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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 classical theory of total positivity concerns matrices in which all minors are non-negative. In the past decade Lusztig has extended this notion by introducing the totally nonnegative variety \(G_{\geq 0}\) in an arbitrary reductive group \(G\) as well as the totally nonnegative part \((G/P)_{\geq 0}\) of a real flag variety \(G/P\). In [\textit{K.~Rietsch}, Total positivity and real flag varieties, Ph.D. Dissertation, MIT (1998)], a cell decomposition for the totally nonnegative part \((G/P)_{\geq 0}\) of an arbitrary flag variety was constructed and the order relation for closures of cells was described. Let us denote by \(L^J\) the corresponding poset of cells.
The goal of the article under review is the study of the poset \(L^J\) by combinatorial methods. The author proves that this poset is graded, thin and EL-shellable. Using the result of \textit{A.~Björner} [Eur. J. Comb. 5, 7--16 (1984; Zbl 0538.06001)] it is shown that \(L^J\) is the face poset of a regular CW complex homeomorphic to a ball. Notice that it is unknown whether \((G/P)_{\geq 0}\) itself is homeomorphic to a ball. In an appendix, the author gives a detailed explanation of the combinatorics of \(L^J\) for Grassmannians of type A. total positivity; Grassmannians; cells; posets; CW complexes L. K. Williams. ''Shelling totally nonnegative flag varieties''. J. Reine Angew. Math. 609 (2007), pp. 1--21.DOI. Grassmannians, Schubert varieties, flag manifolds, Combinatorics of partially ordered sets, Reflection and Coxeter groups (group-theoretic aspects) Shelling totally nonnegative 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 The book under review is the third and last volume of a treatise on projective spaces over a finite field. This trilogy is the most complete work to date on this subject, and is an indispensable reference for anyone working in the area. The first book in the series [the first author, `Projective geometries over finite fields', Oxford Univ. Press, New York (1979; Zbl 0418.51002)] dealt mainly with projective planes over the finite field \(GF (q)\), although some general introductory material was also presented. The second book in the series [the first author, `Finite projective spaces of three dimensions', Oxford Univ. Press, New York (1985; Zbl 0574.51001)] dealt primarily with finite projective 3- space as its title implies. The present book deals with \(PG(n,q)\) for arbitrary dimension \(n\). In all cases the approach taken is one that might reasonably be called ``finite algebraic geometry''. That is, the group theoretic point of view is not emphasized, but rather a combinatorial approach is taken to characterize various curves and collections of subspaces in finite projective space. The main proof techniques thus involve algebraic manipulations of coordinates over finite fields and various counting strategies. It should be noted that complete proofs are given for almost all results in the three volumes. For a more group theoretic approach to finite geometry one is referred to the classic book by \textit{P. Dembowski}, `Finite geometries', Springer, Berlin (1968; Zbl 0159.500), although many results in the latter reference are not proven.
The main topics discussed in this third volume are quadrics, various varieties (Hermitian, Grassmann, Veronese, Segre), polar spaces, generalized quadrangles, partial geometries, arcs and caps. The chapter on quadrics extends some work done in the previous two volumes, where the properties of quadrics in two, three and five dimensions were carefully developed. The chapters characterizing Hermitian and Grassmannian varieties over finite fields are quite different than what one would see in the classical setting, where, for instance, a Hermitian manifold over the complex numbers is not an algebraic variety. However, the chapter developing the properties of Veronese and Segre varieties closely follows the classical model. The chapter on polar spaces and generalized quadrangles is one of the few places where a number of results are stated without proof. The volume concludes with an appendix listing the known results for the existence of ovoids and spreads in the finite classical polar spaces.
There are a few topics in finite geometry that are intentionally omitted in this treatise. For instance, nondesarguesian planes are not at all discussed, and the interested reader is referred to a book such as by \textit{H. Lüneburg}, `Translation planes', Springer, Berlin (1980; Zbl 0446.51003) for an account of this important subject. Other topics of interest today that are not discussed in this treatise include flocks of quadrics in \(PG(3,q)\), generalized \(n\)-gons for \(n>4\), and Buekenhout diagram geometries. Nonetheless, this is a very encompassing piece of work, and the topics covered are discussed with incredible detail. This is certainly true for the volume under review.
The compilation of the bibliography in and of itself is a tremendous accomplishment. There are over 2000 references given in the three volumes with the most recent publications cited appearing in print in 1991. finite geometry; Hermitian; quadrics; varieties; Grassmann; Veronese; Segre; partial geometries; arcs; caps; polar spaces; generalized quadrangles; ovoids; spreads; bibliography J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford Math. Monogr., Oxford University Press, New York, 1991. Research exposition (monographs, survey articles) pertaining to geometry, Other finite linear geometries, Generalized quadrangles and generalized polygons in finite geometry, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Blocking sets, ovals, \(k\)-arcs, Finite partial geometries (general), nets, partial spreads, Grassmannians, Schubert varieties, flag manifolds, Combinatorial structures in finite projective spaces General Galois geometries | 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 \(M(n,k)\) be the \(n\times n\)-matrices with entries in an algebraically closed field \(k\) and let \(V\) be a vector space over \(k\) of dimension \(n\). \textit{N. Spaltenstein} [Nederl. Akad. Wet., Proc., Ser. A 79, 452--456 (1976; Zbl 0343.20029)] established a bijection between the irreducible components of the space of full flags \(\mathcal F_x\) fixed by a nilpotent element \(x\in M(n,k)\) and the standard tableaux associated to the Young diagram of \(x\).
The main result of the present article is to determine, when \(x\) is of hook type, for each irreducible component \(X\) of \(\mathcal F_x\), the unique Schubert cell \(\mathcal C_X\) of the full flag manifold \(\mathcal F(V)\), such that \(\mathcal C_X\cap X\) is a dense subspace of \(X\). flag manifolds; nilpotent element; standard tableaux; Young diagram; Schubert cell; hook type; irreducible components Pagnon, NGJ, On the spaltenstein correspondence, Indag. Mathem., 15, 101-114, (2004) Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields On the Spaltenstein correspondence. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper provides a classification of the simple integrable modules of double affine Hecke algebras via perverse sheaves. Let \(\underline G\) be a simple connected simply connected linear algebraic group. Let \(\underline{\text{Lie}}\,\underline G\) denote the Lie algebra of \(\underline G\), let \(\underline{\text{Lie}}\,\underline H\subset\underline{\text{Lie}}\,\underline G\) be a Cartan subalgebra and let \(\underline{\text{Lie}}\,\underline B\subset\underline{\text{Lie}}\,\underline G\) be a Borel subalgebra containing \(\underline{\text{Lie}}\,\underline H\). Let \(\underline\Phi\) be the root system of \(\underline G\) and let \(\Phi^\vee\) be the root system dual to \(\underline\Phi\). Let \(\{\alpha_i:i\in\underline I\}\), \(\{\alpha_i^\vee:i\in\underline I\}\) be the set of simple roots and of simple coroots, respectively. Let \(I:=\underline I\sqcup\{0\}\). Let \(\underline W\), \(W\) be the Weyl group and the affine Weyl group of \(\underline G\). We identify \(\underline I\) (resp. \(I\)) with the set of simple reflections in \(\underline W\) (resp. \(W\)). Let \(s_i\in W\) be the simple reflection corresponding to \(i\in I\). For all \(i,j\in I\), let \(m_{ij}\) denote the order of the element \(s_is_j\) in \(W\). Let \(\underline X\) be the weight lattice of \(\underline\Phi\) and let \(Y^\vee\) be the root lattice of \(\underline\Phi v\). Let \(\{\omega_i:i\in\underline I\}\) be the set of fundamental weights. Consider the lattices \(Y=\bigoplus_{i\in I}\mathbb{Z}\alpha_i\subset X=\mathbb{Z}\delta\oplus\bigoplus_{i\in I}\mathbb{Z}\omega_i\), \(Y^\vee=\bigoplus_{i\in I}\mathbb{Z}\alpha_i^\vee\), where \(\delta\) is a new variable. There is unique pairing \(X\times Y^\vee\to\mathbb{Z}\) such that \((\omega_i:\alpha_j^\vee)=\delta_{ij}\) and \((\delta:\alpha_j^\vee)=0\).
The double affine Hecke algebra \(\mathbf H\) is the unital associative \(\mathbb{C}[q,q^{-1},t,t^{-1}]\)-algebra generated by \(\{t_i,x_\lambda:i\in I\), \(\lambda\in X\}\) modulo the following defining relations:
\[
x_\delta=t,\quad x_\lambda x_\mu=x_{\lambda+\mu}(t_i-q)(t_i+1)=0,
\]
\[
t_it_jt_i\cdots=t_jt_it_j\cdots\text{ if }i\neq j\;(m_{ij}\text{ factors in both products),}
\]
\[
t_ix_\lambda-x_\lambda t_i=0\text{ if }(\lambda:\alpha_i^\vee)=0,\quad t_ix_\lambda-x_{s_i(\lambda)}t_i=(q-1)x_\lambda\text{ if }(\lambda:\alpha_i^\vee)=1,
\]
for all \(i,j\in I\), \(\lambda,\mu\in X\).
One important step of the proof is the construction of a ring homomorphism from \(\mathbf H\) to a ring defined via the equivariant \(K\)-theory of an affine analogue \(\mathcal Z\) of the Steinberg variety. \(\mathcal Z\) is an ind-scheme of ind-infinite type. It comes with a filtration by subsets \({\mathcal Z}_{\leq y}\) with \(y\) in the affine Weyl group \(W\). The subsets \({\mathcal Z}_{\leq y}\) are reduced separated schemes of infinite type, and the inclusions \({\mathcal Z}_{\leq y'}\subset{\mathcal Z}_{\leq y}\) with \(y'\leq y\) are closed immersions. The set \(\mathcal Z\) is endowed with an action of a torus \(A\) which preserves each term of the filtration. For a well-chosen element \(a\in A\), the fixed point set \({\mathcal Z}^a\subset{\mathcal Z}\) is a scheme locally of finite type. Hence there is a convolution ring \(\mathbf K^A({\mathcal Z}^a)\): it is the inductive limit of the system of \({\mathbf R}_A\)-modules \(\mathbf K^A(({\mathcal Z}_{\leq y})^a)\) with \(y\in W\). (Here \({\mathbf R}_A\) means \({\mathbf K}_A(\text{point})\).) The author defines a ring homomorphism \(\Psi_a\colon{\mathbf H}\to\mathbf K^A({\mathcal Z}^a)_a\), where the subscript \(a\) means specialization at the maximal ideal \(J_a\subset{\mathbf R}_A\) associated to \(a\). The map \(\Psi_a\) becomes surjective after a suitable completion of \(\mathbf H\). It is certainly not injective. Using \(\Psi_a\), a standard sheaf-theoretic construction, due to Ginzburg in the case of affine Hecke algebras, provides a collection of simple \(\mathbf H\)-modules. These are precisely the simple integrable modules. -- The paper also give some estimates for the Jordan-Hölder multiplicities of induced modules. simple integrable modules; double affine Hecke algebras; perverse sheaves; linear algebraic groups; Lie algebras; Cartan subalgebras; Borel subalgebras; root systems; affine Weyl groups; simple reflections; pairings; equivariant \(K\)-theory; Jordan-Hölder multiplicities of induced modules Vasserot, Eric, Induced and simple modules of double affine Hecke algebras, Duke Math. J., 126, 2, 251-323, (2005) Hecke algebras and their representations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Lie algebras of linear algebraic groups, Grothendieck groups, \(K\)-theory, etc., Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Induced and simple modules of double affine Hecke 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 See the review of the announcement of this paper \((=\) part I) [Proc. Natl. Acad. Sci. USA 82, 2217-2219 (1985)] in Zbl 0601.58023. harmonic maps; complex Grassmann manifolds Chern S. S. and Wolfson J. G., Harmonic maps of the two-sphere into a complex Grassmann manifold. II, Ann. of Math. (2) 125 (1987), no. 2, 301-335. Harmonic maps, etc., Grassmannians, Schubert varieties, flag manifolds Harmonic maps of the two-sphere into a complex Grassmann manifold. II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author provides the definitions of counter-geometric counterparts of divided difference (Demazure) operators from the Schubert calculus and representation theory. These operators are used to capture Demazure characters of reductive group representations. Technically speaking such operators inductively define certain polytopes leading to Demazure characters. In particular the author gives an uniform construction for Gelfand-Zetlin polytopes and twisted cubes of Grossberg-Karshon. Gelfand-Zetlind polytope; divided difference operator; Demazure character Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Reflection and Coxeter groups (group-theoretic aspects) Divided difference operators on polytopes | 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 over \(\mathbb{C}\), \(e\) be a nilpotent element in the Lie algebra \(\mathrm{Lie} (G)\), \(P\) be a parabolic subgroup of \(G\), and \(\mathfrak{i}\) be a \(P\)-stable subspace in \(\mathrm{Lie} (P)\). Define a subvariety of \(G/P\) by
\[
\mathcal{P}_{e,\mathfrak{i}}=\{gP\in G/P: g^{-1}e\in \mathfrak{i} \}.
\]
The author proved that if the minimal Levi subalgebra of \(\mathrm{Lie} (G)\) containing \(e\) has no non-regular component of exceptional type (see the definition on p. 422), then \(\mathcal{P}_{e,\mathfrak{i}}\) admits an affine paving, which means that it is covered by a finite subset of closed subvarieties, each of them isomorphic to an affine space. This generalizes a theorem of \textit{C. de Concini} et al. [J. Am. Math. Soc. 1, No. 1, 15--34 (1988; Zbl 0646.14034)], which asserts the statement when \(P\) is a Borel subgroup and \(\mathfrak{i}=\mathrm{Lie}(P)\), i.e., when \(\mathcal{P}_{e,\mathfrak{i}}\) is the Springer fiber over \(e\). This affine-paving property is desirable because it guarantees good cohomological properties for the subvariety.
We give a few words about the proof. After recalling some basic facts in the first two sections about parabolic orbits of partial flag varieties and smooth pavings, the author reduces the proof to the case when \(G\) is of classical type and \(e\) is distinguished. During the reduction process, we apply a useful theorem of Bialynicki-Birula, which was also used in [loc. cit.]. In the last three sections, he proceeds with explicit calculations using root spaces of \(\mathrm{Lie} (G)\). affine pavings; partial flag varieties L. Fresse, Existence of affine pavings for varieties of partial flags associated to nilpotent elements, Int. Math. Res. Not. IMRN 2016 (2016), 418--472. Classical groups (algebro-geometric aspects), Homogeneous spaces and generalizations, Coadjoint orbits; nilpotent varieties, Grassmannians, Schubert varieties, flag manifolds Existence of affine pavings for varieties of partial flags associated to nilpotent elements | 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 give a global vector field naturally defined over the universal Grassmannian manifold (UGM) of Sato [see \textit{M. Sato}, Proc. Symp. Pure Math. 49, Pt. 1, 51-66 (1989; Zbl 0688.58016)] and \textit{M. Sato} and \textit{Y. Sato} [Nonlinear partial differential equations in applied science, Proc. U.S.-Jap. Semin., Tokyo, 1982, North-Holland Math. Stud. 81, 259-271 (1983; Zbl 0528.58020)]. With this, the UGM becomes a differential-algebraic variety in the sense of Kolchin. The main results in the paper deal with cohomological properties (projectivity and duality) of some \({\mathcal D}\)-modules on UGM arising in the KP hiearchy (see loc. cit.). The paper also contains a differential-algebraic interpretation of the KP hierarchy. pseudo-differential operator; projectivity; global vector field; universal Grassmannian manifold; differential-algebraic variety; duality; \({\mathcal D}\)-modules; KP hiearchy Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), KdV equations (Korteweg-de Vries equations), Infinite-dimensional manifolds, Equations in function spaces; evolution equations, Theta functions and abelian varieties, Grassmannians, Schubert varieties, flag manifolds Structure and duality of \(\mathcal D\)-modules related to 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 Let \(G\) be a connected, simply-connected, semisimple linear algebraic group over an algebraically closed field \(k\), \(B\) a Borel subgroup containing a maximal torus \(T\), \(W = N_G(T)/T\) the Weyl group, \(S\subset W\) the set of simple reflections with respect to \(B\). A simple reflection \(s\in S\) determines a minimal parabolic subgroup \(P_s = BsB\cup B\supset B\). Consider, also, the maximal parabolic subgroup \(Q_s\) which is the union of the double cosets \(BwB\) with \(w\) in the subgroup of \(W\) generated by \(S\setminus \{s\}\). If \({\mathfrak w}=(s_1,\dots ,s_n)\) is a sequence of simple reflections and \(1\leq i \leq n\), one denotes by \({\mathfrak w}[i]\) the truncated sequence \((s_1,\dots ,s_{n-i})\) and by \({\mathfrak w}(i)\) the subsequence of \({\mathfrak w}\) obtained by deleting \(s_i\).
The Bott-Samelson variety \(Z_{\mathfrak w}\) is defined as follows : \(B^n\) acts from the right on \(P_{s_1}\times \dots \times P_{s_n}\) by : \((p_1,\dots ,p_n)(b_1,\dots b_n) = (p_1b_1,b_1^{-1}p_2b_2,\dots , b_{n-1}^{-1}p_nb_n)\) and \(Z_{\mathfrak w} := P_{s_1}\times \dots \times P_{s_n}/B^n\). Since \(P_s/B\simeq {\mathbb P}^1\), \(\forall s\in S\), one has a tower of \({\mathbb P}^1\)-bundles : \(Z_{\mathfrak w}\rightarrow Z_{{\mathfrak w}[1]}\rightarrow \cdots \rightarrow Z_{{\mathfrak w}[n-1]}=P_{s_1}/B\simeq {\mathbb P}^1.\) For \(1\leq i\leq n\), the natural projection \({\pi}_i : Z_{\mathfrak w}\rightarrow Z_{{\mathfrak w}[i]}\) admits a section and \(Z_{{\mathfrak w}(i)}\) embeds as a divisor into \(Z_{\mathfrak w}\). The multiplication of \(G\) induces a morphism \(Z_{\mathfrak w}\rightarrow G/B\) and if the word \({\mathfrak w}\) is reduced then \(Z_{\mathfrak w}\) is a Demazure-Hansen desingularisation of the Schubert variety \(X(s_1\dots s_n)\) which is the Zariski closure in \(G/B\) of the (left) \(B\)-orbit of \(s_1\dots s_nB\).
Let \({\mathcal O}_{\mathfrak w}(1)\) denote the pull-back on \(Z_{\mathfrak w}\) of the ample generator of \(\text{Pic}(G/Q_{s_n})\simeq {\mathbb Z}\) (when \(G = \text{SL}(n)\), \(G/Q_{s_n}\) is a Grassmannian) by the composite map \(Z_{\mathfrak w}\rightarrow G/B\rightarrow G/Q_{s_n}\). One sees easily that \({\pi}_i^{\ast}{\mathcal O}_{{\mathfrak w}[i]}(1)\), \(i=0,\dots ,n-1\), is a \({\mathbb Z}\)-basis of \(\text{Pic}Z_{\mathfrak w}\). The authors show, firstly, that \({\mathcal O}_{\mathfrak w}(m_1)\otimes {\pi}_1^{\ast}{\mathcal O}_{{\mathfrak w}[1]}(m_2)\otimes \dots \otimes {\pi}_{n-1}^{\ast}{\mathcal O}_{{\mathfrak w}[n-1]}(m_n)\) is very ample (resp., globally generated) on \(Z_{\mathfrak w}\) iff \(m_i>0\) (resp., \(\geq 0\)) for \(i=1,\dots n\). Moreover, if \({\mathfrak w}\) is reduced then \({\mathcal O}(\mathop{\sum}_{i=1}^nm_iZ_{{\mathfrak w}(i)})\) is effective iff \(m_i\geq 0\) for \(i=1,\dots ,n\).
Then, using the Frobenius splitting technique of \textit{V. Mehta} and \textit{A. Ramanathan} [Ann. Math. (2) 122, 27--40 (1985; Zbl 0601.14043)], the authors show that if, for some \(1\leq t\leq q\leq n\), the subsequence \((s_t,\dots ,s_q)\) is reduced then \(\text{H}^i(Z_{\mathfrak w},{\mathcal L}\otimes {\mathcal O}(-\mathop{\sum}_{j=t}^q Z_{{\mathfrak w}(j)}))=0\), for all \(i>0\) and all globally generated line bundles \({\mathcal L}\) on \(Z_{\mathfrak w}\). This generalizes a result of \textit{S. Kumar} [Invent. Math. 89, 395--423 (1987; Zbl 0635.14023)] who assumed that \(\text{char}k = 0\) and that \({\mathcal L}\) is the line bundle associated to a dominant weight. Schubert varieties; cohomology of line bundles; semisimple algebraic groups; Frobenius splitting Lauritzen, N.; Thomsen, J. F., Line bundles on Bott-Samelson varieties, J. Algebraic Geom., 13, 461-473, (2004) Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Classical groups (algebro-geometric aspects), Vanishing theorems in algebraic geometry Line bundles on Bott-Samelson 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 [For the entire collection see Zbl 0564.00013.]
Let Z be a smooth hyperquadric in the projective complex space \({\mathbb{P}}^ n\), \(Y\subset Z\) be a smooth subvariety of Z. The author establishes that the normal bundle of Y in Z is non-ample if and only if there exists a line \(\ell \subset Y\) such that the polar set \(\ell^{\perp}\subset {\mathbb{P}}^ n\) of \(\ell\) with respect to Z contains all the linear projective subspaces of \({\mathbb{P}}^ n\) which are tangent to Y. This criterion of non-ampleness is applied to constructions of surfaces Y in the Plücker quadric with non-ample normal bundle. These examples refute a statement of \textit{A. Papantonopoulou} in Proc. Am. Math. Soc. 77, 15-18 (1979; Zbl 0424.14017) which was corrected [ibid. 95, 330 (1985; 0575.14043)]. normal bundle; non-ampleness; Plücker quadric Projective techniques in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds Manifold having non-ample normal bundles in quadrics | 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 0527.00041.]
This paper has two purposes. One is to call attention to a recent campaign to systematically study linear fractional parameterizations of function spaces. The second purpose is to give a new proof of the Youla- Jabr-Bongiorno parameterization of all compensators which stabilize a given plant. This is a linear fractional parameterization and is one specialized example of our general study of linear fractional maps. Further examples of linear fractional parameterizations are those of Nevanlinna type (scalar case), Adamajan-Arov-Krein [A-A-K] and Arsene- Ceausescu-Foias [A-C-F] type (vector case) which are obtained in designing certain digital filters, and in many given equalization studies. The first section of this note gives an outline of our general Lie group approach to the subject and the second focuses on the [Y-J-B] parameterization in a way as analogous to classical circuit theory as possible. Much of classical circuit theory (e.g. Darlington's theorem) amounts to a study of rational functions with values in the matrix Lie group U(n,n); the analogous results for the Lie group SL(n,\({\mathbb{C}})\) give the [Y-J-B] parameterization. We focus on the [Y-J-B] results rather than other aspects of our study, not because they are more representative, but because they are more unexpected. stabilizing compensators; Youla-Jabr-Bongiorno parameterization; linear fractional maps Algebraic methods, Representations of Lie and linear algebraic groups over real fields: analytic methods, Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators, Grassmannians, Schubert varieties, flag manifolds, \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables, Harmonic analysis on homogeneous spaces, Synthesis problems, Stabilization of systems by feedback Linear fractional parameterizations of matrix function spaces and a new proof of the Youla-Jabr-Bongiorno parameterization for stabilizing compensators | 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 problem that is frequently encountered in a variety of mathematical contexts is to find the common invariant subspaces of a single or of a set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea consists of finding common eigenvectors for exterior powers of the matrices concerned. A convenient formulation of the Plücker relations is then used to ensure that these eigenvectors actually correspond to subspaces or provide the initial constraints for eigenvectors involving parameters. A procedure for computing the divisors of a totally decomposable vector is also provided. Several examples are given for which the calculations are too tedious to do by hand and are performed by coding the conditions found into Maple. Our main motivation lies in Lie symmetry, where the invariant subspaces of the adjoint representations for the Lie symmetry algebra of a differential equation must be known explicitly and comprehensively in order to determine all the ideals of the Lie symmetry algebra. invariant subspace; totally decomposable multivector; Grassmann manifold; Plücker relations Invariant subspaces of linear operators, Grassmannians, Schubert varieties, flag manifolds, Exterior algebra, Grassmann algebras A novel procedure for constructing invariant subspaces of a set of matrices | 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 1940, \textit{L. A. Santaló} proved a Helly-type theorem for line transversals to boxes in \(\mathbb{R}^d\) [Publ. Inst. Mat., Rosario 2, 49--60 (1940; Zbl 0025.36803)]. An analysis of his proof reveals a convexity structure for ascending lines in \(\mathbb{R}^d\) that is isomorphic to the ordinary notion of convexity in a convex subset of \(\mathbb{R}^{2d-2}\). This isomorphism is through a Cremona transformation on the Grassmannian of lines in \(\mathbb{P}^d\), which enables a precise description of the convex hull and affine span of up to \(d\) ascending lines; the lines in such an affine span turn out to be the rulings of certain classical determinantal varieties. Finally, we relate Cremona convexity to a new convexity structure that we call frame convexity, which extends to arbitrary-dimensional flats in \(\mathbb{R}^d\). Helly's theorem; convexity; Grassmannians; Cremona transformations 23.J.E. Goodman, A.~Holmsen, R.~Pollack, K.~Ranestad, F.~Sottile, Cremona convexity, frame convexity, and a theorem of Santaló. Adv. Geom. 6, 301-322 (2006) Helly-type theorems and geometric transversal theory, Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties Cremona convexity, frame convexity and a theorem of Santaló | 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 use the Thom polynomial theory developed by \textit{L. Fehér} and \textit{R. Rimányi} [Duke Math. J. 114, No.2, 193--213 (2002; Zbl 1054.14010)] to prove the component formula for quiver varieties conjectured by \textit{A. Knutson, E. Miller} and \textit{M. Shimozono} [Four positive formulae for type A quiver polynomials, preprint, \texttt{http://arxiv.org/abs/math.AG/0308142}]. This formula expresses the cohomology class of a quiver variety as a sum of products of Schubert polynomials indexed by minimal lace diagrams, and implies that the quiver coefficients of \textit{A. Buch} and \textit{W. Fulton} [Invent. Math. 135, 665--687 (1999; Zbl 0942.14027)] are non-negative. We also apply our methods to give a new proof of the component formula from the Gröbner degeneration of quiver varieties, and to give generating moves for the KMS-factorizations that form the index set in \(K\)-theoretic versions of the component formula. Buch, Anders S.; Fehér, László M.; Rimányi, Richárd, Positivity of quiver coefficients through Thom polynomials, Adv. Math., 197, 1, 306-320, (2005) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Singularities of differentiable mappings in differential topology, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Positivity of quiver coefficients through Thom polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author considers the Schubert polynomials \(X_{\pi}\in {\mathbb{Z}}[x_1,x_2,\ldots]\) associated with the permutations \(\pi\) contained in the symmetric groups \(S_n\). There are many possible ways to introduce the Schubert polynomials via divided difference operators, recursive generation without divided differences based on the Monk rule and the Bruhat order on permutations, via nil-Coxeter relations, the formula of Billey-Jockusch-Stanley, via sums of mixed shift and multiplication operators, via balanced labeled tableaux, via configurations of labeled pseudo-lines, via flagged Schur modules associated to a Rothe diagram, etc. The theme of the paper is the combinatorial generation of Schubert polynomials via sets of box diagrams. The main reasons to expect a combinatorial rule in terms of box diagrams are: the coefficients in \(X_{\pi}\) are non-negative integers and should count some discrete objects; in the special case of Grassmannian permutations the Schubert polynomial is equal to a Schur function in a finite number of variables and the well-known combinatorial properties of Schur functions should extend to Schubert polynomials. The main result of the paper is the proof of a very elegant and easily applicable combinatorial rule for the generation of Schubert polynomials conjectured in 1990 in the thesis by Kohnert. A similar type of combinatorial rule was given by Bergeron. As an intermediate step in the proof of the Kohnert conjecture the author also obtains a simplified proof of the Bergeron rule. In particular, he shows that the Bergeron rule may be also simplified to a version which is very similar to the recent combinatorial rule of Magyar proved with algebro-geometric methods. The author obtains a direct combinatorial proof of the Magyar rule as well. This shows that there is an algebro-geometric meaning of the Bergeron rule. On the other hand it makes apparent the possibility to give fully combinatorial proofs of other results concerning, e.g., the fact that the Schubert polynomial is the character of the flagged Schur module associated to a Rothe diagram. box diagrams; diagram rules; tableaux; Schubert polynomials; symmetric functions; Schur functions R. Winkel. ''Diagram rules for the generation of Schubert polynomials''. J. Combin. Theory Ser. A 86 (1999), pp. 14--48.DOI. Symmetric functions and generalizations, Polynomials, factorization in commutative rings, Grassmannians, Schubert varieties, flag manifolds Diagram rules for the generation of 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 We use the Fox-Neuwirth cell structure for one-point compactifications of configuration spaces as the starting point for understanding our recent calculation of the mod-two cohomology of symmetric groups. We then use that calculation to give short proofs of classical results on this cohomology due to Nakaoka and to Madsen. Fox-Neuwirth cell structures; configuration spaces; cohomology of symmetric groups Chad Giusti and Dev Sinha, Fox-Neuwirth cell structures and the cohomology of symmetric groups. Preprint, arXiv:1110.4137, 2011. Cohomology of groups, Geometric structures on manifolds of high or arbitrary dimension, Classifying spaces of groups and \(H\)-spaces in algebraic topology, Symmetric groups, Grassmannians, Schubert varieties, flag manifolds Fox-Neuwirth cell structures and the cohomology of symmetric 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 This article deals with the quantum cohomology of the classical flag manifold \(\text{GL}(n,\mathbb C)/B\), where \(B\) is the subgroup of upper triangular nonsingular matrices. The cohomology ring can be obtained as a factor ring of the polynomials in \(n\) variables modulo the ideal generated by the elementary symmetric polynomials. A distinguished basis for the classical cohomology ring is given by the Schubert polynomials. The quantum cohomology ring again is a factor ring of the polynomial ring but now with \(n-1\) additional variables, the deformation parameters, modulo the ideal generated by the ``quantum elementary polynomials''. The author relates the quantum Schubert polynomial and some other related basis with problems in algebraic combinatorics. For the proofs and further details he mainly refers to the following publications of the author [Adv. Math. 136, No. 2, 224-250 (1998; Zbl 0920.05069) and ``On algebraic and combinatorial properties of Schur and Schubert polynomials'' (Bayreuther Math. Schr. 59) (2000; Zbl 0958.05001)]. quantum cohomology; flag manifold; Schubert polynomial; elementary symmetric polynomial; standard elementary monomial 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), Enumerative problems (combinatorial problems) in algebraic geometry From quantum cohomology to algebraic combinatorics: The example of flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(L\) be any standard Levi subgroup which acts by left multiplication on a Schubert variety \(X(w)\) in the Grassmannian. We give a complete classification of the pairs \(L\) and \(X(w)\), where \(X(w)\) is a spherical variety for the action of \(L\). Schubert varieties; Grassmann varieties; spherical varieties Grassmannians, Schubert varieties, flag manifolds, Compactifications; symmetric and spherical varieties A classification of spherical Schubert varieties in 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 produce new combinatorial methods for approaching the tropical maximal rank conjecture, including inductive procedures for deducing new cases of the conjecture on graphs of increasing genus from any given case. Using explicit calculations in a range of base cases, we prove this conjecture for the canonical divisor, and in a wide range of cases for \(m = 3\), extending previous results for \(m = 2\). tropical geometry; tropical independence; maximal rank conjecture Algebraic combinatorics, Tropical geometry Combinatorial and inductive methods for the tropical maximal rank conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The problem of optimal packing in the real Grassmannian \(G_{\mathbb R}(m, n)\) of \(n\)-planes in \(\mathbb R^m\) as originally introduced in [\textit{J. H. Conway} et al., Exp. Math. 5, No. 2, 139--159 (1996; Zbl 0864.51012)] consists of finding a set of points \(U_1, U_2, \dots , U_N\) in the Grassmannian (i.e. \(n\)-dimensional subspaces of \(\mathbb R^m\)) such that the minimal distance between pairs of points, \(\min_{i\neq j} d (U_i, U_j)\), is maximal. The distance used is the chordal distance \(d (U, V)\) that can be expressed in terms of projection operators onto \(n\)-planes in \(\mathbb R^m\) as \(d^2(U, V) = \frac 1 2\mathrm{tr}\, (\mathcal P_U - \mathcal P_V)^2 = n - \mathrm{tr}\, (\mathcal P_U \mathcal P_V),\) where \(\mathcal P_U, \mathcal P_V\) are the matrices of the orthogonal projections on \(U\) and \(V.\) A projection matrix is closely related to the reflection matrix in the same plane, namely \(R_U = 2 \mathcal P_U - I\). The identification of an \(n\)-plane (a point in \(G_{\mathbb R}(m, n)\) or \(G_{\mathbb C}(m, n)\)) with the operator of orthogonal projection in \(\mathbb R^m\) or \(\mathbb C^m\) onto that plane defines an inclusion (actually, an isometric equivariant embedding) into the Euclidean space of real symmetric (resp. Hermitian) \(m \times m\) matrices equipped with the standard trace metric, the image belonging to a certain sphere centered at \(\frac{n}{m} I\). (See the above paper or the reviewer's [Publ. Inst. Math., Nouv. Sér. 59(73), 131--137 (1996; Zbl 0965.53038)]). Because of that, the classical simplex and orthoplex Rankin bounds for spherical packings can be translated to Grassmannian packings. In the space of \(m\times m\) real symmetric matrices, which has the dimension \(D = m(m+1)/2,\) the authors define an \(m \times m\) real (orthogonal) matrix orthoplex as a collection of \(2D\) orthogonal, real symmetric \(m\times m\) matrices \(R_1, R_2, \dots , R_{2D}\) satisfying
\[
\mathrm{tr}\, (R_i^T R_j) = \begin{cases} m \quad &\text{if }i\neq j,\\ -m \quad &\text{if }\{i, j \} = \{2k-1, 2k \},\\ 0 \quad &\text{otherwise.}\end{cases}
\]
Similarly, an \(m\times m\) complex (unitary) matrix orthoplex is defined. In [Conway et al., loc. cit.], several optimal packings are determined, including examples of optimal orthoplices of planes in \(\mathbb R^4\) and 4-dimensional subspaces in \(\mathbb R^8\).
In [\textit{P. W. Shor} and \textit{N. J. A. Sloane}, J. Algebr. Comb. 7, No. 2, 157--163 (1998; Zbl 0904.52009)], certain optimal collections of \(2^{2k} + 2^k - 2\) real subspaces of dimension \(2^{k-1}\) in \(\mathbb R^{2^k}\) were produced. The investigation of Grassmannian packings was subsequently extended to the complex Grassmannian \(G_{\mathbb C}(m, n)\) and the existence of the optimal configurations of \(2^{2k+1} - 2\) complex subspaces of dimension \(2^{k-1}\) in \(\mathbb C^{2^k}\) was established in [\textit{A. Ashikhmin} and \textit{A. R. Calderbank}, ``Space-time Reed-Muller codes for noncoherent MIMO transmission'', in: Proceedings of the international symposium on information theory, ISIT 2005, 4--9 September 2005. SA: Adelaide. 1952--1956 (2005; \url{doi:10.1109/ISIT.2005.1523686})]. All of these results combined settled the question of existence of optimal orthoplex Grassmannian packings for real and complex Grassmannians of subspaces of mid-dimension in real resp. complex Euclidean space whose dimension is a power of 2.
