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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 notion of Castelnuovo-Mumford regularity is a very useful tool in the study of vector bundles on projective spaces. In particular it allows to characterize direct sums of line bundles (Evans-Griffith splitting criterion). Regularity has been first introduced by Mumford in the 60s and more recently it has been generalized by many authors to other ambient spaces, such as Grassmannians, products of projective spaces, quadrics.
The authors of the paper under review propose a new notion of regularity in the case of Grassmannians of lines. Their definition is based on the analogue of the Koszul exact sequence, obtained taking the exterior powers of the universal exact sequence on the Grassmannian. This notion is less restrictive than the other previous definitions of regularity, and allows to obtain a characterization of the direct sums of line bundles on the Grassmannian. Moreover the authors find a cohomological characterization of exterior and symmetric powers of the universal bundles of the Grassmannian. universal bundles on Grassmannians; Castelnuovo-Mumford regularity Costa, L., Miró-Roig, R.M.: Homogeneous Ulrich bundles on Grassmannians of lines (2012) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Cohomological characterization of vector bundles on Grassmannians of 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 We construct affine pavings of Springer-type fibers over the enhanced nilpotent cone. This resolves a question of Achar-Henderson and implies the existence of perverse parity sheaves on the enhanced nilpotent cone. Coadjoint orbits; nilpotent varieties, Grassmannians, Schubert varieties, flag manifolds Affine pavings and the enhanced nilpotent cone | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of \textit{H. Clemens} and \textit{Z. Ran} [Am. J. Math. 126, No. 1, 89--120 (2004; Zbl 1050.14035)] to prove that a very general hypersurface of degree \(\frac{3n+1}{2} \leq d \leq 2n - 3\) in \(\mathbb{P}^n\) contain lines but no other rational curves. Hypersurfaces and algebraic geometry, Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Grassmannians, Schubert varieties, flag manifolds Rational curves on general type hypersurfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 smooth integral curves of bidegree \((1, 1)\), called \textit{smooth conics}, in the flag threefold \(\mathbb{F}\). The study is motivated by the fact that the family of smooth conics contains the set of fibers of the twistor projection \(\mathbb{F}\rightarrow\mathbb{CP}^2\). We give a bound on the maximum number of smooth conics contained in a smooth surface \(S\subset\mathbb{F}\). Then, we show qualitative properties of algebraic surfaces containing a prescribed number of smooth conics. Last, we study surfaces containing infinitely many twistor fibers. We show that the only smooth cases are surfaces of bidegree \((1, 1)\). Then, for any integer \(a > 1\), we exhibit a method to construct an integral surface of bidegree \((a, a)\) containing infinitely many twistor fibers. flag threefold; integral curves Twistor theory, double fibrations (complex-analytic aspects), Grassmannians, Schubert varieties, flag manifolds, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Rational and ruled surfaces Surfaces in the flag threefold containing smooth conics and twistor fibers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Chow motive, a concept introduced by Grothendieck, which has become a fundamental tool for investigating the structure of algebraic varieties, provides one of the few tools for studying the geometry of certain varieties. Computing it has also proved to be valuable for addressing questions on other different topics. In this paper, authors develop two general methods to calculate the Chow motives of some twisted flag varieties over an arbitrary field, for which that computation has not been previously possible. The first of them deals with shells and generalizes \textit{A. Vishik}'s [Lect. Notes Math. 1835, 25--101 (2004; Zbl 1047.11033)] shells of quadratic forms and extends \textit{N. A. Karpenko}'s [J. Reine Angew. Math. 677, 179--198 (2013; Zbl 1267.14009)] results on the upper motives. The second one is based on a formula of \textit{V. Chernousov} and \textit{A. Merkurjev} [Transform. Groups 11, No. 3, 371--386 (2006; Zbl 1111.14009)] and provides a broad generalization on the generic point diagram, which is, as it is well-known, a standard tool in the theory of Chow motives. Both methods are complementary to each other. Indeed, the first of them is designed to eliminate certain decomposition types and the second one is designed to prove that the remaining decomposition types are realizable. Finally, the authors also settle a a 20-year-old conjecture of Markus Rost [Letter to Jean-Pierre Serre (1992)] about the Rost invariant for groups of type \(E_7\). linear algebraic groups; twisted flag varieties; Rost invariant; Chow motives; equivariant Chow groups Garibaldi, S.; Petrov, V.; Semenov, N., \textit{shells of twisted flag varieties and the rost invariant}, Duke Math. J., 165, 285-339, (2016) (Equivariant) Chow groups and rings; motives, Grassmannians, Schubert varieties, flag manifolds, Exceptional groups Shells of twisted flag varieties and the Rost invariant | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 that in finite, simply laced types, every Schubert variety indexed by an involution which is not the longest element of some standard parabolic subgroup is singular. Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry Smoothness of Schubert varieties indexed by involutions in finite simply laced types | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Algebraic varieties over finite fields are considered from the point of view of their invariants such as the number of points of a variety that are defined over the ground field and its extensions. The case of curves has been actively studied over the last thirty-five years, and hundreds of papers have been devoted to the subject. In dimension two or higher, the situation becomes much more difficult and has been little explored. This survey presents the main approaches to the problem and describes a major part of the known results in this direction. algebraic varieties over finite fields; zeta functions; points on surfaces; error-correcting codes; arithmetic statistics; explicit formulae in arithmetic Finite ground fields in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Varieties over finite and local fields, Grassmannians, Schubert varieties, flag manifolds, Curves over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Varieties over finite fields: quantitative 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 [For the entire collection see Zbl 0741.00020.]
The paper summarizes recent results by Matsuki, Oshima, Uzawa, Brion, Vinberg on the space \(H\backslash G/P\) of double cosets of a connected real semisimple Lie group \(G\). Here \(P\) is a minimal parabolic subgroup of \(G\), \(G/P\) is the corresponding flag manifold, and \(H\) is (almost) the fixed point subgroup of an involutive automorphism of \(G\). Among the topics discussed here (often without proof) are:
--- the ``symbol'' of a double coset, when \(G\) is a complex classical group (\(G=GL(n,\mathbb{C})\), \(SO(n,\mathbb{C})\) or \(Sp(n,\mathbb{C})\)).
--- Uzawa's function and vector field on \(G/P\),
--- spherical subgroups of a complex semisimple Lie group \(G\). double cosets; connected real semisimple Lie group; parabolic subgroup; flag manifold; complex semisimple Lie group Matsuki, T.: Orbits on flag manifolds. In: Proceedings of the International Congress of Mathematicians, Kyoto 1990, Vol. II. Springer, pp. 807-813 (1991) Semisimple Lie groups and their representations, Grassmannians, Schubert varieties, flag manifolds, Differential geometry of homogeneous manifolds Orbits on 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 Every embedding of a Grassmann space in a projective space can be obtained - up to collineations -- by projecting the associated Grassmannian. This is a special case of a more general result on homomorphic images of Grassmann spaces [the reviewer, ibid. 16, 152-167 (1981; Zbl 0463.51003); ibid. 16, 168-180 (1981; Zbl 0463.51004)] and it has been proved independently by \textit{A. L. Wells jun.} [Q. J. Math., Oxf. II. Ser. 34, 375-386 (1983; Zbl 0537.51008)]. Whenever this projection is different from identity, the image of the embedding will be called a ``projected Grassmannian''.
In the present paper a result by Wells (loc.cit.) on the existence of projected Grassmannians is substantially improved as follows: Every Grassmannian representing the \(h\)-flats in \(\text{PG}(n,F)\) (over a commutative field \(F\)) with \(n > 2\) and \(0 < h < n-1\) can be projected if and only if \(n > 5\). Thus exactly the Grassmannians representing the lines in \(\text{PG} (n,F)\), \(n = 3,4 \) cannot be projected.
The tools to obtain this result are the quadratic relations among the Grassmann coordinates of subspaces and a recursive construction of Grassmannians.
Moreover, the author mentions without proof that the Grassmann space representing the lines \(\text{PG} (5,F)\) is embeddable into \(\text{PG}(12,F)\) when \(F = \text{GF} (2)\), but fails to have this property when \(F\) is the real or complex number field. embedding; Grassmann space Zanella, C., Embeddings of Grassmann spaces, J. Geom., 52, 193-201, (1995) Incidence structures embeddable into projective geometries, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Grassmannians, Schubert varieties, flag manifolds Embeddings of Grassmann 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 Modern Representation Theory has numerous applications in many mathematical areas such as algebraic geometry, combinatorics, convex geometry, mathematical physics, probability. Many of the object and problems of interest show up in a family. Degeneration techniques allow to study the properties of the whole family instead of concentrating on a single member. This idea has many incarnations in modern mathematics, including Newton-Okounkov bodies, tropical geometry, PBW degenerations, Hessenberg varieties. During the mini-workshop Degeneration Techniques in Representation Theory various sides of the existing applications of the degenerations techniques were discussed and several new possible directions were reported. Collections of abstracts of lectures, Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to nonassociative rings and algebras, Proceedings, conferences, collections, etc. pertaining to combinatorics, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Universal enveloping (super)algebras, Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Symmetric functions and generalizations, Combinatorial aspects of representation theory Mini-workshop: Degeneration techniques in representation theory. Abstracts from the mini-workshop held October 6--12, 2019 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for \({\mathfrak{sl}_2}\). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra \({\mathfrak{gl}_n}\). For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules. Categorification; quantum groups; Lie algebras; canonical bases; flag varieties I. Frenkel, M. Khovanov and C. Stroppel, \textit{A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products}, \textit{Selecta Math. (N.S.)}\textbf{12} (2006) 379431 [math/0511467]. Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Representations of associative Artinian rings A categorification of finite-dimensional irreducible representations of quantum \({\mathfrak{sl}_2}\) and their tensor products | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 equivariant localization of intersection cohomology complexes on Schubert varieties in Kashiwara's flag manifold. Using moment graph techniques we establish a link to the representation theory of Kac-Moody algebras and give a new proof of the Kazhdan-Lusztig conjecture for blocks containing an antidominant element. Kashiwara thick flag variety; Kac-Moody Lie algebra; intersection cohomology; moment graph Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds Localization of IC-complexes on Kashiwara's flag scheme and representations of Kac-Moody algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An algebraic approach is developed to define and study infinite-dimensional Grassmannians. Using this approach, a quantum deformation (i.e. a deformation of the coordinate ring) is obtained for both the ind-variety union of all finite-dimensional Grassmannians \(G_{\infty}\), and the Sato Grassmannian \(\widetilde{UGM}\) introduced by Sato. They are both quantized as homogeneous spaces, that is together with a coaction of a quantum infinite dimensional group. At the end, an infinite-dimensional version of the first theorem of invariant theory is discussed for both the infinite-dimensional special linear group and its quantization. R. Fioresi and C. Hacon, \textit{On infinite-dimensional Grassmannians and their quantum deformations}, Rend. Sem. Mat. Univ. Padova, 111 (2004), pp. 1--24. Grassmannians, Schubert varieties, flag manifolds, Group structures and generalizations on infinite-dimensional manifolds, Geometry of quantum groups On infinite-dimensional Grassmannians and their quantum deformations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 each integer \(k\geq 4\), we describe diagrammatically a positively graded Koszul algebra \(\mathbb {D}_k\) such that the category of finite dimensional \(\mathbb {D}_k\)-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type \(\mathrm{D}_k\) or \(\mathrm{B}_{k-1}\), constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra. Properties such as graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools. positively graded Koszul algebra; Khovanov arc algebra Ehrig, M.; Stroppel, C., Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians, Selecta Math. (N.S.), 22, 3, 1455-1536, (2016) Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Lie algebras of linear algebraic groups, Equivariant homology and cohomology in algebraic topology, Hecke algebras and their representations Diagrammatic description for the categories of perverse sheaves on isotropic 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 explore the geometric space parametrized by (tree level) Wilson loops in SYM \(\mathcal{N}=4\). We show that this space can be seen as a vector bundle over a totally non-negative subspace of the Grassmannian, \(\mathcal{W}_{k,n}\). Furthermore, we explicitly show that this bundle is non-orientable in the majority of the cases, and conjecture that it is non-orientable in the remaining situation. Using the combinatorics of the Deodhar decomposition of the Grassmannian, we identify subspaces \(\Sigma(W)\subset\mathcal{W}_{k,n}\) for which the restricted bundle lies outside the positive Grassmannian. Finally, while probing the combinatorics of the Deodhar decomposition, we give a diagrammatic algorithm for reading equations determining each Deodhar component as a semialgebraic set. SYM \(\mathcal{N}=4\); positive Grassmannians; Deodhar decomposition Supersymmetric field theories in quantum mechanics, Grassmannians, Schubert varieties, flag manifolds, Feynman diagrams, Vector distributions (subbundles of the tangent bundles), Decomposition methods Wilson loops in SYM \(\mathcal{N}=4\) do not parametrize an orientable space | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 under review is concerned with the small quantum Schubert Calculus on flag varieties, namely homogeneous spaces of the form \(G/B\), where \(G\) is a connected simply connected complex Lie group and \(B\) is a Borel subgroup of it. Each homogeneous space carries a natural cellular decomposition and so, by general results of algebraic topology, the homology classes of the closure of the affine cells provide a basis of the homology over the integers, whose elements are said to be Schubert cycles. Poincaré duality holds in \(G/B\), because of its smoothness, and so the Schubert cocycles, Poincaré dual of the Schubert cycles, form a basis of the cohomology ring of the flag variety.
Schubert calculus of \(G/B\) amounts to knowing the \textsl{Schubert structure constants}, according to the authors' terminology, i.e., the structural constants of the cohomology algebra with respect to the basis of Schubert (co)cycles. In the case when \(G=SL(n+1, {\mathbb C})\) and \(B\) is the Borel subgroup of the triangular matrices, the picture is very well understood, as one falls in the usual intersection theory of the Grassmann variety: the structure constants are traditionally known as Littlewood--Richardson coefficients.
The two main theorems concern the small quantum cohomology \(QH^*(G/B)\) of flag varieties, i.e., the determination of what the authors call the \textsl{quantum Schubert constants}. Here \(QH^*(G/B)\) is a deformation of \(H^*(G/B)\): the support is \(H^*(G/B)\otimes_{\mathbb Z}{\mathbb C}[[t]]\) while the product structure is obtained by ``correcting'' the classical product by means of the appropriate Gromow-Witten invariants, which basically count number of rational curves in \(G/B\) having suitable incidence properties with respect to some given configuration of Schubert varieties.
The first main theorem computes the quantum Schubert structural constants of \(G/B\) by means of a certain function involving some combinatorially defined quantities, which are rational functions in simple roots. Its precise and detailed explanation in a review would not be more helpful for the interested reader. The second theorem computes, through a very explicit formula, the structural constants of the Pontryagin products in \(H^T_*(\Omega K)\), which is the equivariant (Borel-Moore) homology, with respect to the action of a \(n\)-dimensional torus \(T\), of the based loop group of the maximal compact subgroup \(K\) of \(G\).
The beginning of the paper consists in a careful introduction followed by a review of general knowledges about Kac--Moody algebras and a set up of notation (Section 2). Section 3 is devoted to deduce an explicit formula for the Pontryagin product on \(H^T_* (\Omega K)\) while Section 4 uses the formula to prove the Main Theorem. The sequence of instructive examples of Section 5 give the measure of how is the formula effective. The appendix is finally devoted to provide the proofs of some properties stated in Section 4. Quantum Schubert Calculus of homogeneous spaces; Pontryagin product, loop groups, Equivariant homology and cohomology of homogeneous spaces, Gromov-Witten invariants Leung, N. C.; Li, C., Gromov-Witten invariants for \(G / B\) and Pontryagin product for {\(\omega\)}\textit{K}, Trans. Amer. Math. Soc., 5, 2567-2599, (2012) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Infinite-dimensional Lie groups and their Lie algebras: general properties Gromov-Witten invariants for \(G/B\) and Pontryagin product for \(\Omega K\) | 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 The category of Cohen-Macaulay modules of an algebra \(B_{k,n}\) is used in [\textit{B. T. Jensen} et al., Proc. Lond. Math. Soc. (3) 113, No. 2, 185--212 (2016; Zbl 1375.13033)] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of \(k\)-planes in \(n\)-space. In this paper, we find canonical Auslander-Reiten sequences and study the Auslander-Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen-Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac-Moody algebra in the tame cases. Auslander-Reiten sequences; Cohen-Macaulay modules Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Cohen-Macaulay modules in associative algebras, Root systems Cluster categories from Grassmannians and root 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 \(G\) be a complex semi-simple connected Lie group and \(P\) its parabolic subgroup. Then the maximal torus \(T\) of \(G\) acts on the homogeneous space \(G/P\) and this action can be naturally lifted to the moduli space (stack) of the stable maps \(\overline {\mathcal M}_{g,n}(G/P,\beta)\), where \(\beta\) is an element in \(H_2(G/P, \mathbb{Z})\). In this paper, we investigate the \(T\)-equivariant Gromov-Witten invariants of \(G/P\) and vertical invariants of flag bundles. \textit{M. Kontsevich} used the fixed point localization method for the first time to compute the Gromov Witten invariants of the projective space and its hypersurfaces [in: The moduli space of curves, Prog. Math. 129, 335--368 (1995; Zbl 0885.14028)]. There he mentioned that his computational scheme works well also for homogeneous spaces and we will follow his method to give a formula of (gravitational) Gromov-Witten invariants of the homogeneous space \(G/P\). The fixed point localization method enables us to obtain the information of the equivariant cohomology \(H^*_T(\overline{\mathcal M}_{0,n}(G/P,\beta))\) from the data on the fixed points of the action of \(T\). In our case, the set of fixed points \(\overline{\mathcal M}_{0,n}(G/P,\beta)^T\) can be described in terms of the Bruhat ordering of the Weyl group \(W\) of \(G\). Integration of an element in \(H_T^*(\overline{\mathcal M}_{0,n}(G/P,\beta))\) can be expressed as a sum of the contributions from the components of the fixed locus. Such formula is known as Bott's fixed point formula. Then, the Gromov-Witten invariants of the flag variety \(G/B\) can be computed effectively by using Bott's fixed point formula for smooth stacks. \textit{W. Fulton} [Duke Math. J. 65, 381--420 (1992; Zbl 0788.14044)] showed that degeneracy loci associated to a flag bundle over a variety \(X\) are expressed as double Schubert polynomials in the Chow group. We give a formula for vertical Gromov-Witten invariants of the flag bundle in terms of double Schubert polynomials. The localization of the virtual fundamental class was developed by \textit{T. Graber} and \textit{R. Pandharipande} [Invent. Math. 135, 487--518 (1999; Zbl 0953.14035)], so this method can be applied to the moduli space of stable maps of higher genus and to other target spaces with torus action. In the final section we briefly see the higher genus invariants. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Vertical Gromov-Witten invariants of flag 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 This monograph is based on the author's lectures at the CBMS conference at North Carolina State University in 2001. It covers the application of symmetric functions to algebraic identities related to the Euclidean algorithm. It does not require extensive knowledge of symmetric functions, although some familiarity with the basic properties would be useful.
Many texts on symmetric functions view the functions as polynomials. In constrast, this monograph uses the method of \(\lambda\)-rings, in which symmetric functions are viewed as operators on the ring of polynomials. This approach is particularly natural for these applications, because the construction of the symmetric functions and all necessary properties follow from a few fundamental results.
An arbitrary polynomial can be viewed as a symmetric function in terms of its roots. In this framework, the successive remainders in the Euclidean algorithm can be expressed in terms of symmetric functions; examples include Sturm sequences and continued fractions. Divided difference operators also act on polynomials, and operations involving partial or full symmetrization can be expressed in terms of divided differences; one example is a symmetric function identity which arises in the cohomology of Grassmannians. Another viewpoint is that of orthogonal polynomials; if the ``moments'' are the complete symmetric functions, the resulting orthogonal polynomials are Schur functions indexed by square partitions, and other Schur functions appear in contexts such as Christoffel determinants.
A similar approach can be used to study generalizations of symmetric functions. Schubert polynomials are constructed in two ways: by generalizing Newton's interpolation to multiple variables, and from a non-symmetric Cauchy kernel. The book concludes with a brief discussion of further generalizations to non-commutative Schur functions and Schubert polynomials. Schur functions; \(\lambda\)-rings; Cauchy kernel; Euclidean algorithm; continued fractions; Padé approximation; divided differences; cohomology of Grassmannian; orthogonal polynomials; Schubert polynomials Lascoux, A.: Symmetric functions \& combinatorial operators on polynomials. CBMS reg. Conf. ser. Math. 99 (2003) Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups, Approximation by polynomials, Padé approximation, Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis, Orthogonal polynomials [See also 33C45, 33C50, 33D45] Symmetric functions and combinatorial operators on 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 construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a ``categorical version'' of Kazhdan-Lusztig-Ginzburg's construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković in the course of the proof of Lusztig's conjectures on equivariant \(K\)-theory of Springer fibers. braid groups; reductive algebraic groups; Lie algebras; Springer resolutions; affine Hecke algebras; dg-schemes; Fourier-Mukai transform Berukavnikov, R; Riche, S, Affine braid group actions on derived categories of Springer resolutions, Ann. Sci. l'Éc. Norm. Supèr. Quatr. Sér. 4, 45, 535-599, (2012) Representation theory for linear algebraic groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Braid groups; Artin groups, Modular Lie (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Derived categories, triangulated categories Affine braid group actions on derived categories of Springer resolutions. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 paper is a characterization of those Schubert varieties for the general linear group that are local complete intersection (lci). The authors prove that a Schubert variety \(X_w\) associated to a permutation \(w \in S_m\) is lci if and only if \(w\) avoids the six patterns 53241, 52341, 52431, 35142, 42513, and 35162.
The paper is organized as follows. Section 2 presents the basic definitions and set-up. Sections 3 and 4 prove that Schubert varieties associated to permutations avoiding the given patterns are lci. The proof involves showing that the variety \(X_w\) is lci at the identity point by identifying a minimal set generators for its defining ideal. This uses the notion of the \textit{essential set} of \(w\) introduced in [\textit{W. Fulton}, Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)] and Schubert varieties \textit{defined by inclusions} from work of \textit{V. Gasharov} and \textit{V. Reiner} [J. Lond. Math. Soc., II. Ser. 66, No. 3, 550--562 (2002; Zbl 1064.14056)].
Section 5 proves the necessity of the pattern avoidance. The strategy is to identify a collection of intervals \([u,v]\) in the Bruhat order for which the corresponding slice is not lci. The authors then show that if \(w\) contains one of the six given patterns, then \(w\) \textit{interval contains} one of the intervals \([u,v]\) above and they conclude that \(X_w\) is not lci.
Section 6 contains a number of applications, including formulas for Kostant polynomials and presentations of cohomology rings for lci Schubert varieties. Section 7 concludes with a number of open questions. Schubert varieties; local complete intersection; pattern avoidance \beginbarticle \bauthor\binitsH. \bsnmÚlfarsson and \bauthor\binitsA. \bsnmWoo, \batitleWhich Scubert varieties are local complete intersections? \bjtitleProc. Lond. Math. Soc. (3) \bvolume107 (\byear2013), page 1004-\blpage1052. \endbarticle \endbibitem Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Complete intersections, Combinatorial aspects of commutative algebra, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Permutations, words, matrices Which Schubert varieties are local complete intersections? | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 product monomial crystal was defined by \textit{J. Kamnitzer} et al. [Proc. Lond. Math. Soc. (3) 119, No. 5, 1179--1233 (2019; Zbl 1451.14143)] for any semisimple simply-laced Lie algebra \(\mathfrak{g}\), and depends on a collection of parameters \(\mathcal{R}\). We show that a family of truncations of this crystal are Demazure crystals, and give a Demazure-type formula for the character of each truncation, and the crystal itself. This character formula shows that the product monomial crystal is the crystal of a generalised Demazure module, as defined by \textit{V. Lakshmibai} et al. [Compos. Math. 130, No. 3, 293--318 (2002; Zbl 1061.14051)]. In type \(A\), we show the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram depending on \(\mathcal{R}\). monomial crystal; generalised Schur module; Demazure crystal Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations A Demazure character formula for the product monomial crystal | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ``Let \(A=\begin{pmatrix} 2&-2\\-2&2 \end{pmatrix}\), \({\mathfrak g}(A)\) the associated Kac-Moody Lie-algebra and G(A) \((=\hat SL_ 2\)) the associated Kac-Moody group. Let P be a (maximal) parabolic subgroup of G(A). Let W (resp. \(W_ P)\) be the Weyl group of \(G(A)\) (resp. P). For \(\tau \in W/W_ P\), let \(X(\tau)\) be the Schubert variety in G(A)/P associated to \(\tau\). We construct explicit bases for \(H^ 0(X(\tau),L^ m)\), \(m\in {\mathbb{Z}}^+\), in terms of ``standard monomials'' where L denotes the tautological line bundle on \(P^ N\) [as well as its restriction to \(X(\tau)\)] for some canonical projective embedding \(X(\tau)\hookrightarrow P^ N\). As a consequence, we obtain similar results for Schubert varieties in \(\hat SL_ 2/B\). Kac-Moody Lie-algebra; ŜL\({}_ 2\); Kac-Moody group; parabolic subgroup; Weyl group; Schubert variety; standard monomials; tautological line bundle V. Lakshmibai and C. S. Seshadri, ?Thèorie monomiale standard pour \(\widehat{\mathfrak{s}\mathfrak{l}}_2 \) ,? C. R. Acad. Sci. Paris Ser. I,305, 183-185 (1987). Infinite-dimensional Lie groups and their Lie algebras: general properties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) Théorie monomiale standard pour \(\hat SL_ 2\). (Standard monomial theory for \(\hat SL_ 2)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We pose and solve the equivalence problem for subspaces of \(P_n\), the \((n+1)\) dimensional vector space of univariate polynomials of degree \(\leq n\). The group of interest is \(SL_{2}\) acting by projective transformations on the Grassmannian variety \(G_kP_n\) of \(k\)-dimensional subspaces. We establish the equivariance of the Wronski map and use this map to reduce the subspace equivalence problem to the equivalence problem for binary forms. polynomial subspaces; projective equivalence Grassmannians, Schubert varieties, flag manifolds, Vector and tensor algebra, theory of invariants, Linear ordinary differential equations and systems, Critical points of functions and mappings on manifolds On projective equivalence of univariate polynomial 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 study the torus equivariant Schubert classes of the Grassmannian of non-maximal isotropic subspaces in a symplectic vector space. We prove a formula that expresses each of those classes as a \textit{sum} of multi Schur-Pfaffians, whose entries are equivariantly modified special Schubert classes. Our result gives a proof to Wilson's conjectural formula, which generalizes the Giambelli formula for the ordinary cohomology proved by Buch-Kresch-Tamvakis, given in terms of Young's raising operators. Furthermore we show that the formula extends to a certain family of Schubert classes of the symplectic partial isotropic flag varieties. Schubert classes; symplectic Grassmannians; torus equivariant cohomology; Giambelli type formula; Wilson's conjecture; double Schubert polynomials Ikeda, T.; Matsumura, T., \textit{Pfaffian sum formula for the symplectic Grassmannians}, Math. Z., 280, 269-306, (2015) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Pfaffian sum formula for the symplectic 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 article a unified description of the structure of the small cohomology rings for all projective homogeneous spaces \(SL_n(\mathbb C)/P\) (with \(P\) a parabolic subgroup) is given. First the results on the classical cohomology rings are recalled. Then the algebraic structure of the quantum cohomology ring is studied. Important results are the general quantum versions of the Giambelli and Pieri formulas of the classical cohomology (classical Schubert calculus). They are obtained via geometric computations of certain Gromov-Witten invariants, which are realized as intersection numbers on hyperquot schemes. quantum cohomology; Gromov-Witten invariants; Schubert calculus; small cohomology rings; homogeneous spaces; Pieri formulas Ionuţ Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485 -- 524. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Homogeneous spaces and generalizations, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On quantum cohomology rings of partial flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the cohomology or Chow ring of a homogeneous space $G/P$. The classes of Schubert varieties of $G/P$ form a ``geometric'' basis. In Schubert calculus one studies the structure constants of the ring with respect to this basis -- c.f. Littlewood-Richardson coefficients. Much is known about cohomological (Chow) Schubert calculus, and even about its $K$-theory generalization. \par The paper under review studies an even more general cohomology theory, in fact the universal oriented algebraic cohomology theory: algebraic cobordism. In such generality the classes of Schubert varieties are not well defined, they depend on choices. The authors make their choices (Bott-Samelson resolution), and prove a formula for the product of the class of a smooth Schubert variety with an arbitrary Bott-Samelson class -- for type A Grassmannians. The last sections of the paper also establish some results on polynomials (``generalized Schubert polynomials'') representing Schubert varieties for hyperbolic formal group laws. Schubert calculus; cobordism; Grassmannian; generalized cohomology Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Connective \(K\)-theory, cobordism, Bordism and cobordism theories and formal group laws in algebraic topology Smooth Schubert varieties and generalized Schubert polynomials in algebraic cobordism of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a \(k\)-dimensional irreducible algebraic subvariety of the \(n\)- dimensional complex projective space \(\mathbb{P}^ n\). The author proves that if \(X\) has codimension bigger than two and contains a \((2k - 4)\)- dimensional family of lines, then \(X\) is either a one-dimensional family of quadrics, or a two-dimensional family of \((k-2)\)-dimensional projective spaces, or a linear section of the Grassmann variety of lines of \(\mathbb{P}^ 4\). codimension bigger than two; Grassmann variety E. Rogora, Varieties with many lines. Manuscripta Math. 82 (1994), no. 2, 207--226. Projective techniques in algebraic geometry, Low codimension problems in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Varieties with many 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 Let \(Q\to Y\) be a family of 2-dimensional quadrics over a 3-dimensional base with \(Q\) and \(Y\) smooth and consider its relative Fano scheme of lines \(\rho: M\to Y\). In this paper, a criterion for the smoothness of \(M\) is given and the bounded derived category \(\mathcal D^b(M)\) of coherent sheaves on \(M\) is studied.
Let \(D_r\subset Y\) be the locus of quadrics of corank at least \(r\). Under the assumptions that the generic fibre of \(Q\to Y\) is smooth, \(D_3=\emptyset\), and \(D_2\) consists of finitely many points \(y_1,\dots,y_N\) all of which are ordinary double points of \(D_1\), the author proves that \(M\) is smooth and there is a semiorthogonal decomposition
\[
\mathcal D^b(M)=\bigl\langle \mathcal D^b(X^+),\mathcal D^b(Y,\mathcal B_0),\{\mathcal O_{\Sigma_i^+}\}_{i=1}^N\bigr\rangle
\]
obtained as follows. For \(i=1,\dots,N\), the fibre \(\rho^{-1}(y_i)\) is the union of two planes and \(\mathbb P^2\cong \Sigma_i^+\subset M\) is chosen as one of them. Then it is shown that the structure sheaves \(\mathcal O_{\Sigma_i^+}\) form a completely orthogonal exceptional collection. The Stein factorisation \(M\to X\to Y\) of \(\rho\) consists of a generically conic bundle \(M\to X\) and the double cover \(X\to Y\) ramified over \(D_1\). Let \(M^+\) be the flip of \(M\) in the planes \(\Sigma_i^+\). The author proves that the induced map \(M^+\to X\) factors as \(M^+\to X^+\to X\) where \(M^+\to X^+\) is a \(\mathbb P^1\)-fibration and \(X^+\to X\) is a small resolution of singularities. It follows that \(\mathcal D^b(X^+)\) can be embedded into \(\mathcal D^b(M)\) via the pull-back along \(M^+\to X^+\) followed by the canonical embedding \(\mathcal D^b(M^+)\subset \mathcal D^b(M)\). Finally, \(\mathcal B_0\) is the sheaf of the even part of the Clifford algebra associated to the quadric fibration \(Q\to Y\) and it is shown that \(\mathcal D^b(Y,\mathcal B_0)\) is equivalent to the left orthogonal complement of \(\mathcal D^b(X^+)\) in \(\mathcal D^b(M^+)\). The results are used by \textit{C. Ingalls} and the author in [``On nodal Enriques surfaces and quartic double solids'', \url{arXiv:1012.3530}] to provide a description of the derived category of a nodal Enriques surface. quadric fibration; Fano scheme of lines; derived category; semiorthogonal decomposition; Clifford algebra A. Kuznetsov, Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category, math. AG/arXiv:1011. 4146. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Brauer groups of schemes, Quadratic spaces; Clifford algebras, Fibrations, degenerations in algebraic geometry, Families, moduli, classification: algebraic theory, Grassmannians, Schubert varieties, flag manifolds Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{E. Arrondo} and \textit{J. Caravantes} [On the Picard group of low-codimensional subvarieties, e-print, \url{arXiv:math. AG/0511267}] sketched a program to decide whether a divisor \(D\) in a smooth subvariety \(X\) of a smooth even dimensional variety \(Y\) is equivalent to a multiple of a hyperplane section of \(X\). In the paper under review, the author discusses the validity of this method in the case of homogeneous \(Y\) with \(\text{Pic}(Y)=\mathbb{Z}\) and shows that it is closely related to the signature \(\sigma_Y\) of the Poincaré pairing on the middle cohomology of \(Y\). Under some topological assumptions a bound on the rank of \(\text{Pic}(X)\) in terms of \(\sigma_Y\) is given. For Grassmannians, these assumptions are removed to recover the main result of E. Arrondo and J. Caravantes. subvariety; cohomology; Grassmannian; Picard group Perrin, N, Small codimension smooth subvarieties in even-dimensional homogeneous spaces with Picard group \({\mathbb{Z}}\), C. R. Acad. Sci., 345, 155-160, (2007) Divisors, linear systems, invertible sheaves, Picard groups, Grassmannians, Schubert varieties, flag manifolds Small codimension smooth subvarieties in even-dimensional homogeneous spaces with Picard group \(\mathbb Z\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian. Jordan algebras; symmetric matrices; complex numbers; Chow form; reciprocal variety Idempotents, Peirce decompositions, Grassmannians, Schubert varieties, flag manifolds Jordan algebras of symmetric 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 Let \({\mathbb{C}}^{p+q}\) be endowed with a Hermitian form H of signature (p,q) and \(M_ r\) be the manifold of r-dimensional subspaces of \({\mathbb{C}}^{p+q}\) on which H is positive-definite. Let E be the determinant bundle of the tautological bundle on \(M_ r\). Starting from the oscillator representation of SU(p,q), it is shown that there is an invariant subspace of \(H^{r(p-r)}(M_ r,{\mathcal O}(E(p+k)))\) which defines a unitary representation of SU(p,q). For \(W\in M_ p\), Gr(r,W) is the subvariety of r-dimensional subspaces of W, and integration over Gr(r,W) associates to an r(p-r)-cohomology class \(\alpha\), a function P(\(\alpha)\) on \(M_ p\). It is shown that this map is injective and provides an intertwining operator with representations of SU(p,q) on spaces of holomorphic functions on Siegel space. determinant bundle; tautological bundle; oscillator representation; invariant subspace; unitary representation; cohomology; intertwining operator; SU(p,q); holomorphic functions; Siegel space C. Patton and H. Rossi, Unitary structures in cohomology, to appear. Semisimple Lie groups and their representations, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties Unitary structures on cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 complex semisimple adjoint group, with maximal compact subgroup \(K\) and associated symmetric space \(X:=G/K\). Let \(\Delta\) be the associated Weyl chamber. The space \(X\) can be given a so-called \(\Delta\)-distance, and a fundamental question is to determine which triples of elements in \(\Delta\) can occur as side-lengths of a geodesic triangle in \(X\). This collection turns out to be a convex homogeneous polyhedral cone \(\mathcal C\) which is also the solution to the generalized eigenvalues of a sum problem. The cone \(\mathcal C\) (which depends only on the root system) was first described by a set of (generally) redundant `triangle inequalities', and later refined to a smaller set of `restricted triangle inequalities' by \textit{P. Belkale} and \textit{S. Kumar} [Invent. Math. 166, No. 1, 185-228 (2006; Zbl 1106.14037)]. For a type \(A\) root system, it is known that the two sets of inequalities coincide and are in fact irredundant.
The first main result of this paper is an explicit computation of the restricted triangle inequalities for a root system of type \(D_4\) (i.e., \(G=\text{PSO}(8)\)). This is obtained via some explicit computations in the cohomology of \(G/P\) for a maximal parabolic subgroup \(P\). Further, with the aid of a computer program, it is verified that the inequalities are irredundant.
A related problem involves the Langlands dual group \(G^\vee\). Consider a triple of dominant weights and the tensor product of the three irreducible modules having corresponding highest weights. The problem is to determine those triples such that the \(G^\vee\)-invariants of this tensor product is non-zero. Denote this set by \(\mathcal R\). It is conjectured for simply-laced root systems that for a triple of weights \((\lambda,\mu,\nu)\) (whose sum lies in the root lattice) and a positive integer \(N\), if \((N\lambda,N\mu,N\nu)\) lies in \(\mathcal R\), then \((\lambda,\mu,\nu)\) necessarily lies in \(\mathcal R\). (The conjecture can equivalently be stated in terms of relationship between \(\mathcal R\) and the cone \(\mathcal C\) discussed above.) The conjecture is known to be true in type \(A\) by work of \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055-1090 (1999; Zbl 0944.05097)]. (It is also known that the conjecture can fail in non-simply laced cases.)
The second main result is a verification via the aid of computer computations that the conjecture holds for a root system of type \(D_4\) and \(G^\vee=\text{Spin}(8)\). A final application is given to structure constants of the spherical Hecke algebra associated to \(G=\text{PSO}(8)\). triangle inequalities; generalized eigenvalues; Langlands dual; irreducible representations; tensor product decompositions; symmetric spaces; Weyl chambers Kapovich, M.; Kumar, S.; Millson, J. J.: The eigencone and saturation for \(Spin(8)\). Pure appl. Math. Q. 5, No. 2, 755-780 (2009) Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Inequalities involving eigenvalues and eigenvectors, Root systems, Cohomology theory for linear algebraic groups, Differential geometry of symmetric spaces, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) The eigencone and saturation for Spin(8). | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Following the historical track in pursuing \(T\)-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton-Okounkov bodies, PBW-filtrations and cluster algebras come to push the subject forward. flag varieties; spherical varieties; cluster algebras; toric degenerations Fang, X.; Fourier, G.; Littelmann, P., Representation Theory -- Current Trends and Perspectives, 11, On toric degenerations of flag varieties, 187-232, (2017), European Mathematical Society Publishing House Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Fibrations, degenerations in algebraic geometry, Compactifications; symmetric and spherical varieties On toric degenerations 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 This fine paper delivers a combinatorial formula for the characters of the homogeneous components of the coinvariant algebra---\(\mathbb{Q}[x_1,\dots, x_n]\) factored out by those polynomials invariant under the action of the symmetric group with no constant term.
It begins with a clear and concise exposition of all the ingredients needed including Schubert polynomials, Monk's formula, and Kazhdan-Lusztig cells. From here the derivation of the action of the symmetric group on Schubert polynomials and some related inner product results lead to a swift proof of the aforementioned character formula, which is also shown to be equivalent to the decomposition of homogeneous components of the coinvariant algebra into irreducible representations.
Finally we are given a taster for two subsequent works ``Deformation of the coinvariant algebra'' and ``Major index of shuffles and restriction of representations'', the former of which yields a \(q\)-analogue of the character formula in this paper, the latter an algebraic interpretation of the set of all permutations of length \(k\) in the symmetric group. coinvariant algebra; Kazhdan-Lusztig cells; Schubert polynomials; character formula; irreducible representations; symmetric group Y. Roichman, Schubert polynomials, Kazhdan--Lusztig basis and characters, (with an, appendix: On characters of Weyl groups, co-authored with, R. M. Adin, and, A. Postnikov, ), Discrete Math, to appear. Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Symmetric groups Schubert polynomials, Kazhdan-Lusztig basis and characters | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article, we construct \(\mathrm{SL}_k\)-friezes using Plücker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of \(k\)-spaces in \(n\)-space via the Plücker embedding. When this cluster algebra is of finite type, the \(\mathrm{SL}_k\)-friezes are in bijection with the so-called mesh friezes of the corresponding Grassmannian cluster category. These are collections of positive integers on the AR-quiver of the category with relations inherited from the mesh relations on the category. In these finite type cases, many of the \(\mathrm{SL}_k\)-friezes arise from specializing a cluster to 1. These are called unitary. We use Iyama-Yoshino reduction to analyze the nonunitary friezes. With this, we provide an explanation for all known friezes of this kind. An appendix by Cuntz and Plamondon proves that there are 868 friezes of type \(E_6\). frieze pattern; mesh frieze; unitary frieze; cluster category; Grassmannian; Iyama-Yoshino reduction Combinatorial aspects of groups and algebras, Combinatorial aspects of representation theory, Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Representations of quivers and partially ordered sets, Categorical structures Friezes satisfying higher \(\mathrm{SL}_k\)-determinants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review, the authors consider complex Fano manifolds \(X\), having nef tangent bundle \(T_X\). It is a famous conjecture of Campana and Peternell that in this case \(X\) must be a rational homogeneous space. The authors prove this conjecture for those \(X\), which are of \textit{Flag Type} (or \textit{FT} for short), i.e. every Mori contraction of \(X\) has \(1\)-dimensional fibers (to begin with note that all extremal contractions on the initial \(X\) are of fiber type). The authors also treat the so-called \textit{conjecture of initiality of FT-manifolds}. The latter asserts that every Fano manifold \(Y\), which is not a product of varieties and has \(T_Y\) nef, is dominated by some FT-manifold \(X\) with connected Dynkin diagram \(\mathcal{D}(X)\) (cf. below). Note that a partial converse does hold (see Proposition 5 in the text) -- for if \(M \longrightarrow X\) is a contraction, where \(M\) is a Fano manifold with \(T_M\) nef, then there exists a variety \(Y\) such that \(M = X \times Y\).
The authors' method is based on the notion of \textit{Dynkin diagram} (denoted \(\mathcal{D}(X)\)) for \(X\). This is defined in terms of intersection theory on \(X\) (see subsection 3.2) and mimics the similar notion for homogenous spaces. More precisely, the authors show first (Theorem 5) that \(X\) is indeed a homogeneous space, provided it has Picard number two (the argument relies heavily on Theorem 1.1 from \textit{K. Watanabe} [J. Algebra 414, 105--119 (2014; Zbl 1304.14054)]). The latter fact is used (Corollary 1) to show that the intersection matrix between \(1\)- and \((\dim X - 1)\)-cycles on any FT \(X\) is a (generalized) Cartan matrix. Then one defines the Dynkin graph as usual (cf. Definition 3).
After all these preparations the authors prove their main result
Theorem 1. Every (connected) \(\mathcal{D}(X)\) as above is realized by some semi-simple finite-dimensional Lie algebra.
This is proved in Section 4 by induction on the Picard number of \(X\). The authors also prove in Section 5 that if \(G \slash B\) is a homogeneous variety, associated with \(\mathcal{D}(X)\), then \(\dim X \leq G \slash B\). The paper concludes by proving that \(X \simeq \mathbb{P}(T_{\mathbb{P}^n})\), \(n \geq 2\), whenever \(\mathcal{D}(X)\) is of type \(A_n\). Fano manifold; nef tangent bundle; homogeneous space Muñoz, R; Occhetta, G; Solá Conde, LE, Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle, Math. Ann., 361, 583-609, (2015) Grassmannians, Schubert varieties, flag manifolds, Minimal model program (Mori theory, extremal rays), Fano varieties, Algebraic cycles Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Ehresmann-Bruhat order on the symmetric group satisfies a symmetric property that we generalize to a direct graph with permutations as vertices, labeling edges with tableaux of a given shape. symmetric group; Ehresmann-Bruhat order; tableaux; Eulerian structure Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Tableaux and Eulerian properties of the symmetric 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 We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete answer to this question in terms of the underlying affine root system. In particular, this proves the corresponding conjecture for split groups stated in [the first author et al., Compos. Math. 146, No. 5, 1339--1382 (2010; Zbl 1229.14036)]. The question of non-emptiness of affine Deligne-Lusztig varieties is closely related to the relationship between certain natural stratifications of moduli spaces of abelian varieties in positive characteristic. affine Deligne-Lusztig varieties; \(\sigma\)-conjugacy classes; affine Weyl groups Görtz, U.; He, X.; Nie, S., \textit{P}-alcoves and nonemptiness of affine Deligne-Lusztig varieties, Ann. Sci. Éc. Norm. Supér., 48, 647-665, (2015) Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Modular and Shimura varieties, \(p\)-adic cohomology, crystalline cohomology, Loop groups and related constructions, group-theoretic treatment, Classical groups (algebro-geometric aspects) \(\mathbf{P}\)-alcoves and nonemptiness of affine Deligne-Lusztig varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The convolution ring \(K^{\mathrm{GL}_n(\mathcal{O})\rtimes \mathbb{C}^\times}(\mathrm{Gr}_{\mathrm{GL}_n})\) was identified with a quantum unipotent cell of the loop group \(L\mathrm{SL}_2\) in [\textit{S. Cautis} and \textit{H. Williams}, J. Amer. Math. Soc. 32, No. 3, 709--778 (2019; Zbl 1442.22022)]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell. Quantum groups (quantized enveloping algebras) and related deformations, Loop groups and related constructions, group-theoretic treatment, Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Coherent IC-sheaves on type \(A_n\) affine Grassmannians and dual canonical basis of affine type \(A_1\) | 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 We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators with two supercharges, for world-volume gauge theories of \(N\) D-brane probes for both \(N\rightarrow \infty \) and finite \(N\). The techniques are applicable to generic singularities, orbifold, toric, non-toric, complete intersections, et cetera, even to geometries whose precise field theory duals are not yet known. The so-called ``plethystic exponential'' provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace BPS operators and (3) the generating function of multi-trace operators. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies. BPS operators; gauge theories R. Stanley, GL(\textit{n, C}) \textit{for combinatorialists}, in \textit{Surveys in Combinatorics}, London Math. Soc. Lecture Note Series, Cambridge University Press (1983), pp. 187-200. Supersymmetric field theories in quantum mechanics, Relationships between surfaces, higher-dimensional varieties, and physics, Representations of quivers and partially ordered sets, Syzygies, resolutions, complexes and commutative rings, Algebraic combinatorics, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Counting BPS operators in gauge theories: quivers, syzygies and plethystics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 review/announcement of results concerning the connection between certain exactly solvable two-dimensional models of statistical mechanics, namely loop models, and the equivariant \(K\)-theory of the cotangent bundle of the Grassmannian. We interpret various concepts from integrable systems (\(R\)-matrix, partition function on a finite domain) in geometric terms. As a byproduct, we provide explicit formulae for \(K\)-classes of various coherent sheaves, including structure and (conjecturally) square roots of canonical sheaves and canonical sheaves of conormal varieties of Schubert varieties. quantum integrability; loop models; \(K\)-theory Exactly solvable models; Bethe ansatz, Miscellaneous applications of \(K\)-theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Relationships between surfaces, higher-dimensional varieties, and physics, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations Loop models and \(K\)-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 Let \(G/P\) be the Grassmannians of \(k\)-planes in \(\mathbb{C}^ n\). Let \(\theta\) be an involution of \(G\) \((=\text{GL}(n))\), and \(K\) the connected component of the fixed point set of \(\theta\). In this paper, the authors prove that the closures of \(K\)-orbits in \(G/P\) are normal, have rational singularities (and hence are Cohen-Macaulay). For other generalized flag varieties, these orbit closures are not always normal. By studying those orbit closures which are normal, the authors prove a PRV-conjecture type results for real groups. This paper makes an important contribution to representation theory. Grassmannians; fixed point set; generalized flag varieties; orbit closures; PRV-conjecture D. Barbasch, S. Evens, K-orbits on Grassmannians and a PRV conjecture for real groups, J. Algebra 167 (1994), no. 2, 258--283. Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Homogeneous spaces and generalizations \(K\)-orbits on Grassmannians and a PRV conjecture for real 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 Recently, the existence of an amplituhedron for tree level amplitudes in the bi-adjoint scalar field theory has been proved by \textit{N. Arkani-Hamed} et al. [Grassmannian geometry of scattering amplitudes. Cambridge: Cambridge University Press (2016; Zbl 1365.81004)]. We argue that hyperbolic geometry constitutes a natural framework to address the study of positive geometries in moduli spaces of Riemann surfaces, and thus to try to extend this achievement beyond tree level. In this paper we begin an exploration of these ideas starting from the simplest example of hyperbolic geometry, the hyperbolic plane. The hyperboloid model naturally guides us to re-discover the moduli space associahedron, and a new version of its kinematical avatar. As a by-product we obtain a solution to the scattering equations which can be interpreted as a special case of the two well known solutions in terms of spinor-helicity formalism. The construction is done in 1 + 2 dimensions and this makes harder to understand how to extract the amplitude from the dlog of the space time associahedron. Nevertheless, we continue the investigation accommodating a loop momentum in the picture. By doing this we are led to another polytope called halohedron, which was already known to mathematicians. We argue that the halohedron fulfils many criteria that make it plausible to be understood as a 1-loop amplituhedron for the cubic theory. Furthermore, the hyperboloid model again allows to understand that a kinematical version of the halohedron exists and is related to the one living in moduli space by a simple generalisation of the tree level map. scattering amplitudes; differential and algebraic geometry G. Salvatori and S.L. Cacciatori, \textit{Hyperbolic Geometry and Amplituhedra in 1+2 dimensions}, arXiv:1803.05809 [INSPIRE]. \(2\)-body potential quantum scattering theory, Applications of differential geometry to physics, Yang-Mills and other gauge theories in quantum field theory, Quantum field theory on lattices, Grassmannians, Schubert varieties, flag manifolds Hyperbolic geometry and amplituhedra in 1+2 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 Many criteria are available in the literature which give necessary and sufficient conditions for checking if an element in Grassmann space is decomposable. However, those criteria are difficult to check. The purpose of this paper is to present some criteria that allow easy checks on a non-decomposable element in the Grassmann space. The main results are given in Theorem 1 of the paper, which gives a set of five necessary conditions for an element in the Grassmann space to be decomposable. Based on these five necessary conditions, one can easily conclude that an element is not decomposable if any one of the five conditions is violated. non-decomposable element; Grassmann space Exterior algebra, Grassmann algebras, Grassmannians, Schubert varieties, flag manifolds Some criteria for non-decomposable elements in Grassmann 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 gives a classification of smooth congruences of lines of degree \( 9\) in three-dimensional complex projective space. Proofs are given in terms of the associated grassmannian G(1,3), i.e. Klein's quadric. Moreover explicit constructions for these congruences are given. The paper extends a classical result on congruences of degree \(<9\) due to G. Fano (1893). classification of smooth congruences of lines of \(\deg ree\quad 9\); grassmannian Verra, A.: Smooth Surfaces of Degree 9 inG(1, 3). Manuscr. Math.62, 417--435 (1988) Grassmannians, Schubert varieties, flag manifolds Smooth surfaces of degree 9 in G(1,3) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Matroids with coefficients have been introduced by the first author [Adv. Math. 59, 97-123 (1986; Zbl 0656.05025)]. In this note an outline of the theory, restricted to the finite case, is given, taking advantage of whatever simplifications are possible in this particular case. [DW1] A. W. M. Dress and W. Wenzel: Endliche Matroide mit Koeffizienten,Bayreuth. Math. Schr. 26 (1988), 37--98. Combinatorial aspects of matroids and geometric lattices, Combinatorial aspects of finite geometries, Combinatorial geometries and geometric closure systems, Grassmannians, Schubert varieties, flag manifolds Endliche Matroide mit Koeffizienten. (Finite matroids with coefficients.) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 literature on maximal torus orbits in the Grassmannian is vast; in this paper we initiate a program to extend this to diagonal subtori. Our main focus is generalizing portions of Kapranov's seminal work on Chow quotient compactifications of these orbit spaces. This leads naturally to discrete polymatroids, generalizing the matroidal framework underlying Kapranov's results. By generalizing the Gelfand-MacPherson isomorphism, these Chow quotients are seen to compactify spaces of arrangements of parameterized linear subspaces, and a generalized Gale duality holds here. A special case is birational to the Chen-Gibney-Krashen moduli space of pointed trees of projective spaces, and we show that the question of whether this birational map is an isomorphism is a specific instance of a much more general question that hasn't previously appeared in the literature, namely, whether the geometric Borel transfer principle in non-reductive GIT extends to an isomorphism of Chow quotients. Grassmannians, Schubert varieties, flag manifolds, (Equivariant) Chow groups and rings; motives, Parametrization (Chow and Hilbert schemes), Geometric invariant theory Chow quotients of Grassmannians by diagonal subtori | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(X=\mathbf{V}(f)\subset\mathbb P^{n+1}\) be a smooth hypersurface of degree \(\deg f=d\). A classical result of [\textit{P. A. Griffiths}, Am. J. Math. 90, 805--865 (1968; Zbl 0183.25501)] identifies the primitive part of \(H^{n-p+1,p-1}(X)\) with the \(pd-n-2\) homogeneous component of \(\mathbb C[x_0,\ldots,x_{n+1}]/J_f\), where \(J_f=(f_0,\ldots,f_{n+1})\) is the Jacobian ideal generated by the partial derivatives of \(f\).
There exist various generalizations of this statement. For instance to complete intersections in projective space by [\textit{A. Dimca}, Duke Math. J. 78, No. 1, 89--100 (1995; Zbl 0839.14009)], or toric varieties by [\textit{V. V. Batyrev} and \textit{D. A. Cox}, Duke Math. J. 75, No. 2, 293--338 (1994; Zbl 0851.14021)], [\textit{K. Konno}, Compos. Math. 78, No. 3, 271--296 (1991; Zbl 0737.14002)], [\textit{A. R. Mavlyutov}, Pac. J. Math. 191, No. 1, 133--144 (1999; Zbl 1032.14013)], or hypersurfaces of high degree in arbitrary projective manifolds by [\textit{M. L. Green}, Compos. Math. 55, 135--156 (1985; Zbl 0588.14004)], or recently to zero loci of homogeneous vector bundles by [\textit{A. Huang} et al., ``Jacobian rings for homogenous vector bundles and applications'', Preprint, \url{arXiv:1801.08261}].
The main results of this paper are generalizations of Griffiths' result to complete intersections in Grassmann varieties. They do not use primitive cohomology, but a refinement, the vanishing cohomology. For hypersurfaces in projective space the distinction is irrelevant.
For instance if \(X\subset G=\mathrm{Gr}(k,n)\) is a smooth hypersurface of degree \(d\geq n-1\) and dimension \(N-1=k(n-k)-1\), and \(N\) is odd, then the vanishing part of \(H^{N-1-p,p}(X)\) is isomorphic to the degree \((p+1)d-n\) part of \(R^G_f\). The latter is what the authors call the Griffiths ring of \(X\). It is the quotient of the Plücker coordinate ring of \(G\) by the homogeneous ideal generated by \(f\) and its \(\mathfrak{sl}_n\)-orbit. Here \(\mathfrak{sl}_n\) acts by derivations in a concrete way. When \(N\) is even, then we may no longer obtain an isomorphism, but the difference is explicit, determined only by \(G\) and \(p\).
The authors prove a similar result for complete intersections of more than one hypersurface in \(G\). The Griffiths-type ring is now denoted \(\mathcal U\). Its definition is cohomological, motivated by a Cayley trick to reduce from complete intersections in \(G\) to hypersurfaces in a projective bundle over \(G\).
The main advantage of the results of this paper over other generalizations is their explicit nature, mirroring the original result of Griffiths. The authors give concrete presentations of their Griffiths-type rings and use them to compute several explicit examples of Hodge groups. For instance they carry out these computations for Fano 5-folds and 4-folds of genus 6 and degree 10, a Calabi-Yau section of \(\mathrm{Gr}(2,7)\), and for Fano varieties of K3-type. Some of these computations of Hodge groups were known, but the ring structure is new.
Apart from classical methods such as chasing exact sequences in cohomology, the paper makes great use of the Cayley trick for reducing complete intersections to hypersurfaces in a projective bundle, and of the connection between the Hodge theory of a particular class of projectively normal \(X\) and the infinitesimal first-order deformation module of the affine cone \(A_X\). Griffiths ring; Jacobian ideal; complete intersection in Grassmann; Hodge group; Cayley trick Transcendental methods, Hodge theory (algebro-geometric aspects), Hypersurfaces and algebraic geometry, Fano varieties, Calabi-Yau manifolds (algebro-geometric aspects), Complete intersections, Grassmannians, Schubert varieties, flag manifolds A note on a Griffiths-type ring for complete intersections in 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 give a simple proof of a criterion for the existence of global sections for line bundles on Schubert varieties. Unlike previous proofs, which use the nontrivial fact that Schubert varieties are normal, our proof is based on an elementary geometric property of root systems and their Weyl groups. global sections; line bundles; Schubert varieties; root systems; Weyl groups R. Dabrowski, A simple proof of a necessary and sufficient condition for the existence of nontrivial global sections of a line bundle on a Schubert variety, in: Kazhdan-Lusztig Theory and Related Topics (Chicago, IL, 1989), Contemp. Math., Vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 113-120. Linear algebraic groups over arbitrary fields, Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras A simple proof of a necessary and sufficient condition for the existence of nontrivial global sections of a line bundle on a Schubert variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study good packings in Grassmannian space, or, in other words, how to pack \(m\)-dimensional real Euclidean space with \(N\) \(n\)-dimensional subspaces such that these are as far apart as possible. This research was initiated by \textit{J. H. Conway, R. H. Hardin}, and \textit{N. J. A. Sloane} in Exp. Math. 5, No. 2, 139-159 (1996); editor's note ibid. 6, No. 2, 175 (1997; Zbl 0864.51012) where small cases are studied and bounds are derived. One of these bounds (Corollary 5.3) is called the orthoplex bound.
For each \(m\) a power of two, the current paper gives a construction of a family of packings of \(m^2+m-2\) \(m/2\)-dimensional subspaces of \(m\)-dimensional Euclidean space that meet the orthoplex bound, and are therefore optimal.
The construction was discovered by studying a family of subgroups of the real orthogonal group that are connected with quantum codes. optimal packings; Grassmannian manifolds Shor, PW; Sloane, NJA, A family of optimal packings in Grassmannian manifolds, J. Algebraic Comb., 7, 157-163, (1998) Packing and covering in \(n\) dimensions (aspects of discrete geometry), Polyhedra and polytopes; regular figures, division of spaces, Grassmannians, Schubert varieties, flag manifolds A family of optimal packings in Grassmannian 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 Schubert polynomials are the building blocks of several cohomological degeneracy locus formulas. Their role in \(K\)-theory is played by so-called Grothendieck polynomials. In particular, ``stable'' versions of Schubert and Grothendieck polynomials turn up naturally, for example in degeneracy locus problems associated with certain quiver representations.
The paper under review studies the expansion of stable Grothendieck polynomials in the basis of stable Grothendieck polynomials for partitions. This generalizes the result of Fomin-Green in the cohomological setting, where the analogues of stable Grothendick polynomials for partitions are the Schur polynomials. The existence of such a finite, integer linear combination expansion was proved by Buch, and the sign of the coefficients were determined by Lascoux. In this paper the authors give a new, non-recursive combinatorial rule for the coefficients. Namely, they prove that the coefficients, up to explicit sign, count the number of increasing tableau of a given shape, with an associated word having explicit combinatorial properties stemming from the combinatorics of the 0-Hecke monoid. The main ingredient of the proof is a generalized, so-called Hecke insertion algorithm.
The main application showed in the paper is a \(K\)-theoretic analogue of the factor sequence formula of Buch-Fulton for the cohomological quiver polynomials (of equioriented type A). Grothendieck polynomials; \(K\)-theory; 0-Hecke monoid; insertion algorithm; factor sequence formula Buch, A.; Kresch, A.; Shimozono, M.; Tamvakis, H.; Yong, A., Stable Grothendieck polynomials and \textit{K}-theoretic factor sequences, Math. Ann., 340, 359-382, (2008) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], \(K\)-theory of schemes, Symmetric functions and generalizations Stable Grothendieck polynomials and \(K\)-theoretic factor sequences | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Investigations on determinantal ideals and varieties are an interesting part of commutative ring theory and algebraic geometry. The authors avoid geometric methods and develop a purely algebraic approach to determinantal rings using mainly the theory of algebras with straightening law (Hodge algebras) on posets of minors. This is done simultaneously with the treatment of the homogeneous coordinate rings of the Schubert varieties of Grassmannians (so called Schubert cycles), where the combinatorial structure is simpler. [Algebraically every determinantal ring may be considered as a dehomogenization of a Schubert cycle.]
The subjects of this book include results on height and grade, the Cohen- Macaulay property and normality of determinantal rings and Schubert cycles and the computation of their singular locus. Moreover the divisor class groups of Schubert cycles and determinantal rings over a normal ground ring of coefficients are considered. In section \(12\) the authors also discuss Hochster-Eagon's proof of the perfection of determinantal ideals where principal radical systems instead of standard monomials are used. Finally Kähler differentials and representation-theoretical aspects of determinantal rings are described in a systematic way. The dominating example for illustrating properties of determinantal ideals is the ideal generated by the maximal minors.
The book is self-contained and includes most of the details needed for beginners. determinantal ideals; algebras with straightening law; Hodge algebras; Schubert cycles; divisor class groups W. Bruns, U. Vetter, \(Determinantal Rings\). Lecture Notes in Mathematics, vol. 1327 (Springer, New York, 1988) Theory of modules and ideals in commutative rings, Research exposition (monographs, survey articles) pertaining to commutative algebra, Determinantal varieties, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Ideals and multiplicative ideal theory in commutative rings, Relevant commutative algebra Determinantal rings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 concerns an example of Landau Ginzburg mirror symmetry related to Grassmannians and cluster algebras. In this set up, the mirror to a Grassmannian \(X\) is given by a cluster variety \(\breve{X}\), together with a ``superpotential'' \(W:\breve{X} \to \mathbf{C}\).
The author introduces two spaces: ``the decorated Grassmannian'', and the ``decorated configuration space''. In rough terms, the decorated Grassmannian is a complement of an ample divisor in the affine cone over the ordinary Grassmannian. On the other hand the decorated configuration space parametrizes particular configuration of lines in a vector space. The main result of the paper shows that both of these spaces have natural cluster structures. More precisely, the decorated Grassmannian is an \(\mathcal{A}\)-cluster variety, whereas the decorated configuration space is an \(\mathcal{X}\)-cluster variety in the sense of Fock-Goncharov [\textit{V. V. Fock} and \textit{A. B. Goncharov}, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865--930 (2009; Zbl 1180.53081)]. To show this, author applies the key result obtained by Gross-Hacking-Keel-Kontsevich [\textit{M. Gross} et al., J. Am. Math. Soc. 31, No. 2, 497--608 (2018; Zbl 1446.13015)], showing that there is a canonical basis for the coordinate ring of the decorated Grassmannian, parametrized by integral tropical points of the decorated configuration space, to describe this canonical basis and obtain the superpotential \(W\) on the decorated configuration space. Moreover, a comparision of this potential to the superpotential introduced by Rietsch-Williams as mirror of the Grassmannian is provided [\textit{K. Rietsch} and \textit{L. Williams}, Duke Math. J. 168, No. 18, 3437--3527 (2019; Zbl 1439.14142)], and it is shown that they are compatible.
The author also proves a purely combinatorial result about plane partitions, called the ``cyclic sieving phenomenon'', by using the fact that the canonical basis for the coordinate ring of the decorated Grassmannian can be parametrized by integral tropical points. cluster algebra; cluster duality; mirror symmetry; Grassmannian; cyclic sieving phenomenon Cluster algebras, Combinatorial aspects of representation theory, Mirror symmetry (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Tropical geometry, Applications of deformations of analytic structures to the sciences Cyclic sieving and cluster duality of Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review is the first part of a series of two papers, the second one being [\textit{B. Feigin, M. Finkelberg, A. Kusnetsov} and \textit{I. Mirkovic}, in: Differential topology, infinite-dimensional Lie algebras, and applications. D. B. Fuchs' 60th anniversary collection. AMS Transl., Ser. 2, Am. Math. Soc. 194(44), 113--148 (1999; Zbl 1076.14511)], where semi-infinite flag varieties are studied together with perverse sheaves on them. For a Borel subgroup \(B\subset G\), the notion of the semi-infinite flag space \(\mathcal Z\) was introduced by \textit{B. L. Feigin} and \textit{E. V. Frenkel} [Commun. Math. Phys. 128, No. 1, 161--189 (1990; Zbl 0722.17019)] as the quotient of \(G((z))\) modulo the connected component of \(B((z))\). The authors are interested in the space \(\mathcal{PS}\) of perverse sheaves on \(\mathcal Z\) equivariant with respect to the Iwahori subgroup \(I\subset G[[z]]\).
The first part is devoted to the construction of \(\mathcal{PS}\) as an ind-scheme, using spaces \(\mathcal Q^\alpha\) of ``quasi-maps'' from \(\mathbb{P}^1\) to the flag variety of \(G\) (which are Drinfeld compactifications of maps of degree \(\alpha\in H^2(\mathcal B)\). Then, a system of of subvarieties \(\mathcal Z^\alpha \subset \mathcal Q^\alpha\) is considered, which consists of quasi-maps \(f: \mathbb{P}^1\to\mathcal B\) defined at \(\infty\) fulfilling \(f(\infty)=B_-\).
Then \(\mathcal{PS}\) is constructed as collections of perverse sheaves on \(\mathcal Z^\alpha\), together with so-called factorizations
[cf. \textit{R. Bezrukavnikov, M. Finkelberg} and \textit{V. Schechtman}, ``Factorizable sheaves and quantum groups'', Lect. Notes Math. 1691 (1998; Zbl 0938.17016)].
In the third chapter, a convolution functor from the category of perverse sheaves on \(G((z))/G[[z]]\), constant along the orbits of \(I\), to \(\mathcal{PS}\) is constructed. This serves as a geometric counterpart of \textit{V. Ginzburg}'s [Perverse sheaves on a Loop group and Langlands' duality, preprint, \texttt{http://arxiv.org/abs/alg-geom/9511007}] restriction functor to the principal block in the category of modules over the quantum group at a root of unity, which has been conjectured to be equivalent to \(\mathcal{PS}\). Finkelberg, M., Mirković, I.: Semi-infinite flags I. Case of global curve \(\mathbb{P}^1\), in Differential topology, infinite-dimensional Lie algebras, and applications, 81-112, Am. Math. Soc. Transl. Ser. 2, vol. 194, Am. Math. Soc. (1999) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Cohomology theory for linear algebraic groups, Quantum groups (quantized function algebras) and their representations Semi-infinite flags. I: Case of global curve \(\mathbb{P}^1\) | 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 We continue, generalize and expand our study of linear degenerations of flag varieties from [the authors, Math. Z. 287, No. 1--2, 615--654 (2017; Zbl 1388.14145)]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel-Weil theorem for the flat irreducible locus. Grassmannians, Schubert varieties, flag manifolds, Fibrations, degenerations in algebraic geometry Linear degenerations of flag varieties: partial flags, defining equations, and group actions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [\textit{A. Knutson} and \textit{E. Miller}, Gröbner geometry of Schubert polynomials, Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)]. The original proof of the duality was roundabout, relying on the algebra of certain monomial ideals and a recursive characterization of reduced pipe dreams. This paper provides a direct combinatorial proof. Jia, N.; Miller, E.: Duality of antidiagonals and pipe dreams, Sém. lothar. Combin. 58 (2008) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds Duality of antidiagonals and pipe dreams | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Assume that the valuation semigroup \(\Gamma (\lambda )\) of an arbitrary partial flag variety corresponding to the line bundle \(\mathcal{L}_\lambda \) constructed via a full-rank valuation is finitely generated and saturated. We use Ehrhart theory to prove that the associated Newton-Okounkov body -- which happens to be a rational, convex polytope -- contains exactly one lattice point in its interior if and only if \(\mathcal{L}_\lambda\) is the anticanonical line bundle. Furthermore, we use this unique lattice point to construct the dual polytope of the Newton-Okounkov body and prove that this dual is a lattice polytope using a result by Hibi. This leads to an unexpected, necessary and sufficient condition for the Newton-Okounkov body to be reflexive. Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Classical groups (algebro-geometric aspects) Reflexivity of Newton-Okounkov bodies of partial 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 This paper connects two degenerations related to the manifold \(F_n\) of complete flags in \({\mathbb C}^n\). \textit{N. Gonciulea} and \textit{V. Lakshmibai} [Transform.~Groups 1, No.~3, 215--248 (1996; Zbl 0909.14028)] used standard monomial theory to construct a flat sagbi degeneration of \(F_n\) into the toric variety of the Gelfand-Tsetlin polytope, and recently \textit{A. Knutson} and \textit{E. Miller} [Ann.~Math. (2) 161, No.~3, 215--248 (2005; Zbl 1089.14007)] constructed Gröbner degenerations of matrix Schubert varieties into linear spaces corresponding to monomials in double Schubert polynmials. The flag variety is the geometric invariant theory (GIT) quotient of the space \(M_n\) of \(n\) by \(n\) matrices by the Borel group \(B\) of lower triangular matrices. A matrix Schubert variety is an inverse image of a Schubert variety under this quotient. The main result in the paper under review is that this GIT quotient extends to the degenerations. The sagbi degeneration is a GIT quotient of the Gröbner degeneration.\smallskip The nature of this GIT quotient is quite interesting. The authors exhibit an action of the Borel group \(B\) on the product \(M_n\times{\mathbb C}\) of \(M_n\) with the complex line, so that the GIT quotient \(B\backslash\backslash(M_n\times{\mathbb C})\) remains fibred over \({\mathbb C}\) and is the total space of the sagbi degeneration. In this GIT quotient, the total space of the Gröbner degeneration (as in Knutson and Miller) of a matrix Schubert variety in \(M_n\times{\mathbb C}\) covers the total space of the Lakshmibai-Gonciulea degeneration of the corresponding Schubert variety. At the degenerate point, the matrix Schubert variety has become a union of coordinate planes, each of which covers a component of the sagbi degeneration of the Schubert variety indexed by a face of the Gelfand-Tsetlin polytope.
The authors use this to identify which faces of the Gelfand-Tsetlin polytope occur in a given degenerate Schubert variety, and to give a simple explanation of the classical Gelfand-Tsetlin decomposition of an irreducible polynomial representation of \(\text{GL}_n\) into one-dimensional weight spaces; in the degeneration, sections of a line bundle over \(F_n\) become sections of the defining line bundle on the toric variety for the Gelfand-Tsetlin polytope. Flag variety; Schubert variety; sagbi basis; Gelfand-Tsetlin pattern; Borel-Weil theorem Kogan, M., Miller, E.: Toric degeneration of Schubert varieties and Gelfand--Tsetlin polytopes. Adv. Math. 193(1), 1--17 (2005) Grassmannians, Schubert varieties, flag manifolds, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Toric degeneration of Schubert varieties and Gelfand--Tsetlin 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 In this article, we prove a tableau formula for the double Grothendieck polynomials associated to 321-avoiding permutations. The proof is based on the compatibility of the formula with the \(K\)-theoretic divided difference operators. Our formula specializes to the one obtained by \textit{W. Y. C. Chen} et al. [Eur. J. Comb. 25, No. 8, 1181--1196 (2004; Zbl 1055.05149)] for the (double) skew Schur polynomials. symmetric polynomials; Grothendieck polynomials; \(K\)-theory; set-valued tableaux; 321-avoiding permutations Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices A tableau formula of double Grothendieck polynomials for 321-avoiding permutations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 variety of dimension \(m\), with a Kähler metric. If \(A\) is an effective cycle on \(X\) of pure dimension \(p\), we can associate to it a Green current \(g_A\), determined uniquely by the conditions that it be admissible with respect to the given Kähler metric on \(X\), and that its harmonic projection be zero. The same holds for an effective cycle \(B\) of dimension \(q\). If \(p+q=m-1\), then the arithmetic intersection theory of Gillet and Soulé defines a local intersection number \(\langle A,B\rangle\in\mathbb R\).
Let \(\mathcal C_p^\alpha(X)\) denote the Chow variety of dimension-\(p\) cycles on \(X\) in a given cohomology class \(\alpha\), and consider the incidence set consisting of all points in \(\mathcal C_p^\alpha(X)\times\mathcal C_q^\beta(X)\) corresponding to pairs of cycles \(A\) and \(B\) on \(X\) whose supports intersect. This set can contain whole irreducible components of \(\mathcal C_p^\alpha(X)\times\mathcal C_q^\beta(X)\). \textit{B. Mazur} has conjectured that this incidence set is a Cartier divisor on the union of the irreducible components of \(\mathcal C_p^\alpha(X)\times\mathcal C_q^\beta(X)\) not contained in the incidence set. He has further suggested that the local intersection number \(\langle A,B\rangle\) on the complement of the incidence set should be equal to some metric on this divisor.
The present paper answers a large part of this question by showing that there exists a metrized line bundle \(\mathcal L\) and a section \(s\) of \(\mathcal L\) such that \(\log\|s(A,B)\|^2 =(m-1)!\langle A,B\rangle\). The proof uses reduction to the diagonal, and some geometry of Grassmannians. archimedean height pairing; Green current; Chow variety; archimedean intersection number; Grassmannians DOI: 10.1023/A:1000650009973 Arithmetic varieties and schemes; Arakelov theory; heights, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Mazur's incidence structure for projective 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 [For the entire collection see Zbl 0608.00010.]
The wedge and the vertex construction of the fundamental highest weight representations of the group \(GL_{\infty}\) are discussed, and a kind of boson-fermion correspondence is obtained. This is applied to the modified KP hierarchies yielding several definitions of these hierarchies and polynomial solutions. Geometrical interpretations of the modified KP hierarchies are given in terms of an infinite dimensional Grassmann variety and an associated flag variety. Kac-Moody algebra; wedge; vertex construction; highest weight representations; boson-fermion correspondence; KP hierarchies; polynomial solutions; Grassmann variety; flag variety Kac, V. G.; Peterson, D. H.: Lectures on infinite wedge representation and MKP hierarchy. Séminaire de math. Sup., LES presses de l'université de Montreal 102, 141-186 (1986) Partial differential equations of mathematical physics and other areas of application, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Infinite-dimensional Lie groups and their Lie algebras: general properties, Grassmannians, Schubert varieties, flag manifolds, Geometric theory, characteristics, transformations in context of PDEs Lectures on the infinite wedge-representation and the MKP hierarchy | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Some geometric and combinatorial aspects of the solution to the full Kostant-Toda (f-KT) hierarchy are studied, when the initial data is given by an arbitrary point on the totally non-negative (tnn) flag variety of \(\text{SL}_n(\mathbb{R})\). The f-KT flows on the tnn flag variety are complete, and it is shown that their asymptotics are completely determined by the cell decomposition of the tnn flag variety given by Rietsch. These results represent the first results on the asymptotics of the f-KT hierarchy (and even the f-KT lattice); moreover, these results are not confined to the generic flow, but cover non-generic flows as well. The f-KT flow on the weight space via the moment map is defined, and it is shown that the closure of each f-KT flow forms an interesting convex polytope which is called a Bruhat interval polytope. In particular, the Bruhat interval polytope for the generic flow is the permutohedron of the symmetric group \(\mathfrak{G}_n\). An analogous results for the full symmetric Toda hierarchy, by mapping f-KT solutions to those of the full symmetric Toda hierarchy is also proved. In the appendix, it is shown that Bruhat interval polytopes are generalized permutohedra. Kostant-Toda lattice; Grassmannian; flag variety; moment polytope Kodama, Y.; Williams, L., The full Kostant--Toda hierarchy on the positive flag variety, Commun. Math. Phys., 335, 247-283, (2015) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Grassmannians, Schubert varieties, flag manifolds, Special polytopes (linear programming, centrally symmetric, etc.) The full Kostant-Toda hierarchy on the positive 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 Staggered \(t\)-structures are a class of \(t\)-structures on derived categories of equivariant coherent sheaves. In this note, we show that the derived category of coherent sheaves on a partial flag variety, equivariant for a Borel subgroup, admits a staggered \(t\)-structure with the property that all objects in its heart have finite length. As a consequence, we obtain a basis for its equivariant \(K\)-theory consisting of simple staggered sheaves. P. Achar and D. Sage, Staggered sheaves on partial flag varieties, arXiv:0712.1615. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Grassmannians, Schubert varieties, flag manifolds Staggered sheaves on partial 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 In this article, we develop the theory of local models for the moduli stacks of global \(\mathfrak{G}\)-shtukas, the function field analogs for Shimura varieties. Here, \(\mathfrak{G}\) is a smooth affine group scheme over a smooth projective curve. As the first approach, we relate the local geometry of these moduli stacks to the geometry of Schubert varieties inside global affine Grassmannian, only by means of global methods. Alternatively, our second approach uses the relation between the deformation theory of global \(\mathfrak{G}\)-shtukas and associated local \(\mathbb{P}\)-shtukas at certain characteristic places. Regarding the analogy between function fields and number fields, the first (resp. second) approach corresponds to Beilinson-Drinfeld-Gaitsgory (resp. Rapoport-Zink) type local model for (PEL-)Shimura varieties. This discussion will establish a conceptual relation between the above approaches. Furthermore, as applications of this theory, we discuss the flatness of these moduli stacks over their reflex rings, we introduce the Kottwitz-Rapoport stratification on them, and we study the intersection cohomology of the special fiber. Stacks and moduli problems, Modular and Shimura varieties, Grassmannians, Schubert varieties, flag manifolds Local models for the moduli stacks of global \(\mathfrak{G}\)-shtukas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 certain class of finite dimensional algebras the author shows that certain moduli spaces of finite dimensional modules are isomorphic to certain Grassmannian varieties. From this result it follows that the two different varieties associated to a given quiver by \textit{G. Lusztig} [in Adv. Math. 136, No. 1, 141-182 (1998; Zbl 0915.17008)] are isomorphic. The isomorphism the author constructs induces a bijection between the \(\mathbb C\)-points of these varieties constructed already in the above mentioned paper of Lusztig. schemes; Grassmannians; finite-dimensional algebras; moduli spaces; varieties Shipman B. A., Discrete and Continuous Dynamical Systems Representations of associative Artinian rings, Grassmannians, Schubert varieties, flag manifolds On representation schemes and Grassmanians of finite dimensional algebras and a construction of Lusztig. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials \(\mathfrak{S}_w\) represent cohomology classes of Schubert cycles in the full flag variety. Their coefficients are nonnegative integers. There exists a number of combinatorial formulas for computing these coefficients. This paper is devoted to the following question: when are all the coefficients of a Schubert polynomial equal to \(0\) or \(1\)? Such polynomials are called zero-one Schubert polynomials.
To answer this question, the authors first make the following observation: if a permutation \(\sigma\in S_m\) is a pattern of \(w\in S_n\), then the Schubert polynomial \(\mathfrak{S}_w\) equals a monomial times \(\mathfrak{S}_\sigma\) plus a polynomial with nonnegative coefficients. Hence the set of 0-1 Schubert polynomials is closed under pattern containment. Using Magyar's orthodontia, an inductive algorithm for computing \(\mathfrak{S}_w\) in terms of the Rothe diagram of \(w\), they describe the set of twelve avoided patterns, and also formulate equivalent confitions for a Schubert polynomial to be 0-1 in terms of Rothe diagrams and orthodontic sequences of permutations.
According to the recent result of the same authors about the supports of Schubert polynomials (see [\textit{A. Fink} et al., Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)]), this implies that each 0-1 Schubert polynomial is equal to the integer transform of a generalized permutahedron. Schubert polynomial; pattern avoidance; Rothe diagram Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Permutations, words, matrices Zero-one 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 The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group \(S_n\). Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote as \({\hat{\mathfrak S}}_y\) (to be called \textit{involution Schubert polynomials}) and \(\hat{\mathfrak S}^{\mathtt{FPF}}_y\) (to be called \textit{fixed-point-free involution Schubert polynomials}). Our main results are explicit formulas decomposing the product of \({\hat{\mathfrak S}}_y\) (respectively, \(\hat{\mathfrak S}^{\mathtt{FPF}}_y\)) with any \(y\)-invariant linear polynomial as a linear combination of other involution Schubert polynomials. These identities serve as analogues of Lascoux and Schützenberger's transition formula for Schubert polynomials, and lead to a self-contained algebraic proof of the nontrivial equivalence of several definitions of \({\hat{\mathfrak S}}_y\) and \( \hat{\mathfrak S}^{\mathtt{FPF}}_y\) appearing in the literature. Our formulas also imply combinatorial identities about \textit{involution words}, certain variations of reduced words for involutions in \(S_n\). We construct operators on involution words based on the Little map to prove these identities bijectively. The proofs of our main theorems depend on some new technical results, extending work of Incitti, about covering relations in the Bruhat order of \(S_n\) restricted to involutions. Symmetric groups, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Reflection and Coxeter groups (group-theoretic aspects), Compactifications; symmetric and spherical varieties Transition formulas for involution 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 In the paper under review, the author studies interactions among vertex operators, Grassmannians, and Hilbert schemes. The infinite Grassmannian is approximated by finite-dimensional cutoffs, and a family of fermionic vertex operators is introduced as the limit of geometric correspondences on the equivariant cohomology groups with respect to a one-dimensional torus actions on the cutoffs. The author proves that in the localization basis, these operators are the fermionic vertex operators on the infinite wedge representation. Moreover, the boson-fermion correspondence, locality and intertwining properties with the Virasoro algebra are the limits of relations on the cutoffs. The author further shows that these operators coincide with the vertex operators defined by A. Okounkov and the author in an earlier work on the equivariant cohomology groups of the Hilbert schemes of points on the affine plane with respect to a special torus action. Vertex operators; Grassmannians; Hilbert schemes Carlsson, E.: Vertex operators, grassmannians, and Hilbert schemes, Comm. math. Phys. 300, No. 3, 599-613 (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vertex operators; vertex operator algebras and related structures, Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds Vertex operators, Grassmannians, and Hilbert schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson-Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules over the current Lie algebra. Grassmannians, Schubert varieties, flag manifolds, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Beilinson-Drinfeld Schubert varieties and global Demazure modules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The well-known generalized flag manifolds model important geometric situations, such as all the compact Kähler manifolds or the coadjoint orbits of compact semisimple Lie groups. They are quotients \(G/P\) of a complex semisimple Lie group \(G\) by a parabolic subgroup \(P\). The paper under review is concerned with the quantization of the sub-class of (irreducible) compact Hermitian symmetric spaces, within the class of Kähler manifolds.
In particular, the quantization considered here is a subalgebra \(A\) of the quantized function algebra \(\mathbb C[G]_q\). In order to quantize the Kähler metric as well, one is naturally lead to Connes' spectral triples. In particular, one needs to formulate a Dolbeault-Dirac operator as a (non-trivial) equivariant K-homology class and study its spectrum. A spectral triple as such, and a Dolbeault-Dirac operator \(D\) were given by \textit{U. Krähmer} in [Lett. Math. Phys. 67, No. 1, 49--59 (2004; Zbl 1054.58005)], however the formula of \(D\) given there does not allow the computation of its spectrum, or a rigorous proof that it defines an appropriate K-homology class.
The authors construct an element \(D = \overline{\partial} + \overline{\partial}^{\ast}\) in \(U_q(\mathfrak{g}) \otimes Cl_q\), referred to as the Dolbeault-Dirac operator. Here \(U_q(\mathfrak{g})\) is the compact real form of the quantized universal enveloping algebra and \(Cl_q\) is a quantized Clifford algebra. Their construction is quite algebraic and uses the theory of the braided symmetric and exterior algebras of \textit{A. Berenstein} and \textit{S. Zwicknagl} [Trans. Am. Math. Soc. 360, No. 7, 3429--3472 (2008; Zbl 1220.17004); Adv. Math. 220, No. 1, 1--58 (2009; Zbl 1174.17019)]. The main result of the paper is that \(\overline{\partial}\) squares to zero, so that \(D^2 = \overline{\partial}\overline{\partial}^* + \overline{\partial}^* \overline{\partial}\). As the authors state in the introduction, this is a first effort towards a quantum version of Parthasarathy's formula, which in the classical case allows the computation of the spectral properties of the Dolbeault-Dirac operator.
We would like to point out that a thorough geometric insight to the motivation and the ideas of the authors is given in the last section of the paper. quantization; compact Hermitian symmetric spaces; Dolbeault-Dirac operator Kr''ahmer, U., and M. Tucker-Simmons, On the Dolbeault-Dirac Operator of Quantized Symmetric Spaces, Trans. of the London Math. Soc. 2 (2015), 33--56. Noncommutative geometry (à la Connes), Grassmannians, Schubert varieties, flag manifolds, Quadratic and Koszul algebras, Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Harmonic analysis on homogeneous spaces, Spin and Spin\({}^c\) geometry, Geometry of quantum groups On the Dolbeault-Dirac operator of quantized symmetric spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove bounds on intersections of algebraic varieties in \(\mathbb{C}^4\) with Cartesian products of finite sets from \(\mathbb{C}^2\), and we point out connections with several classic theorems from combinatorial geometry. Consider an algebraic variety \(X\) in \(\mathbb{C}^4\) of degree \(d\), such that the polynomials defining \(X\) are not all of the form \(F(x,y,s,t) = G(x,y)H(x,y,s,t) + K(s,t)L(x,y,s,t)\). Let \(P\) and \(Q\) be finite subsets of \(\mathbb{C}^2\) of size \(n\). If \(X\) has dimension one or two, then we prove \(|X\cap (P\times Q)| = O_d(n)\), while if \(X\) has dimension three, then \(|X\cap (P\times Q)| =O_{d,\varepsilon}(n^{4/3+\varepsilon})\) for any \(\varepsilon>0\). Both bounds are best possible in this generality (except for the \(\varepsilon\)). These bounds can be viewed as different generalizations of the Schwartz-Zippel lemma, where we replace a product of ``one-dimensional'' finite subsets of \(\mathbb{C}\) by a product of ``two-dimensional'' finite subsets of \(\mathbb{C}^2\). The bound for three-dimensional varieties generalizes the Szemerédi-Trotter theorem. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries of our two bounds, we obtain bounds on the number of repeated and distinct values of polynomials and polynomial maps of pairs of points in \(\mathbb{C}^2\), with a characterization of those maps for which no good bounds hold. These results generalize known bounds on repeated and distinct Euclidean distances. Erdős problems and related topics of discrete geometry, Algebraic combinatorics, Classification of affine varieties Schwartz-Zippel bounds for two-dimensional products | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The symplectic Grassmannian \(G_w = G_w(3,6)\) is the variety of all 3-spaces in \({\mathbb{C}}^6\) where a non-degenerate 3-form \(w\) vanishes. \(G_w\) is a smooth Fano 6-fold with \(\mathrm{Pic}(G_w) = {\mathbb{Z}}H\), and canonical class \(K_{G_w} = -4H\). The ample generator \(H\) embeds \(G_w\) in \({\mathbb{P}}^{13}\) as a smooth 6-fold of degree 16. In this paper the author studies the geometry of the general linear section \(B\) of \(G_w\) with a codimension 2 subspace \({\mathbb{P}}^{11}\) of \({\mathbb{P}}^{13}\). In Section 2 is shown that the family of lines \(F_B\) of \(B\) is a linear section of the Segre 4-fold \({\mathbb{P}}^1 \times {\mathbb{P}}^1 \times {\mathbb{P}}^1 \times {\mathbb{P}}^1\). In Section 3 is computed the Chow ring of \(B\). In Sections 1 and 4 are constructed four rank two vector bundles \(E_i\) on \(B\) such that \(E_i(1)\) give embeddings \(f_i\) of \(B\) in the Grassmannian \(G(2,6)\). One interesting property of these embeddings is that the isomorphic images of \(B\) in \(G(2,6)\) under \(f_i\), \(i = 1,\dots,4\) can be realized as congruences of lines described by \textit{E. Mezzetti} and \textit{P. De Poi} [Geom. Dedicata 131, 213--230 (2008; Zbl 1185.14042)]. Fano manifold; vector bundle; family of lines; symplectic Grassmannian Han, F, Geometry of the genus 9 Fano 4-folds, Ann. Inst. Fourier (Grenoble), 60, 1401-1434, (2010) Fano varieties, \(4\)-folds, Grassmannians, Schubert varieties, flag manifolds Geometry of the genus 9 Fano 4-folds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the cohomology ring of a homogeneous space, namely, of a partial flag variety. The variety is stratified to geometrically relevant subvarieties, the Schubert varieties. If some version of a characteristic class is assigned to those Schubert varieties, they form a basis in the cohomology ring. The structure constants of the ring with respect to this basis are important numbers (polynomials in the equivariant setting) in geometry and representation theory.
The paper under review gives a formula for such structure constants, when the characteristic class is the so-called Segre-Schwartz-MacPherson characteristic class (which is explicitly calculable from the so-called Chern-Schwartz-MacPherson charactersitic class). This characteristic class can be regarded as a 1-parameter deformation of the fundamental class, and it is known to (essentially) coincide with the Maulik-Okounkov ``stable envelope class'' for the cotangent bundle of the partial flag variety.
The structure constant formula presented in this paper is of ``localization flavor'': it is a sum (symmetrization) of rational functions whose apparent poles diappear in the summation. MacPherson classes; flag variety; Bott-Samelson variety Grassmannians, Schubert varieties, flag manifolds, Homology and cohomology of homogeneous spaces of Lie groups Structure constants for Chern classes of 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 \(G\) be a connected reductive algebraic group over an algebraically closed field and \(B\) a Borel subgroup of it. A variety \(X\) equipped with an action of \(G\) is called spherical, if \(B\) has an open orbit in \(X\). A well-known result of E.~Vinberg and M.~Brion asserts that in this case \(B\) has finitely many orbits in \(X\), so that it is possible to regard the set of \(B\)--orbits as a finite poset. The author studies the specific example, where \(X= \text{GL}_n/B_n\) and \(G= \text{GL}_{n-1}\). Here \(B_n\) is the subgroup of the upper-triangular matrices in \(\text{GL}_n\) and \( \text{GL}_{n-1}\) is the natural subgroup of \(\text{GL}_n\). It is well known (and easy to prove) that \(X\) is spherical as \(\text{GL}_{n-1}\) variety [see e.g. \textit{D. Akhiezer} and {D. Panyushev}, Mosc. Math. J. 2, 17--33 (2002; Zbl 1052.14050)], so that it consists of finitely many \(B_{n-1}\)--orbits. The author gives a combinatorial description of the closure relation for \(B_{n-1}\)--orbits in \(X\). He also considers another natural (``weak'') order on the set of \(B_{n-1}\)--orbits and shows that there is a minimal \(B_{n-1}\)--orbit with respect to the weak order that is not closed. Bruhat-Chevalley order; spherical subgroup; weak order Hashimoto, T, \(B_{n-1}\)-orbits on the flag variety \({{\mathrm GL}}_n/B_n\), Geom. Dedicata, 105, 13-27, (2004) Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] \(B_{n-1}\)-orbits on the flag variety \(\text{GL}_n/B_n\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a permutation \(\omega \in S _{n }\), \textit{B. Leclerc} and \textit{A. Zelevinsky} in [Olshanski, G. I. (ed.), Kirillov's seminar on representation theory. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 181(35), 85--108 (1998; Zbl 0894.14021)] introduced the concept of an \(\omega \)-chamber weakly separated collection of subsets of \(\{1,2,\dots ,n\}\) and conjectured that all inclusionwise maximal collections of this sort have the same cardinality \(\ell (\omega )+n+1\), where \(\ell (\omega )\) is the length of \(\omega \). We answer this conjecture affirmatively and present a generalization and additional results. weakly separated sets; Rhombus tiling; generalized tiling; weak Bruhat order; cluster algebras Danilov, VI; Karzanov, AV; Koshevoy, GA, On maximal weakly separated set-systems, J. Algebraic Comb., 32, 497-531, (2010) Grassmannians, Schubert varieties, flag manifolds, Cluster algebras On maximal weakly separated set-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 We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work [\textit{I. Ciocan-Fontanine}, Int. Math. Res. Not. 1995, No. 6, 263-277 (1995; Zbl 0847.14011)]. We also give a geometric proof of the quantum Monk formula. quantum cohomology; flag varieties; hyperquot schemes; degeneracy loci; quantum Monk formula Ionuţ Ciocan-Fontanine, The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2695 -- 2729. Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry, (Co)homology theory in algebraic geometry, Model quantum field theories The quantum cohomology ring 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 Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper [Compos. Math. 146, No. 5, 1339-1382 (2010; Zbl 1229.14036)] by \textit{T. J. Haines, R. E. Kottwitz, D. C. Reuman}, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic \(\sigma\)-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic \(\sigma\)-conjugacy class is the class of \(b=1\), we also prove the dimension formula predicted in [op. cit.] in almost all cases. affine Deligne-Lusztig varieties; Weyl groups; conjugacy classes; minimal length elements; affine root systems Görtz, U.; He, X., Dimensions of Deligne-Lusztig varieties in affine flag varieties, Doc. Math., 15, 1009-1028, (2010) Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over finite fields, Reflection and Coxeter groups (group-theoretic aspects), Group actions on varieties or schemes (quotients), Linear algebraic groups over local fields and their integers Dimensions of affine Deligne-Lusztig varieties in 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 We study certain pairs of subspaces \(V\) and \(W\) of \(\mathbb{C}^n\) we call supports that consist of eigenspaces of the eigenvalues \(\pm \| M \|\) of a minimal hermitian matrix \(M (\| M \| \leq \| M + D \|\) for all real diagonals \(D)\).\par For any pair of orthogonal subspaces we define a non negative invariant \(\delta\) called the adequacy to measure how close they are to form a support and to detect one. This function \(\delta\) is the minimum of another map \(F\) defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of \(F\) in order to approximate \(\delta\). These results allow us to prove that the set of supports has interior points in the space of flag manifolds. minimal Hermitian matrix; diagonal matrices; flag manifolds; geometry Hermitian, skew-Hermitian, and related matrices, Vector spaces, linear dependence, rank, lineability, Conditioning of matrices, Grassmannians, Schubert varieties, flag manifolds Supports for minimal Hermitian 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 This mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective.
We prove a formula for Whittaker functions of a real semisimple group as an integral over a geometric crystal in the sense of Berenstein-Kazhdan. We explain the connections of this formula to the program of mirror symmetry of flag varieties developed by Givental and Rietsch; in particular, the integral formula proves the equivariant version of Rietsch's mirror symmetry conjecture.
We also explain the idea that Whittaker functions should be thought of as geometric analogues of irreducible characters of finite-dimensional representations. Lam, Th.: Whittaker functions, geometric crystals, and quantum Schubert calculus (2013). arXiv:1308.5451 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Whittaker functions, geometric crystals, and quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We review recent results on the relation between classical solutions of nonlinear \(\sigma\)-models and Toda equations, and point out some relations among Toda equations (two dimensions), the continuum Toda equation (three dimensions), and self-dual Einstein equations (four dimensions). nonlinear \(\sigma\)-models; Toda equations; self-dual Einstein equations DOI: 10.1007/BF00750679 PDEs in connection with relativity and gravitational theory, Einstein's equations (general structure, canonical formalism, Cauchy problems), 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.), Grassmannians, Schubert varieties, flag manifolds, Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory Nonlinear Grassmann \(\sigma\)-models, Toda equations, and self-dual Einstein equations: Supplements to previous papers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 an algebraic group of type \(D_4\) over a field \(F\) of characteristic not equal to \(2\). The paper under review describes the \(F\)-points of twisted flag varieties associated with \(G\). The author's starting point is the fact that these varieties depend only on the \(F\)-isogeny class of \(G\), whence one may consider only the special case in which \(G\) is simply connected. It is known that when this occurs one can associate with \(G\) a triality \(T\) [see \textit{M.-A. Knus, A. Merkurjev, M. Rost} and \textit{J.-P. Tignol}, The book of involutions. Colloq. Publ. 44, AMS, Providence, RI (1998; Zbl 0955.16001)]. The notion of a triality is too technical to be presented here (the corresponding definition can be found in the book referred to). We only note that a triality \(T\) is a \(4\)-tuple \((E,L,\sigma,\alpha)\) satisfying certain conditions, where \(L\) is a cubic étale \(F\)-algebra, \(E\) is a central simple \(L\)-algebra, \(\sigma\) is an orthogonal involution on \(E\), and \(\alpha\) is an isomorphism of the even Clifford algebra \((C_0(E,\sigma),\overline\sigma)\) on the algebra \(^\rho((E,\sigma)\otimes_F\Delta(L))\), \(\overline\sigma\) being the involution of \(C_0(E,\sigma)\) canonically induced by \(\sigma\), \(\Delta(L)\) the discriminant \(F\)-algebra of \(L\), and \(\rho\) a generator of the Galois group of \(L\otimes_F\Delta(L)\) over \(\Delta(L)\). This means that \(L\) is presentable as a direct sum \(\bigoplus_{i=1}^n L_i\) of separable field extensions of \(F\) of degree \(3\), \(E\) is isomorphic to a direct sum \(\bigoplus_{i=1}^n E_i\) of central simple algebras \(E_i\) of degree \(8\) over the field \(L_i\), \(\sigma\) is an involution of \(E\) inducing on \(E_i\) an orthogonal involution \(\sigma_i\), for each index \(i\), and \((C_0(E,\sigma),\overline\sigma):=\bigoplus_{i=1}^n(C_0(E_i,\sigma_i),\overline\sigma_i)\), where \(\overline\sigma_i\) is the involution of the Clifford algebra \((C_0(E_i,\sigma_i),\overline\sigma_i)\) canonically induced by \(\sigma_i\), and \(\overline\sigma\) is a prolongation of each \(\overline\sigma_i\).
The author gives a method of constructing an isotropic right ideal of \((C_0(E,\sigma),\overline\sigma)\) from an arbitrary isotropic right ideal of \((E,\sigma)\) and describes the \(F\)-points of \(G\) in terms involving this correspondence and the triality \(T\). flag varieties; algebraic groups of type \(D_4\); triality; central simple algebras; orthogonal involutions; Clifford algebras Garibaldi, R. S.: Twisted flag varieties of trialitarian groups. Commun algebra 27, No. 2, 841-856 (1999) Finite-dimensional division rings, Linear algebraic groups over arbitrary fields, Galois cohomology of linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Rings with involution; Lie, Jordan and other nonassociative structures, Clifford algebras, spinors Twisted flag varieties of trialitarian groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper is devoted to the study of an intrinsic distribution, called polar, on the space of \(l\)-dimensional integral elements of the higher order contact structure on jet spaces. The main result establishes that this exterior differential system is the prolongation of a natural system of PDEs, named pasting conditions, on sections of the bundle of partial jet extensions. Informally, a partial jet extension is a \(k\)th order jet with additional \((k+1)\)st order information along \(l\) of the \(n\) possible directions. A choice of partial extensions of a jet into all possible \(l\)-directions satisfies the pasting conditions if the extensions coincide along pairwise intersecting \(l\)-directions.
It is further shown that prolonging the polar distribution once more yields the space of \((l,n)\)-dimensional integral flags with its double fibration distribution. When \(l > 1\) the exterior differential system is holonomic, stabilizing after one further prolongation. jet spaces; exterior differential systems; geometry of PDEs Differentiable manifolds, foundations, Exterior differential systems (Cartan theory), Pfaffian systems, Jets in global analysis, Vector distributions (subbundles of the tangent bundles), Natural bundles, Calculus on manifolds; nonