of \(\text{ob}(\mathfrak{C})\) is dual to \(\nabla\) if \(\text{Hom}^{\bullet}(\Delta_n,\nabla^i)=0\) for \(n> i\), and there is an isomorphism \(\Delta_n\cong \nabla^n\text{ mod }\mathfrak{C}_{<n}\). Now let \(I=\mathbb{N}_{>0}\). Then there exists a unique \(t\)-structure \((\mathfrak{C}^{\geq 0}, \mathfrak{C}^{<0})\) on the category \(\mathfrak{C}\) such that \(\nabla^i\in\mathfrak{C}^{\geq 0}\) and \(\Delta_i \in \mathfrak{C}^{\leq 0}\), where \(\mathfrak{C}^{\geq 0} = \langle \{\nabla^i[d]: i\in I, d \leq 0 \}\rangle\) and \(\mathfrak{C}^{<0} = \langle \{\Delta_i[d] : i\in I, d>0 \}\rangle\). The exotic \(t\)-structure on the triangulated category \(D^{\text{b}}\text{Coh}^{\dot{G}\times \mathbb{G}_m}(\widetilde{\mathcal{N}})\) is the \(t\)-structure of the graded, exceptional set \(\nabla^{\lambda,0}\).
An important feature of the exotic \(t\)-structure is that the higher \(t\)-cohomology of every exceptional object vanishes, which implies that the heart of this \(t\)-structure has the structure of a highest weight category (also known as quasi-hereditary category or Kazhdan-Lusztig type category). This fact plays a similar role to the Kempf vanishing theorem for reductive, algebraic groups or to the Artin vanishing theorem, which says if \(X\) is an affine variety over a separably closed field \(\mathbf{k}\) and if \(\mathcal{F}\) is a constructible sheaf on \(X\), then the étale cohomology \(H^i_{\text{ét}}(X,\mathcal{F})\) vanishes for \(i > \dim X\). The exotic \(t\)-structure and higher \(t\)-cohomology vanishing are key tools in the proof of the graded Finkelberg-Mirković conjecture, which relates the principal block of a reductive group to perverse sheaves on the Langlands dual affine Grassmannian, thus also playing important aspects in the proof of the tilting character formula for reductive groups.
Now, let \(I\subseteq S\), and \(\lambda\in \mathbf{X}_I^{+,\text{reg}} := \{\lambda\in \mathbf{X}: \langle \lambda, \alpha_{S}^{\vee}\rangle >0 \text{ for any }s\in I \}\), where \(\leq'\) is given in [\textit{P. N. Achar} and \textit{S. Riche}, Invent. Math. 214, No. 1, 289--436 (2018; Zbl 1454.20095)] (see Section 9, page 81). When \(I=\varnothing\), we have \(\nabla_I(\lambda)=\nabla^{\lambda}\) and \(\Delta_I(\lambda)=\Delta_{\lambda}\) in \(D^{\text{b}}\text{Coh}^{\dot{G}\times \mathbb{G}_m}(\widetilde{\mathcal{N}})\). When \(I=S\), \(\nabla_S(\lambda)\) is the induced module of highest weight \(\lambda-\varsigma_S\), and \(\Delta_S(\lambda)\) is the Weyl module of highest weight \(\lambda-\varsigma_S\). The objects \((\nabla_I(\lambda): \lambda\in \mathbf{X}_I^{+,\text{reg}})\) form a graded exceptional set of objects in the triangulated category \(D^{\text{b}}\text{Coh}^{\dot{G}\times \mathbb{G}_m}(\widetilde{\mathcal{N}}_I)\) on the cotangent bundle \(\widetilde{\mathcal{N}}_I := T^*(\dot{G}/\dot{P})\) of a partial flag variety, with respect to the order \(\leq'\) and the shift functor \(\langle 1\rangle\). That is,
\[
\text{Hom}_{D^{\text{b}}\text{Coh}^{\dot{G} \times \mathbb{G}_m}(\widetilde{\mathcal{N}}_I)}(\nabla_I(\lambda),\nabla_I(\mu)\langle n \rangle[m])=0 \quad \text{ if }\mu \leq\!\!\!\!\!/ ' \lambda, \text{ or if } \lambda=\mu \text{ and } (n,m)\neq(0,0).
\]
Furthermore, \(\text{Hom}_{D^{\text{b}}\text{Coh}^{\dot{G}\times \mathbb{G}_m}(\widetilde{\mathcal{N}}_I)}(\nabla_I(\lambda),\nabla_I(\lambda))=\mathbf{k}\), with the dual exceptional set being \((\Delta_I(\lambda): \lambda \in \mathbf{X}_I^{+,\text{reg}})\), satisfying
\[
\text{Hom}_{D^{\text{b}}\text{Coh}^{\dot{G}\times \mathbb{G}_m}(\widetilde{\mathcal{N}}_I)}(\Delta_I(\lambda),\nabla_I(\mu)\langle n \rangle [m])=0 \quad \text{ if } \mu <' \lambda,
\]
and \(\Delta_I(\lambda)\cong\nabla_I(\lambda)\text{ mod }D^{\text{b}}\text{Coh}^{\dot{G}\times \mathbb{G}_m}(\widetilde{\mathcal{N}}_I)_{<' \lambda}\). This exceptional set determines a bounded \(t\)-structure on the derived category (see Section 2.C, page 5), which is again called the exotic \(t\)-structure. Its heart is denoted by \(\text{ExCoh}(\widetilde{\mathcal{N}}_I)\).
P. Achar, N. Cooney, and S. Riche prove that the higher \(t\)-cohomology of every exceptional object vanishes in the parabolic setting (Theorem 2.2, page 5): for \(\lambda\in \mathbf{X}_I^{+,\text{reg}}\), the objects \(\Delta_I(\lambda)\) and \(\nabla_I(\lambda)\) belong to \(\text{ExCoh}(\widetilde{\mathcal{N}}_I)\). That is, since \(\nabla_I(\lambda)\in D^{\text{b}}\text{Coh}^{\dot{G}\times \mathbb{G}_m}(\widetilde{\mathcal{N}}_I)^{\geq 0}\), the \(i\)-th cohomology \({}^{t}H^i(\nabla_I(\lambda))\) with respect to the exotic \(t\)-structure vanishes, for \(i>0\) (there is a similar vanishing result for the dual \(\Delta_I(\lambda)\)). Theorem 2.2 is proved using constructible sheaves and mixed derived categories in the sense of [\textit{P. N. Achar} and \textit{S. Riche}, Duke Math. J. 165, No. 1, 161--215 (2016; Zbl 1375.14162)]. It follows that its heart is a highest weight category (Corollary 2.3, page 5): the category \(\text{ExCoh}(\widetilde{\mathcal{N}}_I)\) is a graded highest weight category, with weight poset \((\mathbf{X}_I^{+,\text{reg}},\leq')\), standard objects \((\Delta_I(\lambda): \lambda\in \mathbf{X}_I^{+,\text{reg}})\), costandard objets \((\nabla_I(\lambda): \lambda\in \mathbf{X}_I^{+,\text{reg}})\), and shift functor \(\langle 1\rangle\). The notion of a highest weight category is in the sense of the third author's habilitation thesis [Geometric Representation Theory in positive characteristic, \url{https://tel.archives-ouvertes.fr/tel-01431526}].
Finally, the authors prove a parabolic analogue of the Arkhipov-Bezrukavnikov-Ginzburg equivalence (see Theorem 5.5, page 19), and when \(\text{char}(\mathbf{k})\) is greater than the Coxeter number, Achar-Cooney-Riche deduce an analogue of the graded Finkelberg-Mirković conjecture for some singular blocks (Theorem 5.7, page 21). flag varieties; derived category of coherent sheaves; parity complexes; t-structure; exceptional collection; highest weight categories; partial flag varieties; exotic t-structure Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Kac-Moody groups The parabolic exotic t-structure | 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 associate to every matroid \(M\) a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of \(M\), in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincaré polynomials. We also introduce a \(q\)-deformation of the Möbius algebra of \(M\), and use our polynomials to define a special basis for this deformation, analogous to the canonical basis of the Hecke algebra. We conjecture that the structure coefficients for multiplication in this special basis are non-negative, and we verify this conjecture in numerous examples. matroid; Orlik-Terao algebra; intersection cohomology; Kazhdan-Lusztig theory B. Elias, N. Proudfoot, and M. Wakefield, The Kazhdan-Lusztig polynomial of a matroid, \textit{Adv.} \textit{Math.}, 299 (2016), 36--70.Zbl 1341.05250 MR 3519463 Combinatorial aspects of representation theory, Combinatorial aspects of matroids and geometric lattices, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups The Kazhdan-Lusztig polynomial of a matroid | 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 concerns the recursive computation of real Gromov-Witten invariants using WDVV-style relations in a few four and six-dimensional examples.
Classical Gromov-Witten (or GW) invariants are weighted geometric or virtual counts of holomorphic curves in complex projective varieties or symplectic manifolds with an almost complex structure. Real GW theory concerns holomorphic curves with real defining equations in complex projective varieties equipped with a complex conjugation or symplectic manifolds equipped with an anti-symplectic involution. This paper calculates a particular class of genus-zero real GW invariants, known as Welschinger invariants, in certain real symplectic four-folds and six-folds. To do the calculations, they use Jake Solomons's real variant of a classical recursive method, known as WDVV relations. Theorems 1 and 3 state the general form of the WDVV relations in four and six dimensions following their earlier results in [\textit{X. Chen}, ``Steenrod pseudocycles, lifted cobordisms, and Solomon's relations for Welschinger invariants'', Preprint, \url{arXiv:1809.08919}, Theorem~1.1] and [\textit{X. Chen} and \textit{A. Zinger}, Math. Ann. 379, No. 3--4, 1231--1313 (2021; Zbl 1490.53103), Theorem~1.5]. The formulas have two versions, depending on the number of non-real marked points, and involve real and classical GW invariants. Subsequently, the examples worked out in Sections~3-8 includes the complex projective space \(\mathbb{P}^2\), \(\mathbb{P}^1\times \mathbb{P}^1\) with two types of involution, blow-ups of \(\mathbb{P}^2\), \(\mathbb{P}^3\), and \(\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^1\) with two types of involution. Several tables of invariants in low degrees are provided. The Mathematica programs implementing their recursive formulas are available from the Wolfram Foundation's Notebook Archive. Gromov-Witten invariants; Welschinger invariants; WDVV relations Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Real algebraic sets, Enumerative problems (combinatorial problems) in algebraic geometry WDVV-type relations for Welschinger invariants: 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 In recent publications, the same combinatorial description has arisen for three separate objects of interest: non-negative cells in the real Grassmannian (Postnikov, Williams); torus orbits of symplectic leaves in the classical Grassmannian (Brown, Goodearl and Yakimov); and, torus invariant prime ideals in the quantum Qrassmannian (Launois, Lenagan and Rigal). The aim of this meeting was to explore the reasons for this coincidence in matrices and the Frassmannian in particular, and to explore similar ideas in more general settings.
Contents:
Tom Lenagan, Introduction: Non-negativity is a quantum phenomenon
K. R. Goodearl, Poisson algebras and symplectic leaves
Lauren Williams, Introduction to total non-negativity
Gérard Cauchon, Quantum algebras (deleting derivations algorithm)
Robert J. Marsh, The dual canonical basis of a quantized enveloping algebra
Bernard Leclerc, Introduction to cluster algebras
Francesco Brenti, Bruhat intervals
Ken A. Brown, Symplectic cores, symplectic leaves and the orbit method
Laurent Rigal (joint with Stéphane Launois, Tom Lenagan), Links between prime spectra of quantum algebras and the geometry of their ``totally positive'' counterpart: some significant examples
Konstanze Rietsch, On total positivity in flag varieties
Antoine Mériaux (joint with Gérard Cauchon), Cauchon diagrams for quantised enveloping nilpotent algebras
Thomas Lam, Total positivity for loop groups
Milen Yakimov, Weak splittings of surjective Poisson submersions
Karin Baur (joint with Robert Marsh), Geometric construction of cluster categories
Stéphane Launois (joint with Ken Goodearl, Tom Lenagan), Conclusion: The matrix case. Proceedings, conferences, collections, etc. pertaining to combinatorics, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to associative rings and algebras, Proceedings, conferences, collections, etc. pertaining to nonassociative rings and algebras, Proceedings, conferences, collections, etc. pertaining to group theory, Collections of abstracts of lectures Mini-workshop: Non-negativity is a quantum phenomenon. Abstracts from the mini-workshop held March 1st -- March 7th, 2009. | 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 \(E\) be a symplectic vector space of dimension \(2n\) (with the standard antidiagonal symplectic form) and let \(G\) be the Lagrangian Grassmannian over \(\text{Spec} \mathbb{Z}\), parametrizing Lagrangian subspaces in \(E\) over any base field. Equip \(E(\mathbb{C})\) with a hermitian metric compatible with the symplectic form and \(G(\mathbb{C})\) with the Kähler metric induced from the natural invariant metric on the Grassmannian of \(n\)-planes in \(E\). We give a presentation of the Arakelow Chow ring \(\text{CH}(\overline G)\) and develop an arithmetic Schubert calculus in this setting. The theory uses the \(\widetilde Q\)-polynomials of \textit{P. Pragacz} and \textit{J. Ratajski} [Compos. Math. 107, No. 1, 11-87 (1997; Zbl 0916.14026)] and involves `shifted hook operations' on Young diagrams.
As an application, we compute the Faltings height of \(G\) with respect to its Plücker embedding in projective space. symplectic vector space; Lagrangian Grassmannian; Arakelow Chow ring H. Tamvakis : Arakelov theory of the Lagrangian Grassmannian, J. Reine Angew. Math. 516 (1999), 207-223. Arithmetic varieties and schemes; Arakelov theory; heights, Grassmannians, Schubert varieties, flag manifolds Arakelov theory of the Lagrangian 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 We develop the point of view that Schubert classes of the affine Grassmannian of a simple algebraic group \(G\) can be identified with Schur-positive symmetric functions. In particular, we give a geometric proof of the Schur positivity of \(k\)-Schur functions at \(t=1\), together with branching positivity results for the type \(C k\)-Schur functions. Our work is placed in the context of combinatorial Hopf algebras. Lam, T.: Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras. Bull. London Math. Soc. \textbf{43}(2), 328-334 (2011) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Classical problems, Schubert calculus, Connections of Hopf algebras with combinatorics Affine Schubert classes, Schur positivity, and combinatorial Hopf 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 \textit{B. Huber, F. Sottile}, and \textit{B. Sturmfels} [J. Symb. Comput. 26, 767-788 (1998; Zbl 1064.14508)] proposed Pieri homotopies to enumerate all \(p\)-planes in \({\mathbb{C}}^{m+p}\) that meet \(n\) given \((m+1-k_{i})\)-planes in general position, with \(k_1+k_2+\dots+k_n=mp\) as a condition to have a finite number of solution \(p\)-planes. Pieri homotopies turn the deformation arguments of classical Schubert calculus into effective numerical methods by expressing the deformations algebraically and applying numerical path-following techniques. We describe the Pieri homotopy algorithm in terms of a poset of simpler problems. This approach is more intuitive and more suitable for computer implementation than the original chain-oriented description and provides also a self-contained proof of correctness. We extend the Pieri homotopies to the quantum Schubert calculus problem of enumerating all polynomial maps of degree \(q\) into the Grassmannian of \(p\)-planes in \({\mathbb{C}}^{m+p}\) that meet \(mp + q(m+p)\) given \(m\)-planes in general position sampled at \(mp + q(m+p)\) interpolation points. Our approach mirrors existing counting methods for this problem and yields a numerical implementation for the dynamic pole placement problem in the control of linear systems. combinatorial root count; control theory; dynamic pole placement problem; enumerative geometry; Grassmannian; numerical Schubert calculus; Pieri homotopy; quantum Schubert calculus Huber B., SIAM J. Control Optim. 38 (4) pp 1265-- (2000) Enumerative problems (combinatorial problems) in algebraic geometry, Pole and zero placement problems, Numerical computation of solutions to systems of equations, Symbolic computation and algebraic computation, Geometric methods, Grassmannians, Schubert varieties, flag manifolds Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control | 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 To every compact hyperelliptic Riemann surface \(y^2=P(x)\) of genus \(g\) there corresponds the family of double indexed (hyperelliptic) Klein functions \(\wp_{ik}\) where indexes range from \(1\) to \(g\). For \(P(x)\) the polynomial of the third degree we have \(g=1\) and the Klein function \(\wp_{11}\) reduces to the Weierstrass \(\wp\)-function meeting the famous identity \(\wp^{\prime 2} = 4\wp^3 - g_2 \wp - g_3\).
The Klein functions also meet differential identities. Under the normalised condition when one of the branch points of the hyperelliptic cover is moved to the infinity, i.e. \(P(x)\) is of odd degree, many such identities were derived by Baker in early 1900s. Baker also derived the only known until recently covariant identities, i.e. without assuming the normalised condition, considering the case \(g=2\). Recently the theory of such relations was revived by Buchstaber, Enolski, and Leikin who started to study them in the framework of the theory of integrable systems.
In this article the author proposed a method for defining the Klein functions in a covariant way and then deriving the differential identities for them also in the covariant form. Therewith he recovers the known identities for \(g \leq 3\). The new methodology is briefly sketched in the introduction as follows:
``The fundamental observation is that the underlying algebraic curves belong to generic families permuted under \(sl_2\) action... This means that each polynomial identity between derivatives of the \(\wp\)-function belongs to a finite-dimensional representation of \(sl_2\), the knowledge of which depends only upon a highest weight element. It is only necessary to find these highest weight identities to generate the other identities in the representation.'' hyperelliptic Riemann surface; Weierstrass \(\wp\)-function; Klein functions C. Athorne, ''Identities for Hyperelliptic -Functions of Genus One, Two and Three in Covariant Form,'' J. Phys. A: Math. Theor. 41(41), 415202 (2008). Relationships between algebraic curves and integrable systems, Jacobians, Prym varieties, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Identities for hyperelliptic \(\wp\)-functions of genus one, two and three in covariant form | 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 \(\mathcal F(V)\) be the full flag manifold of a vector space \(V\). For a nilpotent endomorphism \(x\) of \(V\) the Springer fiber is
\[
\mathcal F_x= \{(\{0\}=V_0, V_1, \ldots, V_n) \in \mathcal F \mid x(V_i) \subseteq V_{i-1}, \forall i = 1, \ldots, n\}.
\]
To each flag \(\xi \in \mathcal F_x\), Spaltenstein has associated a standard Young tableau \(\sigma_{\xi}\). He showed that the set \(\mathcal F_{x,\sigma}\) of \(\xi\) such that \(\sigma_{\xi}=\sigma\) is a non-singular open irreducible subvariety of \(\mathfrak F_x\). Moreover, the irreducible components of \(\mathcal F_x\) are the closures \(\mathcal F_{\sigma}\) of the \({\mathcal F}_{x, \sigma}\). This paper achieves the following:
(1) It shows that for any standard Young tableau \(\sigma\) of shape \(\lambda\)
\[
\mathcal F_{\sigma} \subseteq \bigcup_{\gamma \preceq \sigma} \mathcal F_{x, \gamma},
\]
where \(\preceq\) is an order previously introduced by the first author;
(2) it shows that the Schubert cell \(\mathcal C_{w_{\sigma}}\) of \(\mathcal F\) intersecting \(\mathcal F_{\sigma}\) in an open dense subset is labelled by the permutation
\[
w_{\sigma} = RS(\sigma_{\max}, \sigma^{\vee}),
\]
where \(\sigma_{\max}\) is the maximal with respect to \(\preceq\) standard Young tableau of shape \(\lambda\), \(^{\vee}\) is the Schützenberger involution, \(RS\) is the Robinson-Schensted correspondence, and the Schubert cell is taken with respect to basis compatible with the Jordan's form of \(x\);
(3) it goes on to show that in the notation above \(\mathcal C_{w_{\sigma}} \cap \mathcal F_{x} \subseteq \mathcal F_{x, \sigma}\);
(4) it gives a sufficient condition for smoothness of \(\mathcal F_{\sigma}\) in terms of \(k\)-adjacency to the so called Richardson components of \(\mathcal F_x\);
(5) it shows that the map \(\sigma \mapsto w_\sigma\) gives an embedding of the adjacency graph of \(\lambda\) into Bruhat order, or as authors call it Bruhat graph of the symmetric group. Robinson-Schensted correspondence; Schubert cell; Spaltenstein correspondence; Bruhat order Pagnon N.G.J., Ressayre N.: Adjacency of Young tableaux and the Springer fibers. Selecta Math. (N.S) 12, 517--540 (2006) Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory Adjacency of Young tableaux and the Springer fibers | 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{I. N. Bernstein}, \textit{I. M. Gel'fand} and the reviewer [Usp. Mat. Nauk 28, No.3(171), 3-26 (1973; Zbl 0286.57025)], described a connection between two approaches to the study of cohomology of the flag variety X of a connected complex reductive Lie group G: one related to Schubert cells in X and another related to Chern classes of line bundles on X. In the present paper the author generalizes these approaches and the above connection to the flag varieties of Kac-Moody groups. cohomology of flag variety; Schubert cells; Chern classes of line bundles; flag varieties of Kac-Moody groups Eugene Gutkin, Schubert calculus on flag varieties of Kac-Moody groups, Algebras Groups Geom. 3 (1986), no. 1, 27 -- 59. Infinite-dimensional Lie groups and their Lie algebras: general properties, Grassmannians, Schubert varieties, flag manifolds, Homology and cohomology of homogeneous spaces of Lie groups Schubert calculus on flag varieties of Kac-Moody groups | 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 a semi-simple algebraic group \(G\) on \(\mathbb C\), the affine Weyl group \(\widehat W\) is defined as \(\widehat W = N(\mathbb C[t,t^{-1}])/T\), where \(T\) is a maximal torus in \(G\) and \(N\) is the normalizer of \(T\). Elements \(w\in \widehat W\) define \textit{affine Schubert varieties} \[ X(w)= \mathop{\dot{\bigcup}}_{\tau \leq w} \mathcal{B}\tau\mathcal{B} \pmod{\mathcal{B}}, \] where \(\mathcal{B}\) is the inverse image of a Borel subgroup \(B\supset T\) in the projection \(G(\mathbb C[[t]])\to G\). These varieties \(X(w)\) are indeed projective. In the paper [Transform. Groups 21, No. 2, 519--530 (2016; Zbl 1390.14148)], the first author showed that there is an embedding of the cotangent bundle \(T^*G_{d,n}\) of a Grassmannian \(G_{d,n}\) into a Schubert variety \(X(\iota)\) (over \(G=\mathrm{SL}(n)\)), that provides a compactification of the cotangent bundle. The result has been generalized by the authors to the cotangent bundles of cominuscule Grassmannian varieties \(G/P\).
The aim of the paper under review is to explore extensions of the previous embeddings to the contangent bundle of quotients \(G/B\), where \(B\) is a Borel subgroup. The authors indeed determine an embedding of \(T^*G/B\) into a suitable variety \(X(\kappa_0)\), which identifies a certain \(\mathrm{SL}_n\)-stable closed subvariety of \(X(\kappa_0)\) as a compactification of \(T^*G/B\). On the other hand, the authors prove that one cannot realize an affine Schubert variety in \(\widehat{\mathrm{SL}}_3/\mathcal B\) as a compactification of \(T^*\mathrm{SL}_3/B\). flag varieties Grassmannians, Schubert varieties, flag manifolds Cotangent bundle to the flag variety. I | 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 a smooth complete toric variety \(X\) given by a fan \(\Sigma\), with \(A\) the set of primitive generators of \(\Sigma\) and \(W_\Sigma\) the Laurent potential, the pair \(((C^*)^n, W_\Sigma)\) is a mirror to the toric variety \(X\), when \(X\) is Fano. \(W_\Sigma\) encodes relevant information to homological mirror symmetry (HMS). First the author reviews some existing results on this topic (HMS theorems for toric varieties). Minor symmetry for toric varieties involves Laurent polynomials (the symplectic topology is related to algebraic geometry). The author shows the existence of a monodromy action on the monomially admissible Fukaya-Seidel categories of these Laurent polynomials (the variation of coefficients corresponds, under homological mirror symmetry, to tensoring by a line bundle associated to the monomials whose coefficients are related). Also included is an example of HMS for ample line bundles. In addition, the author introduces the monomially admissible Fukaya-Seidel category as a new interpretation of the Fukaya-Seidel category for a Laurent polynomial on \((C^*)^n\). This has potential applications and provides the evidence of homological mirror symmetry for non-compact toric varieties. Concluding the paper, in Appendix A, the author shows that the localization approach for defining the Fukaya-Seidel categories is equivalent to the approach using \(A_\infty\)-pre-categories. homological mirror symmetry; Fukaya-Seidel categories; toric mirror symmetry Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric 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 We consider Buch's rule for \(K\)-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Möbius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong. Grassmannian; Richardson varieties; Grothendieck polynomials; Schur multiplicity free Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] A combinatorial approach to multiplicity-free Richardson subvarieties of 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 We give a new proof to V. B. Mehta and A. Ramanathan's theorem that the Schubert subschemes in a flag scheme are all simultaneously compatible split, using the representation theory of infinitesimal algebraic groups. In particular, the present proof dispenses with the Bott-Samelson schemes. Schubert subschemes; flag scheme; representation theory of infinitesimal algebraic groups Kaneda, M., On the Frobenius morphism of flag schemes, Pacific J. Math., 163, 315-336, (1994) Grassmannians, Schubert varieties, flag manifolds, Other algebraic groups (geometric aspects) On the Frobenius morphism of flag 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 [For the entire collection see Zbl 0682.00007.]
Let B be a Borel subgroup and \(T\subset B\) a maximal torus of a semisimple algebraic group. For each element w of the Weyl group W we have a Schubert variety \(X_ w\) in G/B and if \(\lambda\) is a character of T we let \(H^ 0_ w(\lambda)\) denote the sections over \(X_ w\) of the line bundle on G/B induced by \(\lambda\). An excellent filtration of a B-module is then a filtration with quotients of the form \(H^ 0_ w(\lambda)\), \(w\in W\), \(\lambda\) a dominant character of T.
The author uses the normality of \(X_ w\) to prove that \(H^ 0_ w(\lambda)\) is the injective hull of \(w\lambda\) in a certain category of B-modules. This nice characterization has been used by \textit{W. van der Kallen} [Math. Z. 201, No.1, 19-31 (1989; Zbl 0642.20037)] to carry out a systematic investigation of modules with excellent filtrations [resp., more general (so-called Schubert) filtrations]. In turn, the work of van der Kallen comes into the proof of the second main result of the present paper. This result states that in characteristic 0 the B-module obtained by tensoring any finite-dimensional G-module by a dominant character has an excellent filtration. If the characteristic p is positive the analogous result is obtained (for G-modules with good filtrations) provided p is not too small (this restriction on p has recently been removed by further work of the author and O. Mathieu). In particular, this answers a question of A. Joseph about whether certain key tensor products have excellent filtrations.
Finally, the author also obtains a result which includes a verification of the PRV-conjecture for type A. Soon afterwards this conjecture was proved in full generality by \textit{S. Kumar} [Invent. Math. 93, No.1, 117- 130 (1988; Zbl 0668.17008)]. Borel subgroup; maximal torus; semisimple algebraic group; Weyl group; Schubert variety; line bundle; dominant character; excellent filtrations; good filtrations; PRV-conjecture P. Polo : Variétés de Schubert et excellentes filtrations . Preprint (1987). Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Cohomology theory for linear algebraic groups Variétés de Schubert et excellentes filtrations. (Schubert varieties and excellent filtrations.) | 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 Spiral Schubert varieties are conjecturally the only Schubert varieties in type \(\widetilde{A}_{2}\) for which rational smoothness at a torus-fixed point is not detected by the number of torus-invariant curves passing through that point. In this paper we determine the locus of smooth points of a spiral Schubert variety of type \(\widetilde{A}_{2}\). This continues the study begun in [\textit{W. Graham} and \textit{W. Li}, ``The Bruhat order, the lookup conjecture and spiral Schubert varieties of type \(\widetilde{A_2}\)'' (submitted)], where the locus of rationally smooth points was determined. The main result describes the smooth locus in terms of the action of the Weyl group on \(\mathbb{R}^{2}\); using this result, we identify the maximal singular points of these varieties. We make key use of the results of [loc. cit.] relating the Bruhat order to the Weyl group action on \(\mathbb{R}^{2}\). Schubert variety; spiral; Weyl group; Bruhat order; smoothness Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus The smooth loci of spiral Schubert varieties of type \(\widetilde{A}_{2}\) | 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 Let \(k\) be a field, and let \(X\) be a separated, regular, Noetherian \(k\)-scheme of finite Krull dimension. Let \(\pi: \widetilde{X} \to X\) be the blowing-up of \(X\) along a regular equi-codimensional closed subscheme \(Y\), and set \(r\) to be the codimension of \(Y\) in \(X\). Assume \(r \geq 2\). There is a well-known theorem which relates the bounded derived category \(D^b(\text{Coh}(\widetilde{X}))\) to \(D^b(\text{Coh}(X))\) and \(D^b(\text{Coh}(Y))\): roughly, \(D^b(\text{Coh}(\widetilde{X}))\) is a semi-orthogonal decomposition of \(D^b(\text{Coh}(X))\) and \(r - 1\) copies of \(D^b(\text{Coh}(Y))\) (see Theorem 3.4 of the article under review for the precise statement). The main goal of this article is to formulate and prove an analogue of this theorem for categories of matrix factorizations.
Let \(W \in \Gamma(X, \mathcal{O}_X)\), and denote by \(\text{MF}(X, W)\) the triangulated category of \textit{matrix factorizations} of \(W\) over \(X\). In Section 2 of the article under review, the authors provide a comprehensive background on matrix factorization categories in the global setting, including detailed discussions of their dg enhancements and derived functors between them. Section 3 is devoted to a proof of their main theorem, which states, roughly, that \(\text{MF}(\widetilde{X}, W)\) is a semi-orthogonal decomposition of \(\text{MF}(X, W)\) and \(r-1\) copies of \(\text{MF}(Y, W)\) (here, we abuse notation slightly by denoting the pullbacks of \(W\) to \(\widetilde{X}\) and \(Y\) also by \(W\)); see Theorem 3.5 of the article for the precise statement of the main theorem.
The authors also obtain a semi-orthogonal decomposition of the category \(\text{MF}(E, W)\) of matrix factorizations associated to a projective bundle \(E\) over \(X\) (Theorem 3.2), and they discuss several applications of the main theorem (Section 3.3). matrix factorization; semi-orthogonal decomposition; blowing-up; projective space bundle; dg enhancement Lunts, V. A.; Schnürer, O. M., Matrix factorizations and semi-orthogonal decompositions for blowing-ups, J. Noncommut. Geom., 10, 907-979, (2016) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Differential graded algebras and applications (associative algebraic aspects) Matrix factorizations and semi-orthogonal decompositions for blowing-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 We give functorial recipes to get, out of any Hopf algebra over a field, two pairs of Hopf algebras bearing some geometrical content. If the ground field has characteristic zero, the first pair is made by a function algebra \(F[G_+]\) over a connected Poisson group and a universal enveloping algebra \(U({\mathfrak g}_-)\) over a Lie bialgebra \({\mathfrak g}_-\): in addition, the Poisson group as a variety is an affine space, and the Lie bialgebra as a Lie algebra is graded; apart from these last details, the second pair is of the same type, namely \((F[G_-],U({\mathfrak g}_+))\) for some Poisson group \(G_-\) and some Lie bialgebra \({\mathfrak g}_+\). When the Hopf algebra \(H\) we start from is already of geometric type the result involves Poisson duality: the first Lie bialgebra associated to \(H=F[G]\) is \({\mathfrak g}^*\) (with \({\mathfrak g}:=\text{Lie}(G)\)), and the first Poisson group associated to \(H=U(\mathfrak g)\) is of type \(G^*\), i.e. it has \(\mathfrak g\) as cotangent Lie bialgebra. If the ground field has positive characteristic, then the same recipes give similar results, but for the fact that the Poisson groups obtained have dimension 0 and height 1, and restricted universal enveloping algebras are obtained. We show how all these ``geometrical'' Hopf algebras are linked to the original one via 1-parameter deformations, and explain how these results follow from quantum group theory. Hopf algebras; function algebras; connected Poisson groups; universal enveloping algebras; Lie bialgebras; deformations Gavarini, F.: The global quantum duality principle. To appear, 2004 Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Universal enveloping (super)algebras, Affine algebraic groups, hyperalgebra constructions, Universal enveloping algebras of Lie algebras, Lie bialgebras; Lie coalgebras, Poisson algebras, Quantum groups and related algebraic methods applied to problems in quantum theory The crystal duality principle: from general symmetries to geometrical symmetries. | 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 nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby, in amenable cases, nonvanishing is in the complexity class (NP intersection coNP) of problems with ``good characterizations''. This suggests a new algebraic combinatorics viewpoint on complexity theory. This paper focuses on the case of Schubert polynomials. These form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. We give a tableau criterion for nonvanishing, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the \(n \times n\) grid, together with a theorem of \textit{A. Fink} et al. [Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)], which proved a conjecture of \textit{C. Monical} et al. [Sel. Math., New Ser. 25, No. 5, Paper No. 66, 37 p. (2019; Zbl 1426.05175)]. Schubert polynomials; Newton polytopes; computational complexity Combinatorial aspects of representation theory, Symmetric functions and generalizations, Exact enumeration problems, generating functions, Grassmannians, Schubert varieties, flag manifolds, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) Computational complexity, Newton polytopes, and Schubert 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 Given positive integers \(m\) and \(p\) the map that associates homogeneous polynomials \(f_1,\dots ,f_m\) of degree \(m+p-1\) to its Wronskian defines a finite map between Grassmannian varieties
\[
Wr: Gr(m, \mathbb{C}_{m+p-1}[t])\to Gr(1,\mathbb{C}_{mp}[t])=\mathbb{P}(\mathbb{C}_{mp}[t]).
\]
Eisenbud and Harris have computed explicitly the degree \(d_{m,p}\) of this map in the work [\textit{D. Eisenbud} and \textit{J. Harris}, Invent. Math. 74, 371--418 (1983; Zbl 0527.14022)]. When one changes the base field from \(\mathbb{C}\) to \(\mathbb{R}\) the problem of computing the degree \(d_{m,p}'\) of \(Wr\) is known as inverse Wronski problem.
The authors show that \(d_{m,p}\) and \(d_{m,p}'\) are congruent modulo four. They do this using a general framework for congruences modulo four in Schubert calculus which they develop in the present paper. Schubert; Wronskian; Grassmanian \textsc{N.~Hein, F.~Sottile, and I.~Zelenko}, \textit{A congruence modulo four in real Schubert calculus}, J. Reine Angew. Math., 714 (2016), pp.~151-174. Classical problems, Schubert calculus, Semialgebraic sets and related spaces A congruence modulo four in real 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 polynomial of \(n\) variables and \(n\) monomials is determined by an \((n\times n)\)-matrix of exponents. The polynomial is called invertible if the matrix is invertible. Transposing the matrix leads to the Berglund-Hübsch dual. In this note the equivariant monodromy zeta-function for such polynomials is studied. For a finite symmetry group \(G\) this function is an element of the Burnside ring \(K_0(\mathrm{f.}G\mathrm{-sets})\), that is, the Grothendieck ring of finite \(G\)-sets. Dual polynomials have dual zeta-functions, where duality is to be understood in the sense of Saito duality. It can be regarded as a Fourier transformation on Burnside rings.
Finally there is an application of the present results to the \` geometric roots\'\ considered in a previous paper of the authors [Mosc. Math. J. 11, No. 3, 463--472 (2011; Zbl 1257.32028)] Saito duality; burnside ring; invertible polynomial; monodromy; equivariant zeta-function W. Ebeling and S. M. Gusein-Zade, ''Saito duality between Burnside rings for invertible polynomials,'' Bull. London Math. Soc., 44:2 (2012), 814--822. Complex surface and hypersurface singularities, Mirror symmetry (algebro-geometric aspects), Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Frobenius induction, Burnside and representation rings Saito duality between Burnside rings for 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 Let \(\Theta\) be an arbitrary variety of algebras over an infinite field \(P.\) For every \(H\in \Theta\) the author considers two categories: the category \(K_{\Theta}(H)\) of algebraic sets over \(H\) and the category \({\widetilde K}_{\Theta}(H)\) of algebraic varieties over \(H.\) The both categories are geometric invariants of the algebra \(H\) and are in many senses responsible for the geometry in \(H\) (the classical algebraic geometry over the field \(P\) is associated with the variety Com-\(P\) of all commutative and associative algebras with unity over \(P\)). Main questions the paper: When are the categories \(K_{\Theta}(H_{1})\) and \(K_{\Theta}(H_{2})\) isomorphic and the same for the categories \({\widetilde K}_{\Theta}(H_{1})\) and \({\widetilde K}_{\Theta}(H_{2})?\) The author answers the first question in the category Com-\(P\):
The categories \(K_{\Theta}(H_{1})\) and \(K_{\Theta}(H_{2})\) are correctly isomorphic if and only if there exists such \(H\) that \(H\) and \(H_{1}\) are semiisomorphic and \(H\) and \(H_{2}\) are geometrically equivalent. infinite field; variety of algebras; category of algebraic sets; category of algebraic varieties B. Plotkin, Ukr. Math. J. 54(6), 1019 (2002). Varieties and morphisms, Varieties, Noncommutative algebraic geometry, Local categories and functors Some problems in nonclassical algebraic 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 The author shows how the multigraded Hilbert scheme construction of \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] can be used to construct a quasi-projective scheme which parametrize left homogeneous ideals in the Weyl algebra having fixed Hilbert function. Fixing an integral domain \(k\) of characteristic zero, the Weyl algebra \(W=k \langle x_1, \dots, x_n, \partial_1, \dots, \partial_n \rangle\) has a \(k\)-basis consisting of the set \(\mathcal B = \{ x^\alpha \partial^\beta | \alpha, \beta \in \mathbb N^n \}\). If \(A\) is an abelian group, then any \(A\)-grading \(\mathbb N^n \to A\) on the polynomial ring \(S = k[x_1,\dots,x_n]\) extends to an \(A\)-grading \(\mathbb N^{2n} \to A\) on \(W\) by \(\deg (x^\alpha \partial^\beta) = \deg \alpha - \deg \beta\), which induces a decomposition \(W = \bigoplus_{a \in A} W_a\). Given a Hilbert function \(h:A \to \mathbb N\), the corresponding Hilbert functor \(H^h_W\) takes a \(k\)-algebra \(R\) to the set of homogeneous ideals \(I \subset R \otimes_k W\) such that \((R \otimes _k W_a)/I_a\) is a locally free \(R\)-module of rank \(h(a)\) for each \(a \in A\). The main theorem says that \(H^h_W\) is representable by a quasi-projective scheme over \(k\).
The strategy of the proof is similar to that of Haiman and Sturmfels [loc. cit.], but there are some new behaviors regarding monomials in the Weyl algebra \(W\) not seen in the polynomial ring \(S\). An obvious difference is that a product of monomials in \(W\) need not be a monomial. Another difference is that \(W\) has infinite antichains of monomial ideals, unlike the polynomial case: see work of \textit{D. MacLagan} [Proc. Am. Math. Soc. 129, 1609--1615 (2001; Zbl 0984.13013)]. Moreover, the natural extension of Gröbner basis theory for \(S\) to \(W\) does not work well, so the author considers the initial ideal of a left ideal in the associated graded algebra \(\text{gr} W\) and uses Gröbner basis theory for \(W\) developed by \textit{M. Saito} et al. [Gröbner deformations of hypergeometric differential equations. Berlin: Springer (2000; Zbl 0946.13021)]. The arguments are well presented along with examples showing the novel points. Hilbert schemes; Weyl algebras Parametrization (Chow and Hilbert schemes), Rings of differential operators (associative algebraic aspects) Multigraded Hilbert schemes parametrizing ideals in the Weyl 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 paper under review makes precise the relationship between the
Lagrangian subvarieties of the Calogero-Moser space arising in
the representation theory of rational Cherednik algebras of type
\(A\) at \(t=0\) and Schubert cells in the adelic Grassmannian.
It is known, due to \textit{P. Etingof} and \textit{V. Ginzburg} [Invent. Math. 147, No. 2, 243--348 (2002; Zbl 1061.16032)], that the center of the
rational Cherednik algebra of type \(A\), at \(t=0\), is isomorphic to
the coordinate ring of Wilson's completion of the Calogero-Moser phase
space. The paper under review proves that this isomorphism is
compatible with the factorization property of both of these spaces.
Consequently, the space of homomorphisms between certain
representations of the rational Cherednik algebra is identified with
functions on the intersection of some Schubert cells. rational Cherednik algebras; Calogero-Moser space; Schubert calculus; representation theory; adelic Grassmanian Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Deformations of associative rings, Poisson algebras Rational Cherednik algebras and 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 The author proves that Whittaker functions associated to \(\text{GL}(N)\), using its Mellin-Barnes integral representation satisfy a set of new difference equation,
\[
x^{(k)} w(\lambda,q)= (-1)^{k(N- 1)}\exp(q_N+ q_{N-1}+\cdots+ q_{N- k+1}w(\lambda q).
\]
Finally, it is pointed out that these operators \(x^{(k)}\) are limiting cases of an earlier result [cf. \textit{P. Etingof, O. Schiffmann}, and \textit{A. Varchenko}, Traces of intertwiners for quantum groups and difference equations. II. Quantum Algebra QA/0207157 (2002)]. quantum algebra; exactly solvable and integrable systems; Whittaker functions; Mellin-Barnes integral representation; difference equation O. Babelon, Lett. Math. Phys., 65, 229--246 (2003). Difference operators, Groups and algebras in quantum theory and relations with integrable systems, Relationships between algebraic curves and integrable systems, Lattice dynamics; integrable lattice equations, Applications of hypergeometric functions Equations in dual variables for Whittaker 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 Let \(X=G/B\) be a generalized flag manifold, where \(G\) is a semisimple complex algebraic group and \(B\) a Borel subgroup. Let \(W\) be the Weyl group and \(X_w\) be the Schubert variety corresponding to \(w\in W\). There is a surjective homomorphism in cohomology \(H^*(G/B)\to H^*(X_w)\), whose kernel \(I_w\) is generated by the set \(\{\sigma_u\mid u\not\leq w\}\): here \(\sigma_u\) denotes the Kronecker dual of the homology class \([X_u]\) and the considered order is the strong Bruhat order.
In the article under review the authors look for a more efficient system of generators for \(I_w\). The idea is to extend the Borel presentation of \(H^*(G/B,K)\) (\(K\) a field of characteristic \(0\)) as quotient \(K[V]/K[V]^W_{+}\), where \(V=K\otimes_{\mathbb Z}X(T)\) and \(X(T)\) is the coweight lattice of a maximal torus \(T\subset B\).
The main result is Theorem 1.1, which extends a result proved by \textit{E. Akyildiz, A. Lascoux} and \textit{P. Pragacz} for type \(A\) [J. Differ. Geom. 35, No.3, 511--519 (1992; Zbl 0782.14041)]. It says that \(I_w\) is generated by the cohomology classes \(\sigma_u\), such that \(u\not\leq w\) and \(u\) is Grassmannian: this means that the cardinality of the descent set \(\text{Des}(u)\) is at most \(1\). Moreover, let \(\mathcal E(w)\) be the essential set for \(w\), formed by the elements \(u\in W\) such that \(u\) is minimal among those satisfying the condition \(u\not\leq w\). Then the authors prove that \(I_w\) is generated by the classes \(\sigma_u\) with \(u\) Grassmannian and such that it is possible to find \(v\in \mathcal E(w)\), with \(v\) and \(v^{-1}\) Grassmannian, \(u\geq v\) and \(\text{Des}(u) = \text{Des}(v)\).
The proof makes use of combinatorial arguments. An upper bound for the number of generators of \(I_w\) is deduced. In the case \(A_{n-1}\), this bound is \(2^n\). In this case also a more explicit and smaller system of generators for \(I_w\) is given, and the authors conjecture that it is minimal. Schubert calculus; Schubert variety; cohomology presentation; bigrassmannian; essential set V. Reiner, A. Woo, and A. Yong, ''Presenting the cohomology of a Schubert variety,'' Trans. Amer. Math. Soc. 363 (2011), no. 1, 521--543. Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Presenting the cohomology of a Schubert 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 In the Gross-Siebert mirror symmetry program, certain enumerations of tropical disks are encoded in combinatorial objects called scattering diagrams and broken lines. These, in turn, are used to construct a mirror scheme equipped with a canonical basis of regular functions called theta functions. This paper serves to develop the relationships between tropical disks, scattering diagrams, and broken lines in the quantum setting; for example, we express quantum theta functions in terms of refined enumerations of tropical disks. We apply there tropical descriptions to prove a refined version of the Carl-Pumperla-Siebert (Preprint) result on consistency of theta functions, and also to prove the quantum Frobenius Conjecture of Fock and Goncharov (\textit{Ann. Sci. Éc. Norm. Supér.} (4) 42 (2009) 865--930). tropical curves; scattering diagrams; theta functions; descendent log Gromov-Witten invariants Tropical geometry, Cluster algebras, Enumerative problems (combinatorial problems) in algebraic geometry Scattering diagrams, theta functions, and refined tropical curve counts | 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 Double Hurwitz numbers count covers of the sphere by genus \(g\) curves with assigned ramification profiles over 0 and \(\infty \), and simple ramification over a fixed branch divisor. \textit{I. P. Goulden} et al. [Adv. Math. 198, No. 1, 43--92 (2005; Zbl 1086.14022)] have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification, and \textit{S. Shadrin} et al. [Adv. Math. 217, No. 1, 79--96 (2008; Zbl 1138.14018)] have determined the chamber structure and wall crossing formulas for \(g=0\). We provide new proofs of these results, and extend them in several directions. Most importantly, we prove wall crossing formulas for all genera.
The main tool is the authors' previous work expressing double Hurwitz number as a sum over labeled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of
\textit{A. N. Varchenko} [Funct. Anal. Appl. 21, No. 1--3, 9--19 (1987; Zbl 0626.52003)]. This approach to wall crossing appears novel, and may be of broader interest.
This extended abstract is based on a new preprint by the authors. Hurwitz numbers; lattice points; hyperplane arrangements; graphs Coverings of curves, fundamental group, Symmetric functions and generalizations, Families, moduli of curves (algebraic) Chamber structure for double Hurwitz 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 Das Buch ist die Fortsetzung der beiden ersten Bände (1915, 1918; JFM 46.0941.11) und enthält die Theorie der algebraischen Kurven und Funktionen einer Veränderlichen.
Aus dem Inhalt:
Kap. I. \textit{Die linearen Reihen auf einer Kurve}. Einleitung. Die Involutionen \(g^1_n\) auf einer Kurve. Note über die rationalen Funktionen auf einer Riemannschen Fläche. Die linearen Reihen \(g^r_n\). Die vollständige Reihe und die Normalkurven. Addition und Subtraktion der Reihen. Kurven vom Geschlecht 1. Die Jacobischen Reihen und die kanonische Reihe. Ergänzungen; analytische Entwicklungen und Verallgemeinerungen. Anzahl der aus \(r+1\) Punkten bestehenden gemeinsamen Punktgruppen einer Reihe \(g^r_n\) und einer Reihe \(g^1_m\). Dimension der linearen Reihen; der Riemann-Rochsche Satz. Hyperelliptische Kurven. Kanonische Kurve. Das Problem der speziellen Reihen; die Brill-Noetherschen und die Riemannschen Normaltypen.
Kap. II. \textit{Die Geometrie auf einer ebenen Kurve und die Cremonaschen Transformationen; geschichtliche Entwicklung der Gedanken}. Konstruktion der vollständigen Reihe auf einer ebenen Kurve, die keinen mehrfachen Punkt hat. Die vollständige Reihe auf einer ebenen Kurve mit mehrfachen Punkten; die adjungierte Kurve. Der Riemann-Rochsche Satz; Note über eine neue Entwicklung der Theorie nach Severi. Die Geometrie auf einer Kurve in ihrer geschichtlichen Entwicklung, von der projektiven Geometrie zur Geometrie der birationalen Transformationen. Ebene Cremonasche Transformationen. Invarianten der Kurve in bezug auf ebene Cremonasche Transformationen. Note über die Reduktion der linearen Systeme ebener Kurven vom Geschlecht Null und Eins. Die Bedeutung der Transformationen in der Geometrie der Kurve.
Kap. III. \textit{Kurven und Transformationen}. Transformation der Kurven vom Geschlecht Null. Note über die rationalen Kurven mit einer endlichen Anzahl von projektiven Substitutionen; über die allgemeine Theorie der Raumkurven vierten Grades zweiter Art. Kurven mit unendlich vielen projektiven Transformationen. Transformationen der elliptischen Kurven. Note über die endlichen Transformationsgruppen einer elliptischen Kurve. Anwendung auf die projektive Theorie der Raumkurven vierten Grades erster Art. Nachträge: Die syzygetischen Bündel der Raumkurven vierten Grades erster Art. Transformationen der Kurven vom Geschlecht \(p > 1\) und insbesondere der ebenen Kurven vierten Grades. Note über die arithmetischen Irrationalitäten bei Kurventransformationen; Beiträge zur Zahlentheorie. Die Moduln der Kurven vom Geschlecht \(p\) und der Existenzsatz. Note über die Irreduzibilität der Kurvenfamilie vom Geschlecht \(p\) und über den Lüroth-Clebschschen Satz. Ein neuer Gesichtspunkt der Kurvengeometrie, betreffend die Zählung der Moduln. Die Entartungsmethode und die Abzählungsfragen auf der Kurve. Das Entartungsprinzip und die Geometrie auf Kurven von allgemeinem Modul. Die Zyklen der Riemannschen Flächen vom Geschlecht \(p\) und ihre Entartungen.
Kap. IV. \textit{Korrespondenzen zwischen Kurven}. Mehrfache Kurven. Anwendungen; elliptische mehrfache Kurven ohne Verzweigungsstellen. Das Korrespondenzprinzip auf einer Kurve. Anwendungen des Korrespondenzprinzips. Dreifach- und Vierfachschneidende einer Raumkurve. Die irrationalen Involutionen auf einer Kurve; die Sätze von Painlevé und von Castelnuovo-Humbert. Abzählendes Kriterium dafür, daß eine Reihe von Punktgruppen in einer linearen Reihe enthalten sein soll. Die Korrespondenzen auf einer Kurve von allgemeinem Modul. Die singulären Korrespondenzen.
Kap. V. \textit{Über die Theorie der Raumkurven}. Analytische Darstellung. Parameter, von welchen eine Familie von Raumkurven abhängt; Klassifikationsprobleme. Vollständige Durchschnitte zweier Flächen. Raumkurven, welche unvollständige Durchschnitte zweier Flächen sind; Äquivalenzformeln und kanonische Reihe. Postulationsformel; Kurve vom maximalen Geschlecht. Auf einem Monoid von gegebener Ordnung gezogene Raumkurve. Die ebene Darstellung der kubischen Fläche. Auf einer kubischen Fläche gezogene Raumkurven. Bemerkungen über die Einteilung der Raumkurven. Zusätze. (V 5 E.)
Besprechung: L. Godeaux. Bulletin sc. math. (2) 49, 97-100. Research exposition (monographs, survey articles) pertaining to algebraic geometry Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. Vol. III. Pubblicate per cura del O. Chisini. | 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 complex semisimple simply-connected algebraic group and let \(T\) be a maximal torus of \(G\), \(B\) a Borel subgroup of \(G\) containing \(T\) and \(W\) the Weyl group of \(G\). Let \(F = G/B\) be the full flag variety and let \(X_w\subset F\) be the Schubert variety, for any \(w\in W\). In the paper under review, the authors prove that if every irreducible component of \(G\) is of type \(B_n\) or \(C_n\), and \(w_1, w_2\) are two distinct involutions in \(W\), then the tangent cones at the base point \(eB\in X_{w_i}\) to the corresponding Schubert subvarieties \(X_{w_1} , X_{w_2}\) in \(F\) do not coincide as subschemes of the tangent space \(T_{eB}(F)\). Similarly, they also show that if every irreducible component of \(G\) is of type \(A_n\) or \(C_n\), then the reduced tangent cones to \(X_{w_1} , X_{w_2}\) do not coincide as subvarieties of \(T_{eB}(F)\). Their first result is proved by using (what they call) Kostant-Kumar polynomials (so was the following result by Eliseev and Ignatyev). Their second result is proved by using a connection between the tangent cones of \(X_w\) and the geometry of coadjoint orbits of \(B\). In a previous work [J. Math. Sci., New York 199, No. 3, 289--301 (2014); translation from Zap. Nauchn. Semin. POMI 414, 82--105 (2013; Zbl 1312.14116)], \textit{D. Yu. Eliseev} and \textit{M. V. Ignatyev} had proved that \(X_{w_1} , X_{w_2}\) (for distinct involutions \(w_1, w_2\)) do not coincide as subschemes of the tangent space \(T_{eB}(F)\) when the irreducible components of \(G\) are of type \(A_n\), \(F_4\) and \(G_2\) only (partially confirming an earlier conjecture by Panov in 2011 for any \(G\)). flag variety; Schubert variety; tangent cone; reduced tangent cone; involution in the Weyl group; Kostant-Kumar polynomial Bochkarev, M; Ignatyev, M; Shevchenko, A, Tangent cones to Schubert varieties in types \(A_n\), \(B_n\) and \(C_n\), J. Algebra, 465, 259-286, (2016) Grassmannians, Schubert varieties, flag manifolds, Root systems, Classical groups (algebro-geometric aspects) Tangent cones to Schubert varieties in types \(A_{n}\), \(B_{n}\) and \(C_{n}\) | 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 Deodhar introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan-Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context of soliton solutions to the KP equation, as shown by Kodama and the second author. Deodhar components \(S_D\) of the Grassmannian are in bijection with certain tableaux \(D\) called Go-diagrams, and each component is isomorphic to \((\mathbb K^*)^a \times (\mathbb K)^b\) for some non-negative integers \(a\) and \(b\). Our main result is an explicit parameterization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram \(D\) we construct a weighted network \(N_D\) and its weight matrix \(W_D\), whose entries enumerate directed paths in \(N_D\). By letting the weights in the network vary over \(\mathbb K\) or \(\mathbb K^*\) as appropriate, one gets a parameterization of the Deodhar component \(S_D\). One application of such a parameterization is that one may immediately determine which Plücker coordinates are vanishing and nonvanishing, by using the Lindstrom-Gessel-Viennot Lemma. We also give a (minimal) characterization of each Deodhar component in terms of Plücker coordinates. Grassmannian; Deodhar decomposition; networks Talaska, K.; Williams, L.: Network parameterizations of grassmannians, Algebra number theory (2013) Combinatorial aspects of representation theory, Small world graphs, complex networks (graph-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds Network parameterizations 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 An important task in the study of a homogeneous varieties is that of describing the singular locus of its Schubert varieties. There are many results regarding such a problem, but also many aspects needing further investigations. Recall that a homogeneous variety, i.e. a complete variety of the form \(G/P\), where \(P\) is a parabolic subgroup of an algebraic group \(G\), is said of type \(A\) if its Weyl group \(W_P\) is of type \(A_{n}\) for some positive integer \(n\). In case of homogeneous spaces of type \(A\) there are several results regarding the singular locus of Schubert varieties and many guesses and/or conjectures have been formulated in the recent literature.
The paper under review is concerened with a recent conjecture by Woo and Yong regarding the Gorenstein locus of a Schubert variety \(X(w)\) of a type \(A\) homogeneous space. Recall that a point of an algebraic variety is Gorenstein if and only if its local ring is Gorenstein, i.e. if it has finite injective dimension when thought of as a module over itself. Gorenstein singularities are milder than other singularities, because they simulate many behaviours occurring for smooth varieties. The conjecture by Yong and Woo claims that the non-Gorenstein locus of \(X(w)\) is the union of irreducible components whose generic point is non Gorenstein. This amounts to say, in other words, that once one knows the irreducible components of the singular locus, one has enough information to detect the Gorenstein locus.
The author studies the aforementioned conjecture by using a combinatorial tool introduced in [\textit{N.~Perrin}, Compos. Math. 143, No. 5, 1255--1312 (2007; Zbl 1129.14069)]. In fact it is possible to associate to each minuscule Schubert variety a quiver which generalizes Young diagrams. The key notion to be defined is that of holes, virtual holes and essential holes. A quiver is a graph, and a hole is any vertex satisfying certain properties, which is not the case to explain here. However, even in the introduction the author makes very nice and illustrative examples already for the case of grassmannians, the most popular kind of minuscule homogeneous space.
This allows the author to prove the aforementioned conjecture by Yong and Woo for minuscule Schubert varieties. It is indeed a consequence of the following important theorem, the main result of the paper: The generic point of a Schubert subvariety \(X(w')\) of a minuscule Schubert variety \(X(w)\) is in the Gorentsein locus if and only if the quiver of \(X(w')\) contains all the non non-Gorentein holes of the quiver \(X(w)\).
The paper is very well written and very well organized, and the way it adresses to the readers is quite friendly, in the usual author's style. Not only in Section 1 there are all the basics one needs to follow the path drawn by the author, but Section 2 shows through examples how to obtain informations from the quivers associated to Schubert varieties, including the use of the holes introduced in Definition 1.8. Very interesting is the description of the relative canonical model of a Schubert variety in terms of quivers: this is done in Section 3, the final section of the paper, which is richely equipped with illuminating examples as well. minuscule Schubert varieties; singularities; Gorenstein locus Perrin, Nicolas, The {G}orenstein locus of minuscule {S}chubert varieties, Advances in Mathematics, 220, 2, 505-522, (2009) Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds The Gorenstein locus of minuscule 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 This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Gröbner bases) and geometry (via quiver theory). Gröbner bases serve as effective models for computation in algebras of various types. Although the theory of Gröbner bases was developed in the second half of the 20-th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced -- with big impact -- in the 1990s.
Divided into two parts, the book first discusses the theory of Gröbner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Gröbner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line.
While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.
The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to associative rings and algebras, Proceedings, conferences, collections, etc. pertaining to statistics, Collections of articles of miscellaneous specific interest, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Vector bundles on curves and their moduli, Geometric invariant theory, Group actions on varieties or schemes (quotients), Free, projective, and flat modules and ideals in associative algebras, Representations of quivers and partially ordered sets, Quadratic and Koszul algebras, Theories (e.g., algebraic theories), structure, and semantics, 2-categories, bicategories, double categories, Resolutions; derived functors (category-theoretic aspects), Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain, Other special methods applied to PDEs, Momentum maps; symplectic reduction, Exterior differential systems (Cartan theory), Grammars and rewriting systems Two algebraic byways from differential equations: Gröbner bases and 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 Let \(G\) be a finite graph with vertex set \(V(G)\), edge set \(E(G)\), and Betti numbers \(b_0(G)\) and \(b_1(G)\). The Kirchhoff polynomial of \(G\) is a multivariate polynomial defined as \(P_G= \sum_T \prod_{e\not\in T} x_e\), where the sum is taken over all spanning trees \(T\) of \(G\), the product is over all edges of \(G\) not in \(T\), and the \(x_e\) are indeterminates, one for each edge \(e\in E(G)\). Kirchhoff polynomials of graphs first appeared in the 19th century, mainly in the analysis of electrical circuits, and they were intensively studied by G. Kirchhoff, C. Maxwell, C. Borchardt, J. Sylvester, and others, which explains the terminology. In contemporary mathematics and physics, Kirchhoff polynomials also play an important role in the evaluation of Feynman amplitudes in the following sense. Let \(V(P_G)\) be the scheme of zeros of \(P_G\) over \(\mathbb{Z}\), i.e., a hypersurface in the affine scheme \(\mathbb{A}^{E(G)}\), and let \(Y_G\) denote its complement. Feynman amplitudes are then related to period integrals on the schemes \(Y_G\), and this relates Kirchhoff polynomials not only to combinatorics, but also to arithmetic algebraic geometry and quantum field theory. Some years ago, in 1997, M. Kontsevich made a conjecture about the number of points of \(Y_G\) over a finite field, which also has attracted the attention of many leading combinatorialists ever since. Kontsevich's conjecture states that, for every finite graph \(G\), the function \(\# Y_G(\mathbb{F}_q)\) from the set of all prime powers to \(\mathbb{Z}\) is a polynomial in \(q\), i.e., \(Y_G\) is always polynomially countable. This conjecture is obviously equivalent to the conjecture that \(V(P_G)\) in a polynomially countable \(\mathbb{Z}\)-scheme.
Although Kontsevich's conjecture has been verified for special graphs, and partial results have been obtained by combinatorialists, a complete solution of it seemed to be far away.
However, in the utmost fundamental paper under review, the authors disprove the general statement of Kontsevich's conjecture in a stunning way. Namely, instead of exhibiting an explicit combinatorial counterexample, they show that the functions \(\#Y_G(\mathbb{F}_q)\) span a space that includes non-polynomial functions, and that the schemes \(Y_G\) are, with respect to their zeta functions, the most general schemes possible. To this end, the authors use R. Stanley's reformulation of Kontsevich's conjecture [cf.: \textit{R. P. Stanley}, Ann. Comb. 2, No. 4, 351--363 (1998; Zbl 0927.05087)], together with his partial results in this direction, and develop then a framework of combinatorial motives that allows to interpret, and thereby to extent, Stanley's approach in an algebro-geometric setting. Their main theorem states that a certain ring, called the ring of combinatorial motives, equals a certain module over a particular principal ideal domain \(R\), which is (over \(R\)) generated by all functions of the form \(\#Y_G(\mathbb{F}_q)\), where \(q\) runs over all prime powers. The main theorem does not only imply that Kontsevich's conjecture is generally false, which merely appears as a by-product, but the ingredients of its proof give rise to a much more general theory. Namely, the authors show that their proof extends to the setting of the Denef-Loeser ring of geometric motives, therefore to the theory of motivic integration [cf.: \textit{J. Denef} and \textit{F. Loeser}, Invent. Math. 135, No. 1, 201--232 (1999; Zbl 0928.14004)], and that this generalization leads directly back to the mathematical-physical problems that originally motivated-Kontsevich's conjecture. In this more general context, the authors obtain some partial results supporting the conjecture that certain Feynman amplitudes are rational linear combinations of multiple zeta values with respect to the schemes \(Y_G\).
Altogether, this ingenious paper represents a highly substantial contribution toward a central topic of contemporary research in combinatorics, arithmetic algebraic geometry, and mathematical physics, likewise. It offers a wealth of fundamental new ideas for further research in these areas of mathematics and physics, and therefore its value can barely be overestimated. algebraic varieties over finite fields; zeta functions; finite graphs; trees; generating functions; matroids; motives; motivic integration; Feynman integrals P. Belkale, P. Brosnan, \textit{Matroids, motives, and a conjecture of Kontsevich}, Duke Math. Journal \textbf{116} (2003), 147-188. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Feynman diagrams, Exact enumeration problems, generating functions, Combinatorial aspects of matroids and geometric lattices, Trees, Varieties over finite and local fields Matroids, motives, and a conjecture of Kontsevich. | 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 \(Rat_ d\) be the space of degree d rational maps modulo post composition with Moebius transformations, and \(\rho(d):=d^{- 1}\binom{2d-2}{d-1}\) the Catalan number. - The author proves that the number of classes in \(Rat_ d\) with a given branch set is generically equal to the Catalan number \(\rho(d)\).
Let \(Poly_ d\) denote the space of polynomials whose degree is \(\leq d\), and \(G_ 2(Poly_ d)\) the Grassmann manifold of two dimensional subspaces of \(Poly_ d\). There is an embedding of \(Rat_ d\) into \(G_ 2(Poly_ d):\) \(Rat_ d\ni [R]\to X_ R\in G_ 2(Poly_ d),\) where \(R=P/Q\) and \(X_ R\) is generated by P and Q. There is a complex analytic map \(\Phi_ d: G_ 2(Poly_ d)\to {\mathbb{P}}^{2d-2}\) which connects a subspace generated by P and Q to the polynomial \(PQ'-P'Q\) whose roots are the critical points of P/Q. The author computes the degree of \(\Phi_ d\) by transforming the problem to one of enumerative geometry and by using the Schubert calculus. Grassmann manifold of two dimensional subspaces of space of polynomials; rational maps; Catalan number; enumerative geometry; Schubert calculus \beginbarticle \bauthor\binitsL. \bsnmGoldberg, \batitleCatalan numbers and branched coverings by the Riemann sphere, \bjtitleAdv. Math. \bvolume85 (\byear1991), no. \bissue2, page 129-\blpage144. \endbarticle \OrigBibText L. Goldberg, Catalan numbers and branched coverings by the Riemann sphere, Adv. Math. 85 (1991), no. 2, 129-144. \endOrigBibText \bptokstructpyb \endbibitem Coverings of curves, fundamental group, Grassmannians, Schubert varieties, flag manifolds, Polynomials and rational functions of one complex variable, Riemann surfaces; Weierstrass points; gap sequences, Enumerative problems (combinatorial problems) in algebraic geometry Catalan numbers and branched coverings by the Riemann sphere | 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 complex projective variety admitting the following properties:
(1) There exists an algebraic vector field V on X with precisely one zero \(s\in X\), and
(2) there is an algebraic \({\mathbb{C}}^*\)-action \(\lambda: {\mathbb{C}}^*X\to X\) and a positive integer p such that the induced tangent action \(d\lambda\) satisfies the relation \(d\lambda(t)\cdot V=t^ p\cdot V\) for any \(t\in {\mathbb{C}}^*.\)
Examples of such varieties are the projective spaces \({\mathbb{P}}^ n\), the flag varieties G/B defined by a complex semi-simple Lie group G and a Borel subgroup B, the Bott-Samuelson desingularizations of Schubert varieties, and certain Fano threefolds.
In the present paper, the authors prove a product formula for the Poincaré polynomial of varieties satisfying (1) and (2). This formula reads \(P(X,t^{p/2})= \prod^{n}_{i=1}(1-t^{-a_ i+p})/(1-t^{- a_ i}) \); where P denotes the Poincaré polynomial defined by the Betti numbers of X, and \(a_ 1,...,a_ n\) are the weights of \(\lambda\) at the zero s of V, \(n=\dim_{{\mathbb{C}}}X.\)
This explicit product formula for the Poincaré polynomial is then shown to have some amazing consequences: First, in the special case of X being a flag variety G/B of a semi-simple Lie group G, the product formula for the Poincaré polynomial turns out to coincide with the famous Kostant- Macdonald product identity for Lie algebras of maximal tori in B [cf. \textit{B. Kostant}, Am. J. Math. 81, 973-1033 (1959; Zbl 0099.256)] and \textit{I. G. Macdonald}, Math. Ann. 199, 161-174 (1972; Zbl 0286.20062)]. - Secondly, it is deduced that if \(\lambda\) and V arise from an algebraic \(SL_ 2({\mathbb{C}})\)-action on X, where X satisfies (1) and (2), then the second Betti number of X is just the multiplicity of the lagest weight of the induced linear \({\mathbb{C}}^*\)-action on the tangent space of X at the zero \(s\in X\). That means, in particular, \(b_ 2(X)\leq \dim_{{\mathbb{C}}}X=n.\)
Besides these general consequences, the authors discuss, at the end of the paper, several concrete examples. product formula for the Poincaré polynomial; Kostant-Macdonald product; second Betti number Ersan Akyıldız and James B. Carrell, \(A generalization of the Kostant-Macdonald identity. \)Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 11, 3934--3937. Group actions on varieties or schemes (quotients), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) A generalization of the Kostant-Macdonald identity | 0 |
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