In the paper under review, the authors produce, using Hadamard matrices, a new family of optimal orthoplex packings in \(G_{\mathbb R}(8l, 4l)\) and \(G_{\mathbb C}(4l, 2l)\) and a related optimal simplex packings in \(G_{\mathbb R}(8l-1, 4l-1)\) and \(G_{\mathbb R}(8l-1, 4l)\). More precisely, they prove the following results: If there exists a \(4l\times 4l\) Hadamard matrix (\(l \in \mathbb N\)), then there exists an optimal orthoplex packing of \((8l)^2 + 8l - 2\) real subspaces of dimension \(4l\) in the real Grassmannian \(G_{\mathbb R}(8l, 4l)\). Under the same assumption, there exists an optimal orthoplex packing of \(2(4l)^2 - 2\) complex subspaces of dimension \(2l\) in \(G_{\mathbb C}(4l, 2l)\). Moreover, if there exists a \(4l\times 4l\) Hadamard matrix, an \(8l\times 8l\) skew Hadamard matrix and a related 1-factorization of a complete graph \(K_{8l}\), then there exists an optimal simplex packing of \(8l (8l-1)/2\) real subspaces of dimension \(4l-1\) in \(G_{\mathbb R}(8l-1, 4l-1)\), whereas their orthogonal complements form an optimal simplex packing in \(G_{\mathbb R}(8l-1, 4l)\). We recall that an \(n\times n\) Hadamard matrix \(H\) is a matrix with entries equal to \(\pm 1\), satisfying \(H^TH = n I\) and that a skew Hadamard matrix satisfies additionally \(H^T + H = 2 I\). The paper concludes with a construction of a maximal optimal simplex packings in \(G_{\mathbb C}(2l - 1, l - 1)\) and \(G_{\mathbb C}(2l - 1, l)\). Grassmannian packings; optimal packings; Rankin bound; chordal distance; Hadamard matrices; space-time codes Kocák, T; Niepel, M, Families of optimal packings in real and complex Grassmannian spaces, J. Algebraic Comb., 45, 129-148, (2017) Packing and covering in \(n\) dimensions (aspects of discrete geometry), Orthogonal and unitary groups in metric geometry, Polyhedra and polytopes; regular figures, division of spaces, Algebraic systems of matrices, Boolean and Hadamard matrices, Other types of codes, Grassmannians, Schubert varieties, flag manifolds Families of optimal packings in real and complex Grassmannian spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author builds the foundations of the theory of joins and higher secant varieties for subvarieties of a Grassmannian. As a consequence, some properties of general stable bundles on curves are obtained. Grassmannian; higher secant varieties; stable bundles on curves; joins Ballico E., Results Math. 32 pp 29-- (1997) Projective techniques in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Vector bundles on curves and their moduli, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Joins and secant varieties of subvarieties of a 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 Let \(\mathbb{O}\) denote the complex octonion algebra, and let \(\Xi : \mathbb{O}\times \mathbb{O}\times \mathbb{O}\times \mathbb{O}\to \mathbb{R}\) denote the function that assigns the imaginary part to the cross product \(x\times u \times v \times w \in \mathbb{O}\times \mathbb{O}\times \mathbb{O}\times \mathbb{O}\). The Cayley Grassmannian, denoted \(X\), is the variety parametrizing four dimensional subspaces spanned by the vectors \(x,u,v,w\) such that \(\Xi(x,u,v,w)=0\). It is a 12-dimensional, singular, \(\text{Spin}(7,\mathbb{C})\)-stable subvariety of the Grassmann variety, \(Gr(4,\mathbb{O})\). In this article, the author analyzes the geometry of the natural \(\text{Spin}(7,\mathbb{C})\)-action on \(\mathbb{O}\) by restricting it to various subgroups. Briefly stated, the author obtains the following results.
\begin{enumerate}
\item An explicit determination of the \(T\)-fixed points, where \(T\) is the maximal torus of \(\text{Spin}(7,\mathbb{C})\).
\item An explicit characterization of the singular locus of \(X\); it turns out that \(\text{Sing}(X)\) is isomorphic to the partial flag variety corresponding to the long root of the exceptional simple algebraic group, \(\text{G}_2 (\mathbb{C})\).
\end{enumerate}
These results can be seen as natural extensions of the results of the articles by \textit{S. Akbulut} and \textit{M. B. Can} [J. Gökova Geom. Topol. GGT 11, 56--79 (2017; Zbl 1390.22011)] and \textit{L. Manivel} [J. Algebra 503, 277--298 (2018; Zbl 1423.14293)]. Bialynicki-Birula decomposition; Cayley Grassmannian; octonion algebra; \(G_2\); spin(7); torus action; exceptional holonomy Grassmannians, Schubert varieties, flag manifolds, Issues of holonomy in differential geometry, Calibrations and calibrated geometries On the complex Cayley 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 study properties of the exponential Hilbert series of a (partial) flag variety, \(G/P\), where \(G\) is a semisimple, simply-connected complex linear algebraic group and \(P\) is a parabolic subgroup. We prove a relationship between the exponential Hilbert series and the degree and dimension of the flag variety. We then prove a combinatorial formula for the coefficients of an exponential analogue of the Hilbert polynomial. This formula is used in examples to prove further combinatorial identities involving Stirling numbers of the first and second kinds. Hilbert series; Stirling numbers; algebraic groups; representation theory Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Linear algebraic groups over the reals, the complexes, the quaternions Exponential Hilbert series and geometric invariants of 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 The authors introduce here the vertical quantum cohomology for \(\pi:X \to B\) where only vertical rational curves in \(X\) are used to ``deform'' the cohomology structure. This cohomology is then used to expand the computation of the quantum cohomology of the Grassmannian \(\text{Gr} (n,m)= U(n)/U(m) \times U(n-m)\) to the more general case of flag manifold \(F_{n_1, \dots, n_k}= U(n)/U(n_1) \times\cdots \times U(n_k)\). The result is consistent with the quantum cohomology ring computed before by Givental and Kim for the complete flag variety \(F_n=U(n)/U(1) \times \cdots \times U(1)\). vertical quantum cohomology; flag manifold Astashkevich, A.; Sadov, V., Quantum cohomology of partial flag manifolds f (n1 {\dots} nk ), Commun. Math. Phys., 170, 503, (1995) Grassmannians, Schubert varieties, flag manifolds, Geometric quantization, (Co)homology theory in algebraic geometry, Quantization in field theory; cohomological methods Quantum cohomology of partial flag manifolds \(F_{n_ 1\dots n_ k}\) | 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 features a curated selection of research in algebraic combinatorics that explores the boundaries of current knowledge in the field. Focusing on topics experiencing broad interest and rapid growth, invited contributors offer survey articles on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The volume also addresses subjects at the intersection of algebra, combinatorics, and geometry, including the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. All surveys are written at an introductory level that emphasizes recent developments and open problems. An interactive tutorial on Schubert Calculus emphasizes the geometric and topological aspects of the topic and is suitable for combinatorialists as well as geometrically minded researchers seeking to gain familiarity with relevant combinatorial tools.
Featured authors include prominent women in the field known for their exceptional writing of deep mathematics in an accessible manner. Each article in this volume was reviewed independently by two referees. The volume is suitable for graduate students and researchers interested in algebraic combinatorics.
The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to combinatorics, Other designs, configurations, Combinatorial identities, bijective combinatorics, Proceedings, conferences, collections, etc. pertaining to associative rings and algebras, Grassmannians, Schubert varieties, flag manifolds, Proceedings, conferences, collections, etc. pertaining to group theory, Collections of articles of miscellaneous specific interest Recent trends in algebraic combinatorics | 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 symmetrizable Kac-Moody Lie algebra. Associated to two dominant integral weights \(\lambda\), \(\mu \in \mathcal{P}^+\) are the integrable, (irreducible) highest weight representations \(V(\lambda)\) and \(V(\mu)\). Then the content of the tensor product decomposition problem is to express the product \(V(\lambda) \otimes V(\mu)\) as a direct sum of irreducible components, i.e., find the decomposition \(V(\lambda) \otimes V(\mu) = \displaystyle{\bigoplus_{\nu \in \mathcal{P}^+}} V(\nu)^{\oplus m_{\lambda, \mu}^\nu}\), where \(m_{\lambda, \mu}^\nu \in \mathbb{Z}_{\geq 0}\) is the multiplicity of \(V(\nu)\) in \(V(\lambda) \otimes V(\mu)\).
Now, let \(\mathfrak{g}\) be an affine Kac-Moody Lie algebra and let \(\lambda, \mu\) be two dominant integral weights for \(\mathfrak{g}\). The authors prove that for any positive root \(\beta\), \(V(\lambda)\otimes V(\mu)\) contains \(V(\lambda+\mu-\beta)\) as a component, where \(V(\lambda)\) denotes the integrable highest weight (irreducible) \(\mathfrak{g}\)-module with highest weight \(\lambda\), extending the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras (Theorem 9.1, page 841). One crucial ingredient in the proof is the action of Virasoro algebra on the tensor product \(V(\lambda)\otimes V(\mu)\). The authors also prove the corresponding geometric results including the higher cohomology vanishing on the \(\mathcal{G}\)-Schubert varieties in the product partial flag variety \(\mathcal{G}/\mathcal{P}\times \mathcal{G}/\mathcal{P}\) with coefficients in certain sheaves coming from the ideal sheaves of \(\mathcal{G}\)-sub-Schubert varieties (Theorem 11.7, page 854), thereby proving the surjectivity of the Gaussian map. affine Kac-Moody Lie algebra; integrable highest weight modules; root components; Virasoro algebra; higher cohomology vanishing; Schubert varieties; partial flag varieties Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Kac-Moody groups, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) Root components for tensor product of affine Kac-Moody Lie algebra 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 Let \(k\) be an algebraically closed field of characteristic 0 and \(R\) a reduced irreducible curve singularity over \(k\), i.e. \(R \supset k\) is a complete local domain of Krull dimension 1 having \(k\) as residue field. Any torsion free rank one module \(M\) can be seen as a submodule of the normalization \(R'=k[[t]]\) of \(R\) such that \(R \subseteq M \subseteq R'\). The aim of this paper is to construct coarse moduli spaces parametrizing the isomorphism classes of such modules corresponding to some fixed invariants as e.g. \(\delta(M)= \dim_ kR'/M\), \(\Gamma(M)=v(M)\), where \(v\) is the valuation of \(R'\). The idea is to show that the above isomorphism classes can be seen as orbits of the multiplicative group \(R'{}^*\) of invertible elements of \(R'\) (in fact of the Jordan group \(J \cong R'{}^*/k^*)\) acting on some Grassmannians. Then using the authors' results from Proc. Lond. Math. Soc., III. Ser. 67, No. 1, 75-105 (1993) it is enough to see that there exist geometric quotients for a certain stratification given by some invariants. The paper contains an algorithm for computation of these moduli spaces and many useful nice examples. irreducible curve singularity; coarse moduli spaces; Grassmannians Greuel, G.-M., Pfister, G.: Moduli spaces for torsion free modules on curve singularities I. J. of Algebraic Geometry2, 81--135 (1993) Families, moduli of curves (algebraic), Singularities of curves, local rings, Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry Moduli spaces for torsion free modules on curve singularities. 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 The paper continues the authors' investigations [see \textit{K. Altmann} and \textit{J. A. Christophersen}, Manuscr. Math. 115, No. 3, 361--378 (2004; Zbl 1071.13008) and \textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] on the graph \({\mathcal G}\) of monomial ideals in the polynomial ring \(R=k[x_1,\dots,x_n]\), \(k\) a field. \({\mathcal G}\) is the infinite graph with the monomial ideals in \(R\) as vertex set. Two monomial ideals \(M_1\) and \(M_2\) are connected by an edge if there exists an ideal \(I\) in \(R\) such that the set of all initial monomial ideals of \(I\), with respect to all term orders, is precisely \(\{M_1,M_2\}\). \(I\) is called edge providing in this case. It is well known that \(I,M_1,M_2\) have many invariants in common. Each invariant yields a stratification of \({\mathcal G}\). A first proposition concerns the subgraph \({\mathcal G^r}\) obtained by restriction to artinian ideals of colength \(r\): Each such stratum is a connected component of \({\mathcal G}\). The main result (theorem 8) characterizes edge providing ideals as ``very homogeneous'':
There exists upto multiples a single \(c\in\mathbb Z^n\) such that \(I\) is \(A\)-graded for \(A=\mathbb Z^n/c\mathbb Z\).
This allows to define the Schubert scheme \(\Omega_c(M_1,M_2)=\Omega(M_1,M_2)\) of all \(A\)-homogeneous edge providing ideals connecting \(M_1\) and \(M_2\). A thorough analysis of the settings yields an algorithm that computes, for given \(M_1\) and \(M_2\), the direction \(c\) and \(\Omega(M_1,M_2)\) as affine scheme. More generally, the authors consider \(A\)-homogeneous ideals for arbitrary gradings \(\deg\:\mathbb Z^n\to A\) (including the standard one). Define \(h_I\:A\to \mathbb N\) as the Hilbert function of \(I\), i.e., \(h_I(a),\;a\in A\), is the \(k\)-dimension of the \(a\)-homogeneous part of \(I\). In the above situation, for \(I\in \Omega(M_1,M_2)\) the Hilbert functions of \(I\), \(M_1\) and \(M_2\) coincide. For positive gradings, i.e., \(\mathbb N^n\cap \text{ker}(\deg)=(0)\), this is also the general situation:
\(\Omega_c(M_1,M_2)\not=\emptyset\) implies \(\deg(c)=0\) and hence the Schubert schemes describe an essential part of the multigraded Hilbert scheme \(\text{Hilb}_h\) of all \(A\)-homogeneous ideals with given Hilbert function \(h\). This part is sufficient to detect connectedness: Over \(k=\mathbb R\) or \(k=\mathbb C\), \(\text{Hilb}_h\) is connected if and only if the induced subgraph \({\mathcal G}(\text{Hilb}_h)\) is connected.
Section 4 discusses properties of the Schubert schemes for square-free monomial ideals. The results are more technical and continue the investigations started by \textit{K. Altmann} and \textit{J. A. Christophersen} [loc. cit.]. In particular, it turns out that neighboring square-free ideals are connected by a generalization of the bistellar flip construction [see, e.g., \textit{O. Viro}, Proc. Workshop Differential Geometry Topology, Alghero 1992, World Scientific. 244--264 (1993; Zbl 0884.57015) or \textit{D. Maclagan} and \textit{R. R. Thomas}, Discrete Comput. Geom. 27, No. 2, 249--272 (2002; Zbl 1073.14503)]. The paper ends with a list of open problems about the graph \({\mathcal G}\) and the Schubert schemes. graph of monomial ideals; multigraded Hilbert scheme; Schubert scheme; Gröbner bases; Gröbner degenerations; Stanley-Reisner ideals DOI: 10.1016/j.jpaa.2004.12.030 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds The graph of monomial ideals | 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 studies the problem of classifying tuples of linear endomorphisms and linear functions on a finite dimensional vector space up to base change, which can be reinterpreted in the language of quivers, as classifying representations on finite dimensional vector spaces of the two-vertex quiver with as many loops on the first vertex as linear endomorphisms and as many arrows between the two vertices as the number of linear functions.
Such representations can be reinterpreted as framed representations in the sense of [\textit{M. Reineke}, J. Algebra 320, No. 1, 94--115 (2008; Zbl 1153.14033)], where it was proved, for quivers without oriented loops, that the quotient of the stable representations up to isomorphism (i.e. up to base change of the linear maps) is isomorphic to a Grassmannian of subrepresentations of a certain injective representation of the quiver. The general case does not provide a projective quotient, and the author studies the fibers of this quotient in [\textit{S. N. Fedotov}, Trans. Am. Math. Soc. 365, No. 8, 4153--4179 (2013; Zbl 1277.14010)]. It remains to describe a trivializing covering for the quotient map, which is the problem adressed in this article.
By generalizing Reineke's construction, the author associates a finite \textit{skeleton} to each stable representation, and each skeleton carries an open subset of the space parametrizing representations. As there are a finite number of possible skeletons, we find a finite open covering of the parametrizing space. The trivialization comes from the fact that each one of these open pieces defines a normal form of the representation, hence the open pieces are isomorphic to affine spaces. Besides, it is given a criterium to determine whether 2 representations are isomorphic, this is when they have a skeleton in common and the normal form associated to them coincides.
The paper finishes with a collection of examples where all the computation are explicitly described. framed moduli spaces; quiver moduli; tuples of operators; Grassmannians of representations; skeleton Local ground fields in algebraic geometry, Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds Framed moduli spaces and tuples of operators | 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 Author's abstract: We define polar classes associated to a singular holomorphic distribution of tangent subspaces of a projective manifold. We prove that these polar classes can be calculated in terms of the Chern-Mather classes of the tangent sheaf of the distribution and reciprocally. We use their degrees to establish a bound for the degrees of some polar classes associated to an invariant variety. holomorphic foliations; polar classes; Schubert cycles; Chern-Mather classes; invariant varieties Mol (R.S.).- Classes polaires associées aux distributions holomorphes de sous-espaces tangents. Bull. Braz. Math. Soc. (N.S.), 37(1):29-48, (2006). Zbl1120.32019 MR2223486 Singularities of holomorphic vector fields and foliations, Grassmannians, Schubert varieties, flag manifolds Polar classes associated with holomorphic distributions of tangent subspaces | 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 the usual model of hypermaps or, equivalently, bipartite maps, represented by pairs of permutations that act transitively on a set of edges \(E\). The specific feature of our construction is the fact that the elements of \(E\) are themselves (or are labelled by) rather complicated combinatorial objects, namely, the 4-constellations, while the permutations defining the hypermap originate from an action of the Hurwitz braid group on these 4-constellations. The motivation for the whole construction is the combinatorial representation of the parameter space of the ramified coverings of the Riemann sphere having four ramification points. Riemann surface; ramified covering; dessins d'enfants; Belyi function; braid group; Hurwitz scheme; hypermaps; bipartite maps A. Zvonkin, ''Megamaps: Construction and Examples,'' in: \textit{Discr. Math. Theor. Comput. Sci. Proc., AA} (2001), pp. 329-339. Algebraic combinatorics, Topology of Euclidean 3-space and 3-sphere, Topology of Euclidean 2-space, 2-manifolds, Riemann surfaces; Weierstrass points; gap sequences Megamaps: Construction and examples | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a companion paper of [the authors, ibid. 22, No. 5, 1071--1147 (2018; Zbl 1479.81043)]. We study Coulomb branches of unframed and framed quiver gauge theories of type \(ADE\). In the unframed case they are isomorphic to the moduli space of based rational maps from \(\mathbb{P}^1\) to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian. Yang-Mills and other gauge theories in quantum field theory, Representations of quivers and partially ordered sets, Fine and coarse moduli spaces, Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) Coulomb branches of \(3d\) \(\mathcal{N}=4\) quiver gauge theories and slices in the affine 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 Roughly speaking, cluster algebras, introduced recently by \textit{S.~Fomin} and \textit{A.~Zele\-vinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)], are defined by \(n\)-regular trees whose vertices correspond to \(n\)-tuples of cluster variables and edges describing birational transformations between two \(n\)-tuples of variables. Model examples of cluster algebras are coordinate rings of double Bruhat cells. Also every \(m\times n\), \(m\leq n\), integer matrix \(Z\), such that the matrix \([(DZ)_{ij}]_{i,j\leq m}\) is skew-symmetric for a certain positive integer diagonal \(m\times m\) matrix \(D\), defines a natural cluster algebra \({\mathcal A}(Z)\) of geometric type.
The authors introduce in the paper the cluster manifold \(\mathcal X\) as a ``handy'' nonsingular part of \(\text{ Spec}({\mathcal A}(Z))\) and a Poisson structure on \(\mathcal X\) which is compatible with the cluster algebra structure in the sense that the Poisson bracket is homogeneously quadratic in any set of cluster variables. Then edge transformations describe simply transvections with respect to the Poisson structure. Poisson and topological properties of the union of generic orbits of a toric action on this Poisson variety are studied.
The second goal of the paper is to extend calculations of the number of connected components in double Bruhat cells to a more general setting of geometric cluster algebras and compatible Poisson structures. Namely, given a cluster algebra \(\mathcal A\) over the reals, the number of connected components in the union of generic symplectic leaves of any compatible Poisson structure on \(\mathcal X\) is computed. Finally, the general formula is applied to a special case of Grassmannian coordinate ring. cluster algebras; Poisson brackets; toric action; Grassmannians; Poisson-Lie groups; Sklyanin bracket M. Gekhtman, M. Shapiro, A. Vainshtein, \textit{Cluster algebras and Poisson geometry}, Mosc. Math. J. \textbf{3} (2003), no. 3, 899-934, 1199. Poisson manifolds; Poisson groupoids and algebroids, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Associative rings and algebras arising under various constructions, Simple, semisimple, reductive (super)algebras Cluster algebras and Poisson geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Triangulated categories possessing a full exceptional collection are the simplest among the others. Every object of a triangulated category \(\mathcal{T}\) with a full exceptional collection \((E_{1},E_{2}, \dots, E_{n})\) admits a unique (functorial) filtration with \(i\)-th quotient being a direct sum of shifts of \(E_{i}\).
In this paper, the author construct a full exceptional collection of vector bundles in the derived categories of coherent sheaves on the Grassmannian of isotropic two-dimensional subspaces in a symplectic vector space of dimension \(2n\) and in an orthogonal vector space of dimension \(2n+1\) for all \(n\). Grassmannians; derived category; exceptional collections; coherent sheaves; vector bundles A. Kuznetsov, Exceptional collections for Grassmannians of isotropic lines, Proc. Lond. Math. Soc. (3) 97 (2008), 155-182. Grassmannians, Schubert varieties, flag manifolds, Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli Exceptional collections for Grassmannians of isotropic lines | 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 investigate the cohomology modules \(H^i(X(w),L_\lambda)\), for \(\lambda\) non-dominant on a Schubert variety \(X (w)\) in the generalized flag variety, especially, the vanishing or non-vanishing of \(H^i(X(\omega), L_\lambda)\). For a generic \(\lambda\), the authors give a criterion for vanishing (resp. non-vanishing) of \(H^{l(w)}(X (w), L_\lambda)\) (resp. \(H^0(X(w), L_\lambda)\)). For a general \(j\), the authors give partial results on the vanishing of \(H^j(X(w), L_\lambda)\). cohomology; line bundles; Schubert varieties P. Polo, Variétés de Schubert et excellentes filtrations, Astérisque 173{174 (1989), no. 10-11, 281-311.} Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Cohomology of line bundles on Schubert varieties. 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 We extend the work of Fomin and Greene on noncommutative Schur functions by defining noncommutative analogs of Schubert polynomials. If the variables satisfy certain relations (essentially the same as those needed in the theory of noncommutative Schur functions), we prove a Pieri-type formula and a Cauchy identity for our noncommutative polynomials. Our results imply the conjecture of Fomin and Kirillov concerning the expansion of an arbitrary Grothendieck polynomial on the basis of Schubert polynomials; we also present a combinatorial interpretation for the coefficients of the expansion. We conclude with some open problems related to it. Schur functions; Schubert polynomials; conjecture of Fomin and Kirillov; Grothendieck polynomial Lenart, C., Noncommutative Schubert calculus and Grothendieck polynomials.Adv. Math., 143 (1999), 159--183. Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Noncommutative Schubert calculus and Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0614.00006.]
The purpose of the paper is to give information on a certain smooth compactification of the space of all morphisms of a given degree from \({\mathbb{P}}^ 1\) to a Grassmann variety. This scheme is the Grothendieck Quot scheme of quotients of a trivial vector bundle on \({\mathbb{P}}^ 1\). We compute the additve and the multiplicative structure of its Chow ring and identify the ample cone and the corresponding projective embeddings. families of rational curves; smooth compactification; Grassmann variety; Quot scheme; Chow ring Strømme, S A, On parametrized rational curves in Grassmann varieties., Lecture Notes in Math, 1266, 251-272, (1987) Grassmannians, Schubert varieties, flag manifolds, Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Algebraic moduli problems, moduli of vector bundles On parametrized rational curves in Grassmann varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we express the class of the structure sheaves of the closures of Deligne-Lusztig varieties as explicit double Grothendieck polynomials in the first Chern classes of appropriate line bundles on the ambient flag variety. This is achieved by viewing such closures as degeneracy loci of morphisms of vector bundles. Deligne-Lusztig varieties; \(K\)-theory; Grothendieck polynomials; degeneracy loci Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over finite fields, \(K\)-theory of schemes On the \(K\)-theoretic fundamental class 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 Fix an integer partition \(\lambda\) that has no more than \(n\) parts. Let \(\beta\) be a weakly increasing \(n\)-tuple with entries from \(\{1,\dots,n\}\). The flagged Schur function indexed by \(\lambda\) and \(\beta\) is a polynomial generating function in \(x_1,\dots,x_n\) for certain semistandard tableaux of shape \(\lambda\). Let \(\pi\) be an \(n\)-permutation. The type A Demazure character (key polynomial, Demazure polynomial) indexed by \(\lambda\) and \(\pi\) is another such polynomial generating function. \textit{V. Reiner} and \textit{M. Shimozono} [J. Comb. Theory, Ser. A 70, No. 1, 107--143 (1995; Zbl 0819.05058)] and then \textit{A. Postnikov} and \textit{R. P. Stanley} [J. Algebr. Comb. 29, No. 2, 133--174 (2009; Zbl 1238.14036)] studied coincidences between these two families of polynomials. Here their results are sharpened by the specification of unique representatives for the equivalence classes of indexes for both families of polynomials, extended by the consideration of more general \(\beta\), and deepened by proving that the polynomial coincidences also hold at the level of the underlying tableau sets. Let \(R\) be the set of lengths of columns in the shape of lambda that are less than \(n\). Ordered set partitions of \(\{1,\dots,n\}\) with block sizes determined by \(R\), called \(R\)-permutations, are used to describe the minimal length representatives for the parabolic quotient of the \(n\)th symmetric group specified by the set \(\{1,\dots,n-1\}\setminus R\). The notion of 312-avoidance is generalized from \(n\)-permutations to these set partitions. The \(R\)-parabolic Catalan number is defined to be the number of these. Every flagged Schur function arises as a Demazure polynomial. Those Demazure polynomials are precisely indexed by the \(R\)-312-avoiding \(R\)-permutations. Hence the number of flagged Schur functions that are distinct as polynomials is shown to be the \(R\)-parabolic Catalan number. The projecting and lifting processes that relate the notions of 312-avoidance and of \(R\)-312-avoidance are described with maps developed for other purposes. Catalan number; flagged Schur function; Demazure character; key polynomial; pattern avoiding permutation; symmetric group parabolic quotient Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Exact enumeration problems, generating functions, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Symmetric groups Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key 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 These are the notes of the three lectures given by Dave Anderson at the IMPANGA conference in Bȩdlewo in 2010. They provide an introduction to equivariant cohomology in algebraic geometry and their basic properties, as well as some of their applications, especially to Schubert calculus of Grassmannians.
In the first lecture the author introduces the notion of equivariant cohomology \(H^*_GX\) of an algebraic variety with an action of an algebraic group \(G\), lists some of its fundamental properties: functoriality, construction of equivariant Chern classes and fundamental classes, and provides some examples. He also points out two important general facts, which hold in ``nice'' situations: that ordinary cohomology can be recovered from equivariant cohomology, and second, that equivariant cohomology is determined by the information on the fixed points of \(G\) on \(X\) (this is called localization). This approach works particularly nicely when the number of such fixed points is finite.
The second lecture is devoted to a more detailed study of localization, in the case when the group \(G=T\) is a torus, and \(X\) is a smooth projective variety. This situation is ``nice'' in the sense of the previous paragraph: \(H^*_T X\) defines the ordinary cohomology ring, and the localization technique also applies. The main result of this lecture is the Atiyah--Bott--Berline--Vergne integration formula, which allows to compute the integrals of classes from \(H^*_T X\) by summation over the \(T\)-fixed points on \(X\). This theorem allows to solve enumerative problems, e.g. compute the number of twisted cubics on a Calabi--Yau threefold, cf. [\textit{G. Ellingsrud} and \textit{S. R. Strømme}, J. Am. Math. Soc. 9, 175--193 (1996, Zbl 0856.14019)].
In the third lecture the developed technique is applied to Schubert calculus on Grassmannians. The formulas for the \(T\)-equivariant Schubert classes is presented: they can be expressed as double Schur polynomials \(s_\lambda(x_i,t_i)\), where \(x_i\) are the equivariant Chern roots of the dual tautological subbundle on the Grassmannian, and \(t_i\) are the weights of the torus \(T\). The positivity of equivariant Littlewood--Richardson coefficients, originally due to \textit{W.~Graham} [Duke Math. J. 109, 599--614 (2001; Zbl 1069.14055)] is also discussed. The author concludes by a review of some other questions involving equivariant techniques: generalization of equivariant Schubert calculus to the case of an arbitrary homogeneous space \(G/P\), degeneracy locus formulas and computation of Thom polynomials.
The notes are very clear and well written; they can provide a perfect introduction into the subject or serve as a good reference text. equivariant cohomology; Grassmannian; classifying space; Schur polynomial D. Anderson, \textit{Introduction to equivariant cohomology in algebraic geometry}, in: \textit{Contributions to Algebraic Geometry}, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012, pp. 71-92. Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Introduction to equivariant cohomology in 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 paper under review presents a new algebraic approach to quantum cluster algebras based on noncommutative ring theory. The paper proposes a general construction of quantum cluster algebra structures on a broad class of algebras. Initial clusters and mutations are constructed in a uniform and intrinsic way, in particular, avoiding any ad hoc constructions with quantum minors.
The main theorem of the paper asserts that every algebra in a very large, axiomatically defined class of quantum nilpotent algebras admits a quantum cluster algebra structure. Furthermore, for all such algebras, the latter equals the corresponding upper quantum cluster algebra.
This theorem has a broad range of applications and the required axioms are easy to verify. Many classical families of algebras fall within this axiomatic class. In particular, an application of this theorem gives an explicit quantum cluster algebra structures on the quantum Schubert cell algebras for all finite dimensional simple Lie algebras. quantum cluster algebras; quantum nilpotent algebras; iterated Ore extensions; noncommutative unique factorization domain K. R. Goodearl and M. T. Yakimov, \textit{Quantum cluster algebra structures on quantum nilpotent algebras}, Memoirs of the American Mathematical Society \textbf{247} (2017). Ring-theoretic aspects of quantum groups, Cluster algebras, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Quantum cluster algebra structures on quantum nilpotent algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we explicitly compute all Littlewood-Richardson coefficients for semisimple and Kac-Moody groups \(G\), that is, the structure constants (also known as the Schubert structure constants) of the cohomology algebra \(H^*(G/P,\mathbb C)\), where \(P\) is a parabolic subgroup of \(G\). These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is purely combinatorial and is given in terms of the Cartan matrix and the Weyl group of \(G\). However, if some off-diagonal entries of the Cartan matrix are \(0\) or \(-1\), the formula may contain negative summands. On the other hand, if the Cartan matrix satisfies \(a_{ij}a_{ji}\geq 4\) for all \(i\), \(j\), then each summand in our formula is nonnegative that implies nonnegativity of all Littlewood-Richardson coefficients. We extend this and other results to the structure coefficients of the \(T\)-equivariant cohomology of flag varieties \(G/P\) and Bott-Samelson varieties \(\Gamma_{\mathbf i}(G)\). Littlewood-Richardson coefficients; flag varieties; Schubert varieties; semisimple groups; Kac-Moody groups; reflection groups; Cartan matrices; Weyl groups Berenstein, A; Richmond, E, Littlewood-Richardson coefficients for reflection groups, Adv. Math., 284, 54-111, (2015) Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Kac-Moody groups Littlewood-Richardson coefficients for reflection 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 From the Section 2.1: ``Let \(V=\mathbb C^6\) be a 6-dimensional complex vector space, and let \( \alpha:V \times V \to \mathbb C \) be a symplectic form on \(V\). \(\alpha\) is bilinear, skew symmetric, and non degenerate. It is possible to choose a basis of \(V\) such that the Gram matrix of \(\alpha\) is given by a matrix \(J\) with rows \(\left(\begin{smallmatrix} 0 & I_3\\ -I_3 & 0\end{smallmatrix}\right)\) where \(I_3\) is the unit matrix of order \(3\). The complex symplectic group \(\text{Sp}_6(\mathbb C)\) has a natural embedding in \(\text{SL}(6,\mathbb C)\) as the subgroup of all the complex rank-6 matrices that leave the matrix \(J\) invariant:
\[
\text{Sp}_6(\mathbb C)=\{ Z \in \text{SL}(6,\mathbb C): {^t Z J Z}=J \}
\]
A subspace \(U \subset V\) is called isotropic if \(\alpha(U,U)=0\). The maximal dimension of an isotropic subspace in \(V\) is \(3\), and in this case it is called Lagrangian. By definition the complex Lagrangian Grassmannian \(\mathbf{LG}(3,V)\) is the set of Lagrangian subspaces of \(V\).''
In this paper general linear sections of \(\mathbf{LG}(3,V)\) are considered.
In particular here it is generalized a result of Mukai which gives an interpretation of the general complete intersection of \(\mathbf{LG}(3,V)\) in its Plücker embedding in \({\mathbb P}^{13}\) with a linear subspace \({\mathbb P}^{10}\) as a non-abelian Brill-Noether locus of vector bundles on a plane quartic curve. This plane quartic can be interpreted in a very natural way as the orthogonal plane section of the dual variety of \(\mathbf{LG}(3,V)\).
The main goal of this paper is to get similar description for general linear sections of \(\mathbf{LG}(3,V)\) of various dimension.
We give here just two of the results obtained in this paper:
Proposition. Let \(X\) be a general prime Fano threefold of degree \(16\) (which is a linear 3-fold section of \(\mathbf{LG}(3,V)\)). Then the \(\text{Sp}(3)\)-dual to \(X\) is isomorphic to an irreducible component of the moduli space of stable rank \(2\) vector bundles on \(X\) with \(c_1=h\) and \(c_2=C\) where \([h]\) is the class of the hyperplane section of \(X\) and \(C\) is the class of an elliptic sextic curve on \(X\).
Proposition. For the general linear surface section \(S\) of \({\mathbf {LG}}(3,V)\), the moduli \(K3\) surface \(M_S(2,H,4)\) of rank \(2\) stable bundle on \(S\) with \(c_1=H=O_S(1)\) and \(\chi=6\) is isomorphic to the \(\text{Sp}(3)\)-dual quartic surface section. Lagrangian Grassmannian; Brill-Noether loci; vector bundles; \(K3\) surfaces; Fano threefolds A. Iliek, Geometry of the Lagrangian-Grassmannian \(LG(3,6)\) with applications to Brill-Noether Loci, Michigan Math. J., 53, 383, (2005) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Grassmannians, Schubert varieties, flag manifolds Geometry of the Lagrangian Grassmannian LG(3,6) with applications to Brill-Noether loci | 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 bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of \textit{M. Bousquet-Mélou} et al. [Ramanujan J. 1, No. 1, 101--111 (1997; Zbl 0909.05008)], \textit{H. Eriksson} and \textit{K. Eriksson} [Electron. J. Comb. 5, Research paper R18, 32 p. (1998; Zbl 0889.20002); printed version J. Comb. 5, 231--262 (1998)] and \textit{V. Reiner} [Electron. J. Comb. 2, Research paper R25, 21 p. (1995; Zbl 0849.20032); printed version J. Comb. 2, 403--422 (1995)]. In other types the identities appear to be new. For type \(A_n\), the affine colored partitions form another family of combinatorial objects in bijection with \((n + 1)\)-core partitions and \(n\)-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmannians in terms of affine partitions and a generalization of Young's lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted. rationally smooth Schubert varieties; Young's lattice Billey, S., and Mitchell, S., Affine partitions and affine Grassmannians, preprint 46pp: arXiv:0712.2871 (2007), to appear in the Electronic Journal of Combinatorics. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Affine partitions and affine 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 associate with \(n+1\) different points in \(\mathbb{P}^1(\mathbb{C})\) a system of nonlinear differential equations that gives for \(n=1\) the KP-hierarchy. To this geometric configuration there corresponds naturally a basic space \(H^*\) and a group of commuting flows \(\widetilde\Gamma\). Starting from the Grassmann manifold corresponding to \(H^*\), we construct wavefunctions that yield solutions of the differential equations in the derivatives with respect to the flow parameters. KP-hierarchy; Grassmann manifold; wavefunctions KdV equations (Korteweg-de Vries equations), Grassmannians, Schubert varieties, flag manifolds, Meromorphic functions of one complex variable (general theory) A multipoint version of the 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 provide two closed-form geodesic formulas for a family of metrics on Stiefel manifolds recently introduced by \textit{K. Hüper} et al. [J. Geom. Mech. 13, No. 1, 55--72 (2021; Zbl 1477.58006)], reparameterized by two positive numbers, having both the embedded and canonical metrics as special cases. The closed-form formulas allow us to compute geodesics by matrix exponential in reduced dimension for low-rank Stiefel manifolds. We follow the approach of minimizing the square Frobenius distance between a geodesic ending point to a given point on the manifold to compute the logarithm map and geodesic distance between two endpoints, using Fréchet derivatives to compute the gradient of this objective function. We focus on two optimization methods, \textit{gradient descent} and L-BFGS. This leads to a new framework to compute the geodesic distance for manifolds with known geodesic formula but no closed-form logarithm map. We show the approach works well for Stiefel as well as flag manifolds. The logarithm map could be used to compute the Riemannian center of mass for these manifolds equipped with the above metrics. The method to translate directional derivatives using Fréchet derivatives to a gradient could potentially be applied to other matrix equations. Stiefel manifold; geodesic; computer vision; flag manifold; logarithm map; Riemannian center of mass; Fréchet derivative Geodesics in global differential geometry, Grassmannians, Schubert varieties, flag manifolds, Variational problems in applications to the theory of geodesics (problems in one independent variable), Numerical optimization and variational techniques, Real-valued functions on manifolds, Special Riemannian manifolds (Einstein, Sasakian, etc.), Relations of manifolds and cell complexes with engineering, Relations of manifolds and cell complexes with computer and data science, Learning and adaptive systems in artificial intelligence, Machine vision and scene understanding Closed-form geodesics and optimization for Riemannian logarithms of Stiefel and 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 Schubert intersection problem is the problem of counting \(r\)-dimensional subspaces of \(\mathbb{C}^n\) having certain adjacency properties with given subspaces (flags) of \(\mathbb{C}^n\). In certain situations such a Schubert intersection problem can be reduced to another Schubert intersection problem with smaller \(n\). The paper under review extends an earlier such reduction result of Thompson-Therianos to a large family of reduction results. In fact, the reductions presented in this paper are sufficient for the complete solution of a special class of intersection problems, when the Littlewood-Richardson coefficient is 1. The reductions presented are in connection with known multiplicative properties of Littlewood-Richardson coefficients. The method of the proof is a very careful analysis of a combinatorial rule (involving `measures', similar to `puzzles') for Littlewood-Richardson coefficients. Schubert variety; Littlewood-Richardson rule; puzzle; measure Bercovici, H.; Li, W. S.; Timotin, D.: A family of reductions for Schubert intersection problems, J. algebraic combin. 33, 609-649 (2011) Grassmannians, Schubert varieties, flag manifolds A family of reductions for Schubert intersection problems | 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 compute the Witt groups of split Grassmann varieties, over any regular base \(X\). They prove that the total Witt group of the Grassmannian is a free module over the total Witt ring of \(X\). Remark that also the Chow group, or the Grothendieck group, are free over \(X\) with a basis indexed by all Young diagrams.
The authors provide an explicit basis of the total Witt group indexed by a class of Young diagrams which they call even Young diagrams. Recall that the total Witt group is a sum of all the Witt groups depending on a shift \(i\in \mathbb{Z}/4\) and a twist \(L\in \mathrm{Pic} (X)/2\). The cited basis consists of homogeneous elements; moreover the shift and the twist can be read on the corresponding Young diagram. In particular, this fact allows the author to describe the unshifted and untwisted Witt group. The elements of the basis of the total Witt group are defined as push-forwards of the unit form of certain desingularized Schubert varieties. Remark that pushing the unit form is not always possible. The condition for a Young diagram to be even implies the existence of such a push-forward, but it is not necessary.
The computation of Witt groups of a Grassmann variety is harder than the computation of cohomology groups or Chow groups because of the following fact. The classical computation proceeds by induction, using the closed embedding of a smaller Grassmannian \(\mathrm{Gr}_X(d,n-1)\) inside \(\mathrm{Gr}_X(d,n)\), whose open complement \(U\) is an affine bundle over another smaller Grassmannian \(\mathrm{Gr}_X(d-1,n)\). Moreover the restriction morphism from the big Grassmannian \(\mathrm{Gr}_X(d,n)\) to the open \(U\) is split surjective. For Witt groups the restriction morphism is not even surjective. In other words, the connecting homomorphism in the localization long exact sequence in not zero in general. Grassmann variety; Witt group; triangulated category; cellular decomposition Paul Balmer and Baptiste Calmès, Witt groups of Grassmann varieties, J. Algebraic Geom. 21 (2012), no. 4, 601 -- 642. Grassmannians, Schubert varieties, flag manifolds, Algebraic theory of quadratic forms; Witt groups and rings, Derived categories, triangulated categories Witt groups of Grassmann 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 authors study twisted foldings of root systems, generalizing usual involutive foldings corresponding to automorphisms of Dynkin diagrams. Their primary motivating example is the Lusztig projection of the root system of type \(E_8\) onto the subring of icosians of the quaternion algebra which gives the root system of type \(H_4\).
Using moment graph techniques for any such folding, the authors construct a map at the equivariant cohomology level: the \(\tau\)-twisted folding \(\Phi\to \Phi_\tau\) induces ring homomorphisms \(\iota^* \colon \mathcal{Z}(\mathcal{G}) \to \mathcal{Z}(\mathcal{G}_\tau)\) and \(\bar\iota^* \colon \overline{\mathcal{Z}}(\mathcal{G}) \to \overline{\mathcal{Z}}(\mathcal{G}_\tau)\) between the augmented structure algebras that commute with augmentation maps and preserve the gradings (Theorem 6.2, page 79). They show that this map commutes with characteristic classes and Borel maps (Lemma 6.4, page 81). They also introduce and study its restrictions to the usual cohomology of projective homogeneous varieties, to group cohomology and to their virtual analogues for finite reflection groups (Section 6, page 78). moment graph; folding; finite reflection group; equivariant cohomology; structure algebra Grassmannians, Schubert varieties, flag manifolds, Root systems, Exceptional groups Twisted quadratic foldings of root systems | 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 spirit of Alain Lascoux, the authors propose the use of Schubert polynomials for (certain) computations with polynomials in several variables. The idea comes from situations like doing computations with symmetric functions: There, computations are (usually) not done with monomials, but with a basis adapted to the specific problem that we are dealing with, such as Schur functions, for example.
Most of the paper is devoted to survey the background and basic facts about Schubert polynomials. When we regard the complete ring of polynomials in \(x_1,x_2,\dots,x_n\) as a ring over the ring of symmetric polynomials in \(x_1,x_2,\dots,x_n\), then the Schubert polynomials indexed by permutations in \(S_n\) (the symmetric group on \(n\) elements) constitute a linear basis. Similarly, the ring of polynomials in \(x_1,x_2,\dots,x_n\) with coefficients that are polynomials in \(y_1,y_2,\dots,y_n\) has as a linear basis the double Schubert polynomials. In order to use Schubert polynomials efficiently for computations in such rings, one of the first things we need is a rule for multiplying Schubert polynomials. No general formula for multiplying Schubert polynomials has been found yet (in contrast to Schur functions, where we have the Littlewood-Richardson rule). At least, at the very basic level, there is Monk's formula for the multiplication of a Schubert polynomial in \(x_1,x_2,\dots,x_n\) by one of the variables. However, this formula (possibly) involves Schubert polynomials which are indexed by permutations in \(S_{n+1}\) (and not just \(S_n\)). The authors show how to modify the formula so that one obtains, within the ring of polynomials in \(x_1,x_2,\dots,x_n\), regarded as a ring over the symmetric polynomials, expansions consisting of Schubert polynomials indexed by permutations in \(S_n\). A ``Monk's formula'' for double Schubert polynomials is proved as well. Schubert polynomials; symmetric functions; Monk's formula; divided differences Kohnert, A.; Veigneau, S.: Using Schubert basis to compute with multivariate polynomials. Adv. appl. Math. 19, 45-60 (1997) Symmetric functions and generalizations, Polynomials, factorization in commutative rings, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds Using Schubert basis to compute with multivariate polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The results of this paper are motivated by the Geometric Langlands Program and the Geometric Satake Theorem, stating the equivalence of the category of $G^\vee(\mathcal{O})$-equivariant perverse sheaves on the affine Grassmannian associated to a complex algebraic reductive group $G$ and the category of finite dimensional highest weight representations of its Langlands dual group $G^\vee$.
More precisely, it is related to the approach developed in [\textit{D. Gaitsgory}, Invent. Math., 144, 253--280 (2001; Zbl 1072.14055)], who gave a geometric construction of a map from the spherical Hecke algebra to the center of the Iwahori Hecke algebra, via a nearby cycles functor from the category of $G(\mathcal{O})$-equivariant perverses sheaves on the affine Grassmannian to the category of equivariant perverse shaves on the affine flag variety. (Here $\mathcal{K}$ denotes the local field of Laurent series $\mathbb{C}((t))$ and $\mathcal{O}$ the ring of integers of power series $\mathbb{C}[[t]]$; the affine Grassmannian is $G(\mathcal{K})/G(\mathcal{O})$ and the affine flag variety is $G(\mathcal{K})/\mathcal{I}$, where $I\subset G(\mathcal{O})$ is the Iwahori subgroup).
The author studies the algebraic geometry and combinatorics of the central degeneration in type $A$, namely the $T$-equivariant flat degeneration from the affine Grassmannian to the affine flag variety in the global affine flag variety constructed by Gaitsgory. The main results concern the closures of semi-infinite orbits in the affine Grassmannian and their relations with Levi restriction.
Theorem. Let $G =\mathrm{GL}_n(\mathbb{C})$ or any matrix Lie group over $\mathbb{C}$. Given the closure of any $N_w(\mathcal{K})$ orbit $\overline{S^\mu_w}$, for $\mu\in X_*(T)$, in the affine Grassmannian, its special fiber limit is the closure of the corresponding $N_w(\mathcal{K})$ orbit $\overline{S^{(\mu,e)}_w}$, for $(\mu,e)\in W_{\mathrm{aff}}$, in the affine flag variety.
If $r_P$ denotes the restriction map for a Levi factor $G_J$ of a parabolic subgroup $P^J$, then it is shown that the central degeneration commutes with Levi restriction.
Theorem. Let $S^\mu_w$, for $w\in W$, be a semidefinite orbit in the affine Grassmannian of type $A$. Let $P^J\supseteq B_w$ be a parabolic subgroup with Levi factor $G_J$. For any $u$ in the Weyl group $W_J$ of $G_J$, the following diagram commutes. In other words the central degeneration of the closures of semi-infinite orbits commutes with Levi restriction/parabolic retraction.
\[
\begin{tikzcd}
\overline{S^\mu_{wu}}\subset Gr_G \arrow[r, "\text{deg}"] \arrow[d, "r^{\mu,u}"]
& \overline{S^{(\mu,e)}_{wu}}\subset Fl_G \arrow[d, "r^{(\mu,e),u}_P"] \\
\overline{S^\mu_{u,J}}\subset Gr_{G_J} \arrow[r, "\text{deg}"]
& \overline{S^{(\mu,e)}_{u,J}}\subset Fl_{G_J}
\end{tikzcd}
\]
In addition, explicit results on the central degenerations of Mirkovic-Vilonen cycles and related transformations of moment polytopes for $\mathrm{SL}_2(\mathbb{C})$ and for $\mathrm{SL}_n(\mathbb{C})$ are obtained. affine Grassmannian; moment polytope; Levi restriction; Mirković-Vilonen cycles; Gaitsgory's central sheaves; torus equivariance Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Geometric Langlands program (algebro-geometric aspects), Fibrations, degenerations in algebraic geometry, Group schemes, Algebraic cycles, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Convex polytopes for the central degeneration of the affine 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 Let \(G\) be a connected simple algebraic group of the adjoint type over complex numbers, let \(B\) be a Borel subgroup, \(T\subset B\) a maximal torus and let \(W\) be the associated Weyl group. Let \(X:=G/B\) be the full flag variety. For any \(w\in W\), let \(X(w):=\overline{BwB/B}\) be the associated Schubert variety. Choosing a reduced expression \(\bar{w}\) of \(w\) we get the Bott-Samelson-Demazure-Hansen (for short BSDH) desingularization \(Z (\bar{w})\) of \(X (w)\). Let \(T_{Z (\bar{w})}\) be the tangent bundle of \(Z (\bar{w})\). In an earlier paper, \textit{B. N. Chary} et al. [Transform. Groups 20, No. 3, 665--698 (2015; Zbl 1326.14114)] studied the higher cohomoligies \(H^j(Z (\bar{w}), T_{Z (\bar{w})})\) and showed that it vanishes for all \(j\geq 2\). If \(G\) is simply-laced they proved this vanishing for all \(j\geq 1\). This gives the local rigidity of \(Z (\bar{w})\) for simply-laced \(G\). For non simply-laced \(G\), the vanishing of \(H^1(Z (\bar{w}), T_{Z (\bar{w})})\) depends on the choice of the reduced decmposition \(\bar{w}\) of \(w\).
In earlier papers, the first author and \textit{P. Saha} [J. Lie Theory 29, No. 1, 107--142 (2019; Zbl 1415.14016)] (resp. \textit{B. N. Chary} and the first author [ibid. 27, No. 2, 435--468 (2017; Zbl 1429.14014)]) analyzed the vanishing of \(H^1(Z (\bar{w_0}), T_{Z (\bar{w_0})})\) for \(G= PSO(2n+1, \mathbb{C})\) for \(n\geq 3\) (resp. \(G=PSp(2n, \mathbb{C})\)), where \(w_0\) is the longest element of \(W\), i.e., \(X(w)\) is the full flag variety \(G/B\). In these cases they prove that, for some explicit choices of the reduced decompositions \(\bar{w_0}\) of \(w_0\) coming from the Coxeter elements, \(H^1(Z (\bar{w_0}), T_{Z (\bar{w_0})})\) vanishes providing the local rigidity of \(Z (\bar{w_0})\) in these cases.
In the paper under review, they analyze the same question on the vanishing of \(H^1(Z (\bar{w_0}), T_{Z (\bar{w_0})})\) for \(G=F_4\) and \(G_2\). Bott-Samelson-Demazure-Hansen variety; Coxeter element; tangent bundle; local rigidity Grassmannians, Schubert varieties, flag manifolds Rigidity of Bott-Samelson-Demazure-Hansen variety for \(F_4\) and \(G_2\) | 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 We compute the cohomology of the Fano varieties of \(k\)-planes in the smooth complete intersection of two quadrics in \(\mathbb{P}^{2 g + 1}\), using Springer theory for symmetric spaces. Fano variety; hyperelliptic curve; Springer correspondence; symmetric space; Hessenberg variety Chen, T. H.; Vilonen, K.; Xue, T., On the cohomology of Fano varieties and the Springer correspondence, Adv. Math., 318, 515-533, (2017) Fano varieties, Vector bundles on curves and their moduli, Grassmannians, Schubert varieties, flag manifolds On the cohomology of Fano varieties and the Springer correspondence | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(W=(W,S)\) be a Coxeter group with \(S\) its distinguished generator set. Denote by \(\leqslant\) the Bruhat ordering on \(W\). \textit{D. Kazhdan} and \textit{G. Lusztig} defined polynomials \(P_{x,y}\in\mathbb{Z}[q]\) for each \(x\leqslant y\) in \(W\) [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. These polynomials play an important role in representation theory. \textit{M. Dyer} conjectured [Compos. Math. 78, No. 2, 185-191 (1991; Zbl 0784.20019)] that \(P_{x,y}\) depends only on the isomorphism type of the poset \([x,y]=\{z\in W\mid x\leqslant z\leqslant y\}\) for the Bruhat ordering.
The main result of the present paper is to give an affirmative answer to the conjecture for a class of groups, which can be stated as follows. Let \(W,W'\) be two Coxeter groups. Let \(w\in W\) and \(w'\in W'\) be such that the posets \([e,w]\) and \([e',w']\) are isomorphic for the Bruhat orderings on \(W,W'\), where \(e,e'\) are the identity elements of \(W,W'\), respectively. Then any poset isomorphism \(\phi\colon[e,w]\to[e',w']\) preserves Kazhdan-Lusztig polynomials, (in the sense that \(P_{\phi(x),\phi(y)}=P_{x,y}\) for any \(x\leqslant y\) in \([e,w]\)) in the case where one of the two groups \(W,W'\) has the property that the Coxeter graph of each of its irreducible constituents is either a tree or affine of type \(\widetilde A_n\). In particular, the result holds for all finite or affine Coxeter groups.
Note that the above result was also obtained by \textit{F. Brenti} [The intersection cohomology of Schubert varieties is a combinatorial invariant, preprint (2002)] in the case where \(W,W'\) are both of type \(A_n\), as a corollary of a purely combinatorial construction of the \(P_{x,y}\). Bruhat orderings; Kazhdan-Lusztig polynomials; Schubert varieties; Coxeter groups F. du Cloux, ''Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials,'' Adv. in Math. 180 (2003), 146--175. Representation theory for linear algebraic groups, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Combinatorics of partially ordered sets Rigidity of Schubert closures and invariance of Kazhdan-Lusztig 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 a generalized flag manifold \(M=G/K\) of a compact connected simple Lie group \(G\) whose isotropy representation decomposes into more than five isotropy summands, there are only a few results about the homogeneous Einstein metrics on \(M\). Finding the invariant Einstein metrics on generalized flag manifolds, there are two difficulties. One is computing the non-zero structure constants, the other is computing the Gröbner basis of the system of Einstein equations. In this paper, we give a method (Theorem A) which can be used to calculate structure constants of generalized flag manifolds with any number of isotropy summands. In this direction we present invariant Einstein metrics on some flag manifolds of exceptional groups with six isotropy summands. generalized flag manifolds; Einstein metric; isotropy representation; symmetric \(\mathfrak t\)-triples Special Riemannian manifolds (Einstein, Sasakian, etc.), Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Differential geometry of homogeneous manifolds Homogeneous Einstein metrics on certain generalized flag manifolds with six isotropy summands | 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 Here we study the existence of varieties in a Grassmannian \(G(r,n)\) such that all their deformations inside \(G(r,n)\) contain lines and varieties in intersections of quadrics without trisecant lines. Grassmannian; intersections of quadrics; deformations; trisecant lines Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry, Questions of classical algebraic geometry, Formal methods and deformations in algebraic geometry Lines, trisecant lines and subvarieties of Grassmannians and intersections of quadrics | 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 compact connected Lie group, T be a maximal torus in K and T' be its normalizer in K. The flag variety \(X=K/T\) admits a cellular decomposition \(X=\cup X_ w\) with cells indexed by all elements w of the Weyl group \(W=T'/T\). The closures \(\bar X{}_ w\) of these cells determine elements in the dual space to the complex cohomology space \(H^*(X)\) that are called Schubert cycles.
The aim of the present paper is to define and to study the analogues of Schubert cycles for the equivariant cohomology \(H^*_ K(X)\). Identifying (by the Chern-Weil isomorphism) \(H^*_ K(X)\) with the space of polynomial functions on the Lie algebra \({\mathfrak t}\) of T the author gives an explicit formula for the equivariant Schubert cycles in terms of the reduced decompositions of elements of W. polynomial functions on Lie algebra; compact connected Lie group; maximal torus; flag variety; cellular decomposition; Weyl group; Schubert cycles; equivariant cohomology; reduced decompositions A. Arabia, Cycles de Schubert et cohomologie équivariante de \?/\?, Invent. Math. 85 (1986), no. 1, 39 -- 52 (French). Discrete subgroups of Lie groups, Homology with local coefficients, equivariant cohomology, Grassmannians, Schubert varieties, flag manifolds Cycles de Schubert et cohomologie équivariante de K/T. (Schubert cycles and equivariant cohomology of K/T) | 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 in this paper the following results. Let G be a semisimple algebraic group over an algebraically closed field k and Q a parabolic subgroup containing a Borel subgroup B. Let X be a Schubert variety (i.e. the closure of a B orbit) in G/Q. Then (a) If L is a line bundle on G/Q such that \(H^ 0(G/Q,L)\neq 0\) then \(H^ i(X,L)=0\) for \(i>0\) and the restriction map \(H^ 0(G/Q,L)\to H^ 0(X,L)\) is surjective; (b) X is normal; (c) X is projectively normal in any embedding given by an ample line bundle on G/Q. - If we prove the results for fields of positive characteristic they follow for fields of characteristic zero by semicontinuity. When char k\(=0\) we have the absolute Frobenius morphism \(F:X\to X\) defined by raising functions on X to the p-th power. In the preprint ''Frobenius splitting and cohomology vanishing for Schubert varieties'' by \textit{V. B. Mehta} and \textit{A. Ramanathan} it was shown using duality for the Frobenius morphism of the Bott-Samelson-Demazure variety (constructed in the paper of Demazure cited below) that the p-th power map \(0_ X\to F_*0_ X\) admits a section. This quickly gives (a) for ample line bundles L. In this paper we extend this method, by a closer examination of the splitting, to the general case of \(H^ 0(G/Q,L)\neq 0\). We then deduce (b) from (a) by an inductive argument involving the \({\mathbb{P}}^ 1\)-fibrations \(G/B\to G/P\) for suitable minimal parabolic subgroups P containing B.
These results prove the conjectures of \textit{M. Demazure} in his paper in Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009). In particular his character formula for \(H^ 0(X,L)\) for fields of arbitrary characteristic also follows. Incidentally our results uphold the main claims in Demazure's paper in spite of the falsity of proposition 11, {\S}2 of that paper. projective normality; vanishing cohomology groups; Schubert variety; line bundle; Frobenius morphism [RR]Ramanan, S. \&Ramanathan, A., Projective normality of flag varieties and Schubert varieties.Invent. Math., 79 (1985), 217--224. Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) Projective normality of flag varieties 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 The main result of the article is that the Gromov invariants for the scheme of morphisms \(\text{Mor}_ d (C, \text{G} (r,k))\) of sufficiently high degree \(d\) from a Riemannian surface \(C\) into the grassmannian \(\text{G} (r,k)\) of complex \(r\)-planes in \(\mathbb{C}^ k\), can be rigorously defined for the special Schubert cycles, and are realized as an intersection of Chern classes on a projective scheme.
It follows immediately that the Gromov invariants do not depend on the complex structure of the Riemann surfaces, and the algebraic definition of the invariants make them more accessible for computations. -- The author also gives a relation between the Gromov invariants between Riemann surfaces of different genus, enabling him to treat the case \(r=2\).
The main technique of the article is to use the Quot schemes of trivial bundles on Riemann surfaces to obtain a compactification of the scheme \(\text{Mor}_ d (C, \text{G} (r,k))\). The relevant cycles become the intersection with \(\text{Mor}_ d (C, \text{G} (r,k))\) of Chern classes on the Quot schemes. Gromov invariants; scheme of morphisms; special Schubert cycles; Chern classes; trivial bundles on Riemann surfaces; Quot schemes Bertram A.: Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian. Internat. J. Math. 5, 811--825 (1994) Grassmannians, Schubert varieties, flag manifolds, Riemann surfaces; Weierstrass points; gap sequences, Factorization systems, substructures, quotient structures, congruences, amalgams, Schemes and morphisms Towards a Schubert calculus for maps from a Riemann surface to a 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 Let \(G\) be a compact connected Lie group and \(P\subset G\) be the centralizer of a one-parameter subgroup in \(G\). We explain a program that reduces integration along a Schubert variety in the flag manifold \(G/P\) to the Cartan matrix of \(G\).As applications of the program, we complete the project of explicit computation of the degree and Chern number of an arbitrary Schubert variety started in \textit{X. Zhao} and \textit{H. Duan} [J. Symb. Comput. 33, No. 4, 507--517 (2002; Zbl 1046.14027)]. Cartan matrix; flag manifolds; Schubert calculus DOI: 10.1016/j.jsc.2004.03.005 Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symbolic computation and algebraic computation The Cartan matrix and enumerative 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 This paper concerns an interpretation of a special class of holomorphic vector bundles on \(\mathbb P^{2n+1}\) in terms of linear systems on curves. The bundles \({\mathcal E}\) studied are of rank \(2n\) and are generalizations of the bundles on \(\mathbb P^ 3\) which correspond to `t Hooft instantons on \(S^ 4\). Due to the work of \textit{S. M. Salamon} [in Global Riemannian geometry, Proc. Symp., Durham/Engl. 1982, 65--74 (1984; Zbl 0616.53022)] these bundles are also related to Yang-Mills-type equations in higher dimensions, although this is not a principal concern of the authors. The bundles have a number of properties, including the fact that \({\mathcal E}(1)\) has sections and the bundles are simple. Stability unfortunately is not proven.
This particular class of bundles was introduced by \textit{C. Okonek} and \textit{H. Spindler} [J. Reine Angew. Math. 364, 35--50 (1986; Zbl 0568.14009)] and here a study is made of the divisor of jumping lines, which is determined by a hypersurface \(Y\) in a null \(\mathbb P^{2n}\) lying in the Grassmannian \(\text{Gr}_ 2{\mathbb C}^{2n+2}\). The hypersurface \(Y\) is shown to be a Poncelet variety, namely the locus of points on a rational normal curve which are poles of a divisor \(D\) of degree \(r\) such that \(D\leq D'\) for \(D'\) in some fixed linear system of degree \((r+k)\) on the curve. holomorphic vector bundles; linear systems on curves; instantons; Yang-Mills-type equations; Grassmannian Spindler, H., Trautmann, G.: Rational normal curves and the geometry of special instanton bundles on \$\$\{\(\backslash\)mathbb\{P\}\^\{2n+1\}\}\$\$ . Math. Gottingensis 18 (1987) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Constructive quantum field theory, Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Grassmannians, Schubert varieties, flag manifolds Rational normal curves and the geometry of special instanton bundles on \(\mathbb P^{2n+1}\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The article investigates a particular combinatorial structure known as an alternating strand diagram or Postnikov diagram. This is roughly a collection of strands defined on an oriented disc, each with a specified start and end point on the boundary, finitely many intersections with other strands, no self-intersections, and specific laws governing the orientation of intersections. Postnikov diagrams are useful for encoding geometric information, particularly with regard to the Grassmannian, and the main result of the paper concerns the associated dimer algebra of a connected Postnikov diagram.
In short, the main theorem of the article, Theorem 1, is a categorification result, whose underlying algebra is the cluster algebra \(\mathcal{A}_D\) of the associate ice quiver of a Postikov diagram \(D\). The theorem states that the category of Gorenstein-projective modules \(GP(B_D)\) over the boundary \(B_D\) of the dimer algebra of \(D\) can be realised as an additive categorification of this cluster algebra. Since this cluster algebra is closely related to the homogeneous coordinate ring of a positroid variety in the Grassmannian, this result is certainly useful from a geometric point of view. The key step in the argument is the second main result, Theorem 2, which states that the dimer algebra of \(D\) satisfies the appropriate Calabi-Yau property.
The paper begins with a rough outline of the main results and their motivation, focusing heavily on positroid varieties, the current state of their research in the community, and how the results of the article advance this research. After introducing some preliminary material in section 2, including of course the formal definition and basic properties of a Postnikov diagram, its associated ice quiver, the dimer algebra and the cluster algebra, the article goes on in section 3 to explore the Calabi-Yau property. This section focuses on the dimer algebra of a Postnikov diagram, and after describing some algebraic and combinatorial properties of this algebra, it concludes with a complete proof of Theorem 2 (here relabelled Theorem 3.7).
In section 4, the article explores some category theory, and the main theorem (Theorem 4.3) concerns a general algebra satisfying the appropriate Calabi-Yau property, and roughly speaking it states that the associated category of Gorenstein-projective modules is triangulated and satisfies many of the properties required of the categorification of a cluster algebra, and that it contains a specified cluster tilting object. Using Theorem 2, this result of course applies to the dimer algebra.
After proving in section 5 that the quiver associated to the cluster tilting objects in Theorem 4.3 contain no loops or cycles, the article goes on to use this fact to reduce to the situation previously explored by \textit{C. Fu} and \textit{B. Keller} [Trans. Am. Math. Soc. 362, No. 2, 859--895 (2010; Zbl 1201.18007)], and consequently prove Theorem 1 in section 6 (relabelled as Theorem 6.11). Finally, the paper concludes in section 7 with a discussion of the Grassmannian cluster category, and explores how it is closely related to the categorification explored throughout.
Overall, this paper should be of great interest to anyone working with quivers, cluster algebras and interested in their geometric and combinatorial properties. It follows on from previous work of many different authors, none of whom achieved results in as great a generality of this, so it should create a stir in the community. quivers; cluster-algebras; path-algebras; categorification; Grassmannians; triangulated-categories Representations of quivers and partially ordered sets, Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Abelian categories, Grothendieck categories, Derived categories, triangulated categories Calabi-Yau properties of Postnikov diagrams | 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 shown that the Euler characteristic of the variety of Iwahori subalgebras containing a certain nil-elliptic element \(n_t\) is \(t^l\) where \(l\) is the rank of the associated finite type Lie algebra. semisimple groups; Borel subalgebras; Coxeter numbers; Euler characteristic; Iwahori subalgebras; affine flag varieties; Lie algebras Kenneth Fan, C, Euler characteristic of certain affine flag varieties, Transform. Groups, 1, 35-39, (1996) Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Structure theory for Lie algebras and superalgebras Euler characteristic of certain affine 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 reductive algebraic group over a field \(k\) of characteristic \(p\geq 0\). If \(\mathcal L\) is a line bundle on the flag variety \(\mathfrak X\) for \(G\), then the cohomology modules \(H^i(\mathfrak X,\mathcal L)\) have a natural \(G\)-structure. The \(G\)-modules arising in this way play a prominent role in the representation theory of \(G\).
This is for instance illustrated by the following four groups of results: 1) the Chevalley classification of irreducible \(G\)-modules, 2) the strong linkage principle, 3) the sum formula for Weyl modules, 4) restriction to Schubert varieties.
Let \(U_q\) denote the quantum group corresponding to \(G\). Here \(q\) is an arbitrary nonzero element of \(k\) and \(U_q\) is the specialization of the Lusztig integral form of the generic quantum group. Then we have quantized versions of the cohomology modules \(H^i({\mathfrak X},\mathcal L)\) constructed via induction from the quantized Borel subalgebra of \(U_q\). It turns out that there analogues of all the above results. Despite the many efforts leading to the above-mentioned results, there is still an abundance of open questions concerning the cohomology modules \(H^i(\mathfrak X,\mathcal L)\).
In this note, the author collects some of the known facts about these modules and calls attention to several open problems. This is done in such a way that each result and question may easily be quantized. Related to the problem of describing the cohomology of line bundles on \(\mathfrak X\) is the calculation of the Hochschild cohomology groups for \(B\). The author considers the quantized root of unity analogues of these computations and shows that they are related to the cohomology of line bundles on the cotangent bundle of \(\mathfrak X\). reductive group; flag variety; line bundle; vanishing theorem Andersen, Henning Haahr: Cohomology of line bundles, Algebraic groups and homogeneous spaces, 13-36 (2007) Grassmannians, Schubert varieties, flag manifolds, Vanishing theorems in algebraic geometry, Group actions on varieties or schemes (quotients), Divisors, linear systems, invertible sheaves Cohomology of line bundles | 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 \(Y\) be a smooth congruence of lines in complex projective 3-space. Assume that \(Y\) has a curve \(C\) of singular points. The authors give a complete classification of all such congruences. These are four classes which are -- roughly speaking as follows:
(I) \(C\) is a line;
(II) \(Y\) is the set of bisecants of \(C\), with \(C\) a twisted cubic or an elliptic quartic;
(III) \(C\) is a scroll with \(C\) being a conic or a smooth plane cubic;
(IV) \(Y\) is a conic bundle over a smooth plane cubic.
In this classification there is one type of congruences that is missing in \textit{A. Verra}'s paper [ Manuscr. Math. 62, No. 4, 417-435 (1988; Zbl 0673.14026)]. congruence of lines; fundamental curve Special surfaces, Grassmannians, Schubert varieties, flag manifolds On smooth surfaces in Gr\((1,\mathbb{P}^ 3)\) with a fundamental curve | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper is a beautiful contribution to the emerging subject of elliptic geometric representation theory. Its introductory sections review, reformulate, and explain equivariant elliptic cohomology, and the related algebraic constructions (essentially the elliptic Hecke algebras, elliptic quantum groups). The first remarkable result is the identification of two constructions of elliptic Hecke algebras: one defined using convolution products and the other the algebraic elliptic Hecke algebra. After this preparation the main result is Theorem B, the bijection between irreducible representations of the elliptic affine Hecke algebra and certain geometric objects, namely nilpotent Higgs bundles. This identification can be interpreted as the elliptic Deligne-Langlands correspondence. The geometric tool used in the proof is the equivariant elliptic cohomology of a Steinberg variety. Combinatorial aspects of the representations at roots of unity are also studied. Another byproduct of the main results of the paper is the derivation of Demazure-Lusztig operators in equivariant elliptic cohomology. These had been the main tools in elliptic Schubert calculus. equivariant elliptic cohomology; Steinberg variety; elliptic affine Hecke algebra; Higgs bundle, elliptic Demazure-Lusztig operators Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Elliptic cohomology Representations of the elliptic affine Hecke 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 0577.00010.]
For a symmetrizable generalized Cartan matrix \(A=(a_{ij})\), \(1\leq i,j\leq \ell\) we denote by \(\mathfrak g=\mathfrak g(A)\) the corresponding Kac-Moody Lie algebra. With this algebra there is associated a topological group \(G\) called algebraic group associated with \(\mathfrak g\). The paper under review is devoted to the study of the topological structure of \(G\).
Let \(X\subset \{1,...,\ell \}\) be a subset of finite type, i.e. the submatrix \(A_ X=(a_{ij})\), \(i,j\in X\), is a classical Cartan matrix of finite type. We denote: \(\mathfrak p_ X\) the parabolic subalgebra of \(\mathfrak g\) defined by \(X\), \(\mathfrak r_ X\) the reductive part of \(\mathfrak p_ X\), \(P_ X\) the parabolic subgroup of \(G\) associated with \(\mathfrak p_ X\), \(B=P_{\emptyset}\) the Borel subgroup of \(G\), and \(\mathfrak g^ 1=[\mathfrak g,\mathfrak g]\) the derived algebra of \(\mathfrak g\), which in a certain sense is a Lie algebra of \(G\).
The author presents results of three types. First he proves that the Lie algebra cohomology with trivial complex coefficients \(H^*(\mathfrak g^ 1)\) (resp. \(H^*(\mathfrak g,\mathfrak r_ X))\) is isomorphic, as a graded algebra, with the singular cohomology \(H^*(G, \mathbb C)\) (resp. \(H^*(G/P_ X, \mathbb C))\). The isomorphism is explicitly given by an integration map. There are interesting corollaries, i.e. that both \(H^*(\mathfrak g)\) and \(H^*(\mathfrak g^ 1)\) are Hopf algebras, results about low dimensional cohomologies, etc.
Secondly the author shows that the cochain complex \(C(\mathfrak g,\mathfrak r_ X)\) is a formal DG-algebra, and that the quotient space \(G/P_ X\) is a \(\mathbb Q\)-formal space. (Formality of a DG-algebra is the usual formality in the minimal models theory, and \(\mathbb Q\)-formality of a topological space is that from the rational homotopy theory.) In the last, third part the author constructs the minimal model for \(G/B\), and determines the homotopy Lie algebra \(\pi_*(G/B)\otimes\mathbb Q\). flag varieties; Kac-Moody group; generalized Cartan matrix; Kac-Moody Lie algebra; Lie algebra cohomology; formal DG-algebra; \(\mathbb Q\)-formal space; rational homotopy theory; minimal model; homotopy Lie algebra Kumar, S.: Rational homotopy theory of flag varieties associated to Kac-Moody groups. MSRI publications 4, 233-273 (1985) Loop groups and related constructions, group-theoretic treatment, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Rational homotopy theory, Grassmannians, Schubert varieties, flag manifolds, Cohomology of Lie (super)algebras Rational homotopy theory of flag varieties associated to 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 If \(Y\) is an affine symmetric variety for a reductive group \(G\) with Weyl group \(W\), there exists by Lusztig and Vogan a representation of the Hecke algebra of \(W\) in a module which has a basis indexed by the set \(\Lambda\) of pairs \((v,\xi)\), where \(v\) is an orbit in \(Y\) of a Borel group \(B\) and \(\xi\) is a \(B\)-equivariant rank one local system on \(v\). We introduce cells in \(\Lambda\) and associate with a cell a two-sided cell in \(W\). affine symmetric varieties; reductive groups; Weyl group; Hecke algebra; Borel subgroup; local system; two-sided cell Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Representation theory for linear algebraic groups Cells for a Hecke algebra representation. | 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 \(\mathcal O_q(M_{m,n}(k))\) be a generic coordinate \(k\)-algebra on rectangular matrices of size \(m\times n\). It is assumed that \(m\leqslant n\). If \(I=\{i_1<\cdots<i_u\}\), \(K=\{k_1<\cdots<k_v\}\subset M=\{1,\dots,m\}\) and \(J=\{j_1<\cdots<j_u\}\), \(L=\{l_1<\cdots<l_v\}\subset\{1,\dots,n\}\), then \((I,J)\leqslant_{st}(K,L)\) if and only if \(u\leqslant v\) and \(i_s\leqslant k_s\), \(j_s\leqslant l_s\) for all possible indices \(s\). Denote by \(\Pi_{m,n}\) the set of all pairs of indices \((I,J)\) in which \(u=m\).
The quantum Grassmannian \(\mathcal O_q(G_{m,n}(k))\) is the subalgebra in \(\mathcal O_q(M_{m,n}(k))\) generated by all \(m\times m\) quantum minors. In the authors' previous paper [J. Algebra 301, No. 2, 670-702 (2006; Zbl 1108.16026)], it is shown that there is a vector basis in \(\mathcal O_q(G_{m,n}(k))\) consisting of monomials \([I_1\mid M]\cdots [I_t\mid M]\) such that \((I_1,M)\leqslant_{st}\cdots\leqslant_{st}(I_t,M)\).
Take \(\gamma\in\Pi_{m,n}\) and denote by \(\Pi_{m,n}^\gamma\) the set of all \(\alpha\in\Pi_{m,n}\) such that \(\alpha\ngeqslant_{st}\gamma\). The quantum Schubert variety \(\mathcal O_q(G_{m,n}(k))_\gamma\) associated with \(\gamma\) is the algebra \(\mathcal O_q(G_{m,n}(k))\) factorized by the ideal generated by \(\Pi_{m,n}^\gamma\). It is shown that
\[
\mathcal O_q(G_{m,n}(k))[Y^{\pm 1};\varphi]\simeq\mathcal O_q(G_{m,n}(k))_\gamma[\gamma^{-1}].
\]
It is given a criterion under which a quantum Schubert variety has left and right finite injective dimensions. Let \(I_t\) be the ideal in \(\mathcal O_q(M_{m,n}(k))\) generated by all minors of a fixed size \(t\). It is shown that \(\mathcal O_q(M_{m,n}(k))/I_t\) is a normal domain. rings arising in quantum group theory; generic coordinate algebras; quantum Grassmannians; quantum minors; quantum Schubert varieties; normal domains T. H. Lenagan and L. Rigal, Quantum analogues of Schubert varieties in the Grassmannian, Glasg. Math. J. 50 (2008), no. 1, 55 -- 70. , Grassmannians, Schubert varieties, flag manifolds, Divisibility, noncommutative UFDs, Rings arising from noncommutative algebraic geometry, Noetherian rings and modules (associative rings and algebras) Quantum analogues of Schubert varieties in 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 use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type \(\mathbb{A}\), we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type \(\mathbb{A}\) quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation. Syzygies, resolutions, complexes and commutative rings, Determinantal varieties, Representations of quivers and partially ordered sets, Singularities in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Grassmannians, Schubert varieties, flag manifolds Free resolutions of orbit closures of Dynkin quivers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It has been shown recently that the normalized median Genocchi numbers are equal to the Euler characteristics of the degenerate flag varieties. The \(q\)-analogues of the Genocchi numbers can be naturally defined as the Poincaré polynomials of degenerate flag varieties.
We prove that the generating function of the Poincaré polynomials can be written as a simple continued fraction. As an application we prove that the Poincaré polynomials coincide with the \(q\)-version of the normalized median Genocchi numbers introduced by Han and Zeng. Poincaré polynomials of degenerate flag varieties Feigin, E, The Median Genocchi numbers, \(q\)-analogues and continued fractions, Eur. J. Comb., 33, 1913-1918, (2012) Factorials, binomial coefficients, combinatorial functions, \(q\)-calculus and related topics, Grassmannians, Schubert varieties, flag manifolds, Bernoulli and Euler numbers and polynomials The median Genocchi numbers, \(q\)-analogues and continued fractions | 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 give a simple necessary and sufficient condition for a Schubert variety \(X_w\), to be smooth when \(w\) is a freely braided element of a simply laced Weyl group; such elements were introduced by the authors in a previous work [Ann. Comb. 6, No. 3--4, 337--348 (2002; Zbl 1052.20028)]. This generalizes in one direction a result of \textit{C.~K.~Fan} [Transform. Groups 3, No. 1, 51--56 (1998; Zbl 0912.20033)] concerning varieties indexed by short-braid avoiding elements. They also derive generating functions for the freely braided elements that index smooth Schubert varieties. All results are stated and proved only for the simply laced case. Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Schubert varieties and free braidedness | 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 non-degenerate irreducible \(n\)-dimensional variety in \({\mathbb P}^N\). The classical definition of the secant variety of \(X\) has a natural generalization in the \(k\)-Grassmannians of secant planes of \(X\), which are subvarieties of the Grassmannian of \(k\)-planes in \({\mathbb P}^N\). Their study finds application, for instance, in the study of the Waring problem [see \textit{C. Ciliberto}, in: 3rd European congress of mathematics (ECM), Barcelona, Spain, July 10--14, 2000. Volume I. Prog. Math. 201, 289-316 (2001; Zbl 1078.14534)].
The article under review deals with the Grassmannian \(G_{k-1,k}(X)\), i.e. the subvariety of \({\mathbb G}(k-1,{\mathbb P}^N)\) obtained by taking the closure of the set of \(k-1\)-planes of \({\mathbb P}^N\) that are contained in the linear span of \(k+1\) independent points of \(X\). If \(N\geq n+k+1\), then the \(G_{k-1,k}\)-defect of \(X\) is the integer \(a=(k+1)n+k-\text{dim}G_{k-1,k}(X)\geq 0\). In this article, it is shown that the defect is at most \(n-1\) if \(k\geq n\). Furthermore, a characterization of varieties with given defect \(a>0\) is obtained for \(k\geq n\). This is used to give a detailed classification of all smooth non-degenerate varieties \(X\) with defect \(a\geq n-2\) for an integer \(k\geq n\), extending to the case \(n\geq 4\) the results of \textit{F. Cools} [J. Pure Appl. Algebra 203, 204--219 (2005; Zbl 1082.14053)] and \textit{M. Coppens} [Int. J. Math. 15 (7), 651--671 (2004; Zbl 1057.14051)]. DOI: 10.1080/00927870701327807 Classical problems, Schubert calculus, \(n\)-folds (\(n>4\)), Grassmannians, Schubert varieties, flag manifolds On high \(G_{k-1,k}\)-defective 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 \(X\) be a smooth irreducible projective curve of genus \(\geq 2\) over any algebraically closed field \(k\) and let \(G(r,\nu)\) be the Grassmannian of rank \(r\) quotients of \(k^\nu\) with the tautological quotient bundle \(Q\) of rank \(r\) on \(G(r,\nu)\). In the paper under review, it is shown that for \(d\geq\nu +r(g-1)\) and \((\nu,r,d) \neq(4,2,2g+2)\), the set \(\{\varphi^* (Q)\}_\varphi\) forms a dense open subset of the moduli space, \(M(X;r,d)\) of rank \(r\) stable vector bundles on \(X\) with degree \(d\), where \(\varphi: X\to G(r,\nu)\) runs over degree \(d\) embeddings. An analogous result is obtained for all the flag varieties of \(\text{GL}_\nu\) as well. There are similar results obtained for the tangent bundle of \(\mathbb{P}^n\). moduli space of rank \(r\) stable vector bundles; Grassmannian; flag varieties Ballico, E.; Ramella, L., The restricted tangent bundle of smooth curves in grassmanians and curves in flag varieties, Rocky Mountain J. Math., 30, 1207-1227, (2000) Vector bundles on curves and their moduli, Grassmannians, Schubert varieties, flag manifolds, Algebraic moduli problems, moduli of vector bundles The restricted tangent bundle of smooth curves in Grassmannians and curves in 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 Michel's theory of symmetry breaking in its original formulation has some difficulty in dealing with problems with a linear symmetry, due to the degeneration in the symmetry type implied by the linearity of group action. Here, we propose a fully geometric, approach to the problem, making use of Grassmann manifolds. In this way, Michel theory can also be applied to the determination of dynamically invariant manifolds for equivariant nonlinear flows. spontaneous symmetry breaking; Grassman manifolds; Nahm equations Symmetry breaking in quantum theory, Grassmannians, Schubert varieties, flag manifolds Breaking of linear symmetries and Michel's theory: Grassmann manifolds, and invariant subspaces | 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 B denote a Borel subgroup of a semi-simple group G. \(P^-\) stands for the set of B-modules of dimension 1 such that \(H^ 0(G/B,{\mathcal L}(\lambda))\neq 0\). \({\mathcal S}\) denotes the class of B-modules, having a Schubert filtration. The main result is as follows: For every union S of Schubert varieties, every antidominant weight \(\lambda\) and every \(\mu \in P^-\), \(M:=H^ 0(S,\lambda)\otimes \mu\) is a B-module, having a Schubert filtration that allows an explicit description. Furthermore, if S is irreducible, M allows an excellent filtration (meaning that M allows a composition series, with quotients isomorphic to \(H^ 0(X_ w,\lambda)\). This work, - though independently developed, - extends the work of Mathieu (characteristic zero). Borel subgroup; semi-simple group; Schubert filtration; Schubert varieties; antidominant weight; excellent filtration Polo, P, Modules associés aux variétés de Schubert, C. R. Acad. Sci. Paris Sér. I Math., 308, 123-126, (1989) Cohomology theory for linear algebraic groups, Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds Modules associés aux variétés de Schubert. (Modules associated with 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 \(G\) be a reductive linear algebraic group split over \(\mathbb{R}\) with fixed pinning \((T,B^+,B^-,x_i,y_i,i\in I)\) [see \textit{G. Lusztig}, Lie theory and geometry: in honor of Bertram Kostant, Prog. Math. 123, 531-568 (1994; Zbl 0845.20034) and Positivity in Lie theory: open problems, de Gruyter Expo. Math. 26, 133-145 (1998)]. Let \(\mathcal B\) be the variety of all Borel subgroups of \(G\). Let \(W=N_G(T)/T\) be the Weyl group of \(G\) with its simple reflection set corresponding to the pinning. Write \(\ell(w)\) for the length of \(w\in W\) and \(\dot w\) for a representative of \(w\) in \(N_G(T)\). Let \(U^+\) be the unipotent radical of \(B^+\). Denote by \(U^+_{\geq 0}\) the semigroup generated by \(\{x_i(a)\mid a\in\mathbb{R}_{\geq 0},\;i\in I\}\subset U^+\). Let \(U^+_{>0}=B^-\dot w_0B^-\cap U^+_{\geq 0}\) for the longest element \(w_0\) of \(W\). For \(B\in{\mathcal B}\) and \(g\in W\), write \(g\bullet B\) for \(gBg^{-1}\). Then the totally nonnegative part \({\mathcal B}_{\geq 0}\) of the flag variety \(\mathcal B\) is defined to be the closure of \(U^+_{>0}\bullet B^-\) in \(\mathcal B\). Let \({\mathcal C}^-_w=B^-\dot w\bullet B^-\) and \({\mathcal C}^+_w=B^+\dot w\bullet B^-\). Write \({\mathcal R}_{x,y}={\mathcal C}^-_y\cap{\mathcal C}^+_x\). The main result of the paper is to show that for any \(x\leq y\) in \(W\), the intersection \({\mathcal R}_{x,y}\cap{\mathcal B}_{\geq 0}\) is homeomorphic to \(\mathbb{R}^{\ell(y)-\ell(x)}_{>0}\) by a homeomorphism which extends to a real algebraic morphism \((\mathbb{R}\setminus\{0\})^{\ell(y)-\ell(x)}\to{\mathcal R}_{x,y}\). This result was conjectured by \textit{G. Lusztig} [see loc. cit.]. total positivity; varieties of Borel subgroups; reductive linear algebraic groups; Weyl groups; flag varieties; real algebraic morphisms K. Rietsch, An algebraic cell decomposition of the nonnegative part of a ag variety, J. Algebra 213 (1999), 144--154. Linear algebraic groups over the reals, the complexes, the quaternions, Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds An algebraic cell decomposition of the nonnegative part of a flag variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of \textit{A. Weigandt} [J. Comb. Theory, Ser. A 182, Article ID 105470, 52 p. (2021; Zbl 1475.05172)] relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of \textit{K. Motegi} and \textit{K. Sakai} [J. Phys. A, Math. Theor. 46, No. 35, Article ID 355201, 26 p. (2013; Zbl 1278.82042)] to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by \textit{P. J. McNamara} [Electron. J. Comb. 13, No. 1, Research paper R71, 40 p. (2006; Zbl 1099.05078)]. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström-Gessel-Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations. integrable colored six-vertex model; Lindström-Gessel-Viennot lemma Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Double Grothendieck polynomials and colored lattice models | 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 0592.00036.]
The equations of the \(\kappa\)-dimensional invariant subspaces of a nilpotent linear transformation over an arbitrary field are obtained. invariant subspaces; nilpotent linear transformation; algebraic subvariety; Grassmannian variety; Jordan form Canonical forms, reductions, classification, Eigenvalues, singular values, and eigenvectors, Grassmannians, Schubert varieties, flag manifolds On varieties of invariant subspaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the Grassmannian \(\mathcal G_K(k,n)\) over any field \(K\), some flags \(F_i\), \(i=1,\dots,m\), and consider the corresponding sets of conditions \(\Omega(F_i)\) on the intersection of \(k\)-planes with the elements of each \(F_i\). This datum defines a Schubert problem \(\omega=\{\Omega(F_i)\}\). The author studies the case where the expected set of solutions is finite. The problem \(\omega\) is enumerative over \(K\) when one finds the expected number \(\deg(\{\Omega(F_i)\})\) of solutions, formed by distinct \(k\)-planes, all of them defined over \(K\).
The author proves that any Schubert problem is enumerative over \(\mathbb R\), as well as over algebraically closed field of any characteristic. Moreover, he finds that Schubert problems over finite fields are enumerative in a set of positive density. One tool for proving the results is an extension of the Bertini-Kleiman smoothness condition. The author proves that the map from the set of solutions in the universal Grassmannian, to the product of \(m\) copies of the flag variety, is generically smooth.
The author finally studies the Galois group of an enumerative problem, for the Grassmannians of lines and planes. Schubert cycles R. Vakil, Schubert induction. Ann. Math. 164, 489--512 (2006) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Enumerative problems (combinatorial problems) in algebraic geometry Schubert induction | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Codimension two smooth subvarieties in \(\mathbb P^n\) have been studied by many authors, in order to prove that they are the complete intersection of two surfaces or, at least, subcanonical [see \textit{R. Hartshorne,} Bull. Am. Math. Soc. 80, 1017--1032 (1974; Zbl 0304.14005)]. The authors of this paper focus their attention on codimension two smooth subvarieties of the Grassmannian \(G(1,4)\) of lines in \(\mathbb P^4\). In fact they give a complete and detailed classification of the subvarieties whose degree is at most \(25\), proving that they are always subcanonical and either complete intersections or zero loci of some twist of the rank two universal bundle on the Grassmannian. They also give a classification of those subvarieties that are not of general type, proving that they have bounded degree. DOI: 10.1142/S0129167X06003436 Low codimension problems in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Evidence to subcanonicity of codimension two subvarieties of \(\mathbb G(1,4)\) | 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 formulates an approach to supergravity by using the idea of Penrose transformation. Recall that results of Ogevetsky, Sokatchev and Schwarz imply a ''compensation'' of the hyperbolicity of Einstein equations by the non-commutativity of the superextension so that supergravity may be described by means of complex variables. In the author's formulation the main structure is a complex supermanifold, i.e. a sheaf of analytic Grassmann algebras. The examples are superspaces of flags of complex superdimension 4N. At the beginning the author presents supergeometry of the aforementioned flag spaces. Then he defines a so called Frobenius form for general supermanifolds. This form helps him to prove two formulae of decomposition of sheaves of Berezinian's into objects generated by left and right spinors. The author observes that the Lagrangian of simple supergravity is equal to the Berezinian of a transformation between two distinguished frames. In the paper he defines these frames explicitly by means of the Frobenius form, the Ogevetsky- Sokatchev prepotential and the real structure on the base supermanifold. In this way the presented theory becomes mathematically complete. The author notes that his approach leads to a contradiction with physical models of extended supergravity in cases \(N=2,4\). The reviewed paper is the first one where all main data of a supergravity were formulated in terms of advanced global analytic structures. super Schubert cells; analytic supermanifold; Frobenius form; supergravity; Penrose transformation; complex supermanifold; sheaf of analytic Grassmann algebras; flag spaces; Berezinian Applications of global differential geometry to the sciences, Grassmannians, Schubert varieties, flag manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Constructive quantum field theory, General relativity Geometry of supergravity and super Schubert cells | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\Sigma\) be an exterior differential system and let \(B_ k\) be the fiber bundle of ordinary integral k-flags of \(\Sigma\). The main result is: If b is a fixed ordinary integral k-flag then in a neighborhood of b the set of ordinary integral k-elements of \(\Sigma\) is a Schubert manifold (a submanifold of a projective Grassmann manifold). exterior differential system; integral k-flag; Schubert manifold Exterior differential systems (Cartan theory), Grassmannians, Schubert varieties, flag manifolds, Integral geometry Schubert conditions for integral elements of an exterior differential system | 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 Postnikov constructed a decomposition of a totally nonnegative Grassmannian \((\mathrm{Gr}_{kn}) \geq 0\) into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point in \((\mathrm{Gr}_{kn}) \geq 0\) belongs to and to determine affine coordinates of the point within this cell. This simplifies Postnikov's description of the inverse boundary measurement map and generalizes formulas for the top cell given by Speyer and Williams. In addition, we identify a particular subset of Plücker coordinates as a totally positive base for the set of non-vanishing Plücker coordinates for a given positroid cell. positroid; totally nonnegative Grassmannian; Le-diagram Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Combinatorial formulas for \(\lrcorner\)-coordinates in a totally nonnegative Grassmannian, 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 In this paper authors associate a class in the K-theory of the Grassmannian to every matroid. They study this class by using the method of equivariant localization. In particular, they provide a geometric interpretation of the Tutte polynomial and they extend the second author's results concerning the behavior of such classes under direct sum, series and parallel connection, and two-sum. These results were previously only established for realizable matroids, and their earlier proofs were more difficult. matroid; Tutte polynomial; K-theory; equivariant localization; Grassmannian Fink, A.; Speyer, DE, \(K\)-classes for matroids and equivariant localization, Duke Math. J., 161, 2699-2723, (2012) Combinatorial aspects of matroids and geometric lattices, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds \(K\)-classes for matroids and equivariant localization | 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 a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert's algorithm for moving cells down. In this way, for every weak composition one can choose a cell diagram with corresponding row-counts, with each choice giving rise to a combinatorially-defined basis of polynomials. These \textit{Kohnert bases} provide a simultaneous generalization of Schubert polynomials and Demazure characters for the general linear group. Using the monomial and fundamental slide bases defined earlier by the authors, we show that Kohnert polynomials stabilize to quasisymmetric functions that are nonnegative on the fundamental basis for quasisymmetric functions. For initial applications, we define and study two new Kohnert bases. The elements of one basis are conjecturally Schubert-positive and stabilize to the skew-Schur functions; the elements of the other basis stabilize to a new basis of quasisymmetric functions that contains the Schur functions. Schubert polynomials; Demazure characters; key polynomials; fundamental slide polynomials Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric functions and generalizations Kohnert 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 Givental's work on equivariant Gromov-Witten invariants has established the relationship between quantum cohomology and hypergeometric series [see \textit{A. B. Givental'}, Int. Math. Res. Not. 1996, No. 13, 613-663 (1996; Zbl 0881.55006)]. Consider a toral action on a compact Kähler manifold \(X\) with finitely many fixed points \(x_w\). The solutions of the differential equations arising from quantum cohomology are related to equivariant correlators \(Z_w\) associated with the \(x_w\). The correlators \(Z_w\) are the hypergeometric series associated with equivariant quantum cohomology, and can be uniquely determined by linear recursion relations. The main result in the paper under review is the explicit determination of the recursion relations for the \(Z_w\) on flag spaces \(X=G/B\). Here \(G\) is the simply connected algebraic group associated with a finite root system \(R\). Hence \(X\) is a homogeneous space for the action of the maximal torus of \(G\). The set of fixed points is in this case finite and parametrized by the Weyl group of \(R\). A simple explicit formula for the \(Z_w\) is presented in the case \(G=SL(3)\). Givental' recursion relations; hypergeometric functions; quantum cohomology; equivariant Gromov-Witten invariants; flag spaces Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Other hypergeometric functions and integrals in several variables, Quantization in field theory; cohomological methods, Grassmannians, Schubert varieties, flag manifolds On hypergeometric functions connected with quantum cohomology of flag spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Given a finite set of subspaces of \(\mathbb{R}^n\), perhaps of differing dimensions, we describe a flag of vector spaces (i.e. a nested sequence of vector spaces) that best represents the collection based on a natural optimization criterion and we present an algorithm for its computation. The utility of this flag representation lies in its ability to represent a collection of subspaces of differing dimensions. When the set of subspaces all have the same dimension \(d\), the flag mean is related to several commonly used subspace representations. For instance, the \(d\)-dimensional subspace in the flag corresponds to the extrinsic manifold mean. When the set of subspaces is both well clustered and equidimensional of dimension \(d\), then the \(d\)-dimensional component of the flag provides an approximation to the Karcher mean. An intermediate matrix used to construct the flag can also be used to recover the canonical components at the heart of Multiset Canonical Correlation Analysis. Two examples utilizing the Carnegie Mellon University Pose, Illumination, and Expression Database (CMU-PIE) serve as visual illustrations of the algorithm. subspace average; Grassmann manifold; flag manifold; SVD; flag mean; multiset canonical correlation analysis; extrinsic manifold mean B. Draper, M. Kirby, J. Marks, T. Marrinan, and C. Peterson, \textit{A flag representation for finite collections of subspaces of mixed dimensions}, Linear Algebra Appl., 451 (2014), pp. 15--32. Grassmannians, Schubert varieties, flag manifolds, Local Riemannian geometry, Eigenvalues, singular values, and eigenvectors, Statistical aspects of information-theoretic topics, Image processing (compression, reconstruction, etc.) in information and communication theory, Cyclic codes A flag representation for finite collections of subspaces of mixed dimensions | 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 \({\mathcal B}\) be the Borel subgroup of \(SL(2, \mathbb{C})\). Let \(X\) be a smooth projective variety carrying an algebraic action of \({\mathcal B}\) with a unique fixed point. Let \(Z\) be the zero scheme of the nilpotent vector field generated by the maximal unipotent subgroup of \({\mathcal B}\). Then the cohomology algebra \(H^* (X)\) is naturally isomorphic to the coordinate ring \(A(Z)\) [see e.g. \textit{E. Akyildiz} and \textit{J. B. Carrell}, Manuscr. Math. 58, 473-486 (1987; Zbl 0626.14017)]. Let now \(Y\) be a (not necessarily irreducible) \({\mathcal B}\)-invariant subvariety of \(X\). It is natural to ask whether \(H^* (Y)\) is isomorphic to the coordinate ring of the schematic intersection of \(Y\) and \(Z\).
In this paper, it is shown that the answer is positive if, and only if, the cohomology map \(i^* : H^* (X) \to H^* (Y)\) is surjective, and the schematic intersection of \(Y \times \mathbb{C}\) and \({\mathcal Z}\) is reduced. Here \({\mathcal Z}\) is a well-chosen affine curve in \(X \times \mathbb{C}\). -- Let \({\mathcal Z}_Y\) be the subvariety \(Y \times \mathbb{C} \cap {\mathcal Z}\). This is a fundamental invariant of \(Y\) and determines completely this structure of \(H^* (Y)\) when the cohomology map \(i^* : H^* (X) \to H^* (Y)\) is surjective; for instance, in that case, the Poincaré polynomial of \(H^* (Y)\) is symmetric when \({\mathcal Z}_Y\) is Gorenstein.
This has interesting applications to flag varieties. invariant subvariety; cohomology algebra; schematic intersection; flag varieties J. B. Carrell, Deformation of the nilpotent zero scheme and the intersection rings of invariant subvarieties, J. Reine Angew. Math. 460 (1995), 37--54. Classical real and complex (co)homology in algebraic geometry, Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds Deformation of the nilpotent zero scheme and the intersection ring of invariant subvarieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory [Zbl 1035.17016, Zbl 1048.14032, Zbl 1064.14053, Zbl 1165.17302, Zbl 1073.14551]. We also describe related work and recent progress.
After an overview the authors review the construction of complex simple Lie algebras via geometry (Section 2), study triality and exceptional Lie algebras (Section 3), and finally series of Lie algebras via knot theory and geometry (Section 4), among others they investigate Vogel's decompositions and Tits correspondences, Freudenthal geometries and a geometric magic square. Landsberg J. M., ''Representation Theory and Projective Geometry.'' Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Representation theory and projective geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that a von Neumann algebra is finite if and only if its Grassmannians are small in a certain sense related to the atlas of affine coordinates. von Neumann algebra; finite; Grassmannians; affine coordinates General theory of von Neumann algebras, Grassmannians, Schubert varieties, flag manifolds Affine coordinates and finiteness | 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 compact Riemann surface of genus g; \(x_{\infty}\in X\) a point, U a suitable neighborhood of \(x_{\infty}\); \(X_ 0=\overline{X\setminus U}\), \(X_{0\infty}=X_ 0\cap U\), \(U_{0\infty}\) a ``good'' neighborhood of \(X_ 0\). A transfer function \(\phi\) of degree m is a holomorphic map of \(U_{0\infty}\) into GL(m,\({\mathbb{C}})\). To such a function is naturally associated a holomorphic vector bundle \(L_{\phi}\) of rank m over X. Let \({\mathbb{L}}_{\phi}\) be the corresponding locally free sheaf. \({\mathbb{L}}_{\phi}(X_ 0)\) is embedded in \(\ell_ 2({\mathbb{Z}})^ m\). \(W_{\phi}\) denotes the closure of \({\mathbb{L}}_{\phi}(X)\) in \(\ell_ 2({\mathbb{Z}})^ m\) and lies in the Grassmann manifold Gr(\({\mathbb{H}})\) \(({\mathbb{H}}=\ell^ 2({\mathbb{Z}})^ n)\). The space Gr(\({\mathbb{H}})\) is defined with respect to the splitting \({\mathbb{H}}={\mathbb{H}}_+\oplus H_ -\) with \(H_+=\{sequences\) \((a_ k)\in {\mathbb{H}}\), \(a_ k=0\) for \(k<0\}\). It consists of all closed subspaces of \({\mathbb{H}}\) images of an embedding w: \({\mathbb{H}}_+\to {\mathbb{H}}\) that decomposes in \({\mathbb{H}}_+\oplus {\mathbb{H}}_ -\) in \((w_+,w_ -)\), with \(w_+\) Fredholm and \(w_ -\) bounded. The index of \(w_+\) is called the virtual dimensional of the image of w.
The author proves that the virtual dimension of \(W_{\phi}\) is given by
\[
h^ 0(X,{\mathbb{L}}_{\phi})\quad -\quad m\quad -\quad h^ 1(X,{\mathbb{L}}_{\phi})= c(\det {\mathbb{L}}_{\phi})\quad -\quad mg\quad (c(\det {\mathbb{L}}_{\phi})=Chern\quad class).
\]
Grassmann manifold; holomorphic vector bundles; compact Riemann surface; virtual dimensional Helminck, G.: J. Math. Phys. 29 (1983), 1974-1978; with G. Post, Lett. Math. Phys. 16 (1988), 359-364. Holomorphic bundles and generalizations, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Compact Riemann surfaces and uniformization, Pseudodifferential operators as generalizations of partial differential operators, Pseudodifferential and Fourier integral operators on manifolds, Riemann-Roch theorems, Grassmannians, Schubert varieties, flag manifolds Matrix hierarchies and vector bundles over Riemann surfaces | 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 \(G/B\) be the complete flag variety for \(G=\mathrm{GL}_n(C)\), and let \(K\subset G\) be either \(\mathrm{Sp}_n(\mathbb C)\) (for even \(n\)) or \(O_n(\mathbb C)\). The author extends several classical results on Schubert varieties (=closures of \(B\)-orbits) to the closures of \(K\)-orbits (under the left action of \(K\) on \(G/B\)). He obtains recursive formulas for \(S\)-equivariant cohomology classes of such orbit closures in \(H^*_S(G/B)\) where \(S\) is a maximal torus of \(K\). The main tools are divided difference operators and equivariant localization to \(S\)-fixed points. The author also uses that \(\mathrm{Sp}_n(\mathbb C)\) and \(\mathrm{O}_n(\mathbb C)\) are symmetric subgroups of \(\mathrm{GL}_n(\mathbb C)\), and hence, their orbits can be described combinatorially using results of [\textit{R. W. Richardson} and \textit{T. A. Springer}, Geom. Dedicata 35, No. 1--3, 389--436 (1990; Zbl 0704.20039)].
The author interprets formulas for classes of orbit closures in \(H^*_S(G/B)\) as formulas for the following degeneracy loci. Let \(V\to X\) be a rank \(n\) vector bundle over a smooth complex variety \(X\) endowed with a complete flag of subbundles \(F_\bullet=(F_1\subset F_2\subset\ldots\subset F_{n-1}\subset V)\). Furthermore, let \(\gamma \) be a non-degenerate symmetric or skew-symmetric bilinear form on \(V\) with the values in the trivial bundle. For every involution \(b\in S_n\) (assumed fixed point-free if \(\gamma\) is skew-symmetric), define the degeneracy locus
\[
D_b=\{x\in X\;|\;\mathrm{rank}(\gamma|_{F_i(x)\times F_j(x)})\leq r_b(i,j) \;\forall i,j \},
\]
where \(r_b(i,j)\) is a nonnegative integer that depends on \(b\), \(i\) and \(j\). The author describes a recursive procedure for expressing \([D_b]\in H^*(X)\) as a polynomial in the first Chern classes of line bundles \(F_i/F_{i-1}\) for \(i=1,\ldots,n\). This result is motivated by \textit{W. Fulton} [Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)]. degeneracy loci; flag varieties; flagged bundles; symmetric subgroups B. J. Wyser, K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form, Transform. Groups 18 (2013), 557--594, 10.1007/s00031-013-9221-1. Grassmannians, Schubert varieties, flag manifolds, Compactifications; symmetric and spherical varieties \(K\)-orbit closures on \(G/B\) as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear 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 The authors study the cones of ample and numerically effective divisors of moduli spaces of stable maps to Grassmannians \(\overline{M}_{0,m}(G(k,n),d)\). The cones of NEF divisors are described in Theorem 1.1. Let the \(2\leq k<k+2\leq n\) (that is, \(G(k,n)\) is not a projective space; such cases are studied in the previous papers) and let \(m+d\geq 3\). Then there is a injective map
\[
\nu\colon \text{Pic}_{\mathbb Q}(\overline{M}_{0,m+d}/S_d)\to \text{Pic}_{\mathbb Q}(\overline{M}_{0,m}(G(k,n),d))
\]
(where the symmetric group \(S_d\) permutes the last \(d\) marked points). Let \(H_{\sigma_2}\) and \(H_{\sigma_{1,1}}\) be the divisors of maps whose image intersects fixed Schubert varieties \(\Sigma_2\) and \(\Sigma_{1,1}\) respectively. Let \(T\) be the tangency divisor and let \(L_1,\ldots,L_m\) be the pullbacks of the ample generator of \(\text{Pic}(G(k,n))\) under the evaluation maps. Then the cone of NEF divisors of \(\overline{M}_{0,m}(G(k,n),d)\) is the product of cones spanned by the image under \(\nu\) of the NEF cone of \(\overline{M}_{0,m+d}/S_d\) and \(H_{\sigma_{2}}\), \(H_{\sigma_{1,1}}\), \(L_1,\ldots,L_m\). Theorem 5.1 generalizes this theorem to the products of flag varieties. The cones of effective cones are harder to understand. The authors prove that the effective cones of \(\overline{M}_{0,0}(G(k,n),d)\) stabilize as long as \(n\geq k+d\). To study the cone of \(\overline{M}_{0,0}(G(k,k+d),d)\) authors define the particular divisors \(D_{\text{unb}}\) and \(D_{\text{deg}}\) (surprisingly, the construction depends on whether \(k\) divides \(d\)). Theorem 1.2 states that the effective cone of \(\overline{M}_{0,0}(G(k,k+d),d)\) is a simplicial cone generated by \(D_{\text{unb}}\), \(D_{\text{deg}}\) and the boundary divisors. NEF cone; effective cone; Grassmannian; space of stable maps Coskun, Izzet; Starr, Jason, Divisors on the space of maps to Grassmannians, Int. Math. Res. Not., 2006, (2006) Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Divisors on the space of maps to 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 In this note the author gives an explicit formula for the \(b\)-function of certain prehomogeneous vector spaces associated with flag manifolds of the general linear group. These prehomogeneous vector spaces are related to Eisenstein series of the general linear group and the studied \(b\)-functions describe the possible poles of those series. flag manifold; general linear group; prehomogeneous vector space; b-function Sato, F.: B-functions of prehomogeneous vector spaces attached to flag manifolds of the general linear group. Comment. math. Univ. st. Pauli 48, 129-136 (1999) Prehomogeneous vector spaces, Forms of degree higher than two, Analytic theory (Epstein zeta functions; relations with automorphic forms and functions), Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over the reals, the complexes, the quaternions \(b\)-functions of prehomogeneous vector spaces attached to flag manifolds of the general linear group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in \(n\) independent variables and one dependent variable of order one and two, by insisting on the underlying \((2n + 1)\)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a Ph.D course given by two of the authors (G. M. and G. M.). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections. contact and symplectic manifolds; jet spaces; Lagrangian Grassmannians; first and second order PDEs; symmetries of PDEs; characteristics; Monge-Ampère equations; PDEs on complex manifolds Geometric theory, characteristics, transformations in context of PDEs, Parabolic Monge-Ampère equations, Differential geometry of homogeneous manifolds, Global differential geometry of Hermitian and Kählerian manifolds, Symplectic manifolds (general theory), Contact manifolds (general theory), Jets in global analysis, Vector distributions (subbundles of the tangent bundles), Invariance and symmetry properties for PDEs on manifolds, Grassmannians, Schubert varieties, flag manifolds Contact manifolds, Lagrangian Grassmannians and PDEs | 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 \({\mathfrak q}={\mathfrak q}(A)\) be the Kac-Moody-Lie algebra, associated to a symmetrizable generalized Cartan matrix \(A=(a_{ij})_{1\leq i,j\leq\ell }\) and let \(S\subset\{1,...,\ell\}\) be the subset of finite type. Denote by \({\mathfrak p}_ S\) the corresponding ''parabolic''; \({\mathfrak r}\) ''the maximal'' reductive subalgebra of \({\mathfrak p}_ S\) and \({\mathfrak u}^-\) the orthocomplement of \({\mathfrak p}_ S\) (so that \({\mathfrak u}^- \oplus {\mathfrak p}_ S={\mathfrak q})\). Let \(L(\lambda_ 0)\) be the quasi- simple \({\mathfrak q}\)-module, with highest weight \(\lambda_ 0\). Further, let \((\Lambda (\mathfrak u^{-},L(\lambda_ 0)^ t),\partial)\) be the standard chain complex associated to the Lie algebra \(\mathfrak u^{-}\), with coefficients in the right module \(L(\lambda_ 0)^ t\) and let \((C(\mathfrak q,\mathfrak r),d)\) denote the standard co-chain complex associated to the Lie algebra pair \((\mathfrak q,\mathfrak r)\) (with trivial coefficient \(\mathbb C\)). There is associated (in general infinite dimensional) a group \(G\) (resp. a ``parabolic'' subgroup \(p_ S)\) with \(\mathfrak q'\) (resp. \(\mathfrak p_ S\cap\mathfrak q'\)). The flag variety \(G/P_ S\) admits a Bruhat cell decomposition with cells \(\{V_ w\}\), parametrized by \(w\in W_ S\backslash W\cong W^ 1_ S\) (\(W\) is the Weyl group for \(\mathfrak q)\).
In this paper, we explicitly compute the action of the Laplacian \(\Delta =\partial\partial^*+\partial^*\partial\) on \(\Lambda (\mathfrak u^{-}, L(\lambda_ 0)^ t).\) Further, we use this to prove the ``disjointness'' of the operators \(d\) and \(\partial\), acting on \(C(\mathfrak q,\mathfrak r)\). This gives rise to a ``Hodge type'' decomposition, with respect to the pair \(d,\partial\) (\(d,\partial\) are not adjoints of each other), of the space \(C(\mathfrak q, \mathfrak r)\). In particular, every \(d\) cohomology class in \(C(\mathfrak q,\mathfrak r)\) has a unique \(d,\partial\) closed representative. The ``Hodge type'' decomposition also gives, by a slight refinement of the arguments, that \(H^*_ d(\mathfrak q,\mathfrak r)\) is bi-graded; \(H_ d^{p,q}(\mathfrak q,\mathfrak r)=0\) unless \(p=q\) and \(H_ d^{p,p}(\mathfrak q,\mathfrak r)\) is a vector space with a ``canonical'' \(\mathbb C\)-basis \(\{s^ w\}_{w\in W^ 1_ S}\) with \(w=p\).
Finally (and this was our main interest) we prove that, properly defined, \(\int_{V_{w'}}s^ w=0\) unless \(w=w'\) and \(\int_{V_ w}s^ w>0\).
When \(\mathfrak q\) is a finite-dimensional semisimple Lie algebra, all these results are due to Kostant. In the infinite dimensional situation, Garland has computed \(\Delta\) for a special case. Most of the other results are new (as far as is known to the author). action of Laplacian; Hodge type decomposition; Kac-Moody-Lie algebra; chain complex; flag variety Cohomology of Lie (super)algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Hodge theory in global analysis, Grassmannians, Schubert varieties, flag manifolds Geometry of Schubert cells and cohomology of Kac-Moody-Lie algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce genomic tableaux, with applications to Schubert calculus. We report a combinatorial rule for structure coefficients in the torus-equivariant \(K\)-theory of Grassmannians for the basis of Schubert structure sheaves. This rule is positive in the sense of \textit{D. Anderson} et al. [J. Eur. Math. Soc. (JEMS) 13, No. 1, 57--84 (2011; Zbl 1213.19003)]. We thereby deduce an earlier conjecture of \textit{H. Thomas} and \textit{A. Yong} [``Equivariant Schubert calculus and jeu de taquin
'', Ann. Inst. Fourier (to appear)] for the coefficients. Moreover, our rule specializes to give a new Schubert calculus rule in the (non-equivariant) \(K\)-theory of Grassmannians. From this perspective, we also obtain a new rule for \(K\)-theoretic Schubert structure constants of maximal orthogonal Grassmannians, and give conjectural bounds on such constants for Lagrangian Grassmannians. Schubert calculus; equivariant \(K\)-theory; Grassmannians; genomic tableaux Pechenik, O., Yong, A.: Genomic tableaux and combinatorial \(K\)-theory. Discrete Math. Theor. Comput. Sci. Proc. \textbf{FPSAC'15}, 37-48 (2015) Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Genomic tableaux and 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 Soliton solutions of the KP equation have been studied since 1970, when \textit{B. B. Kadomtsev} and \textit{V. I. Petviashvili} [Sov. Phys., Dokl. 15, 539--541 (1970); translation from Dokl. Akad. Nauk SSSR 192, 753--756 (1970; Zbl 0217.25004)] proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. The regular soliton solutions that one obtains in this way come from points of the totally nonnegative part of the Grassmannian. In this paper we explain how the theory of total positivity and cluster algebras provides a framework for understanding these soliton solutions to the KP equation. We then use this framework to give an explicit construction of certain soliton contour graphs and solve the inverse problem for soliton solutions coming from the totally positive part of the Grassmannian. Kadomtsev-Petviashvili equation; Grassmannian; soliton solutions Kodama Y, Williams LK. (2011) KP solitons, total positivity, and cluster algebras. Proc. Natl Acad. Sci. USA 108, 8984-8989. (10.1073/pnas.1102627108) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Cluster algebras, Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Grassmannians, Schubert varieties, flag manifolds KP solitons, total positivity, and cluster algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review contains a description of an operator giving a homology theory of real Grassmann varieties. Several computations of homology groups are performed. homology theory; real Grassmann varieties Grassmannians, Schubert varieties, flag manifolds, Étale and other Grothendieck topologies and (co)homologies, Real algebraic and real-analytic geometry Homology of certain Grassmann varieties | 0 |
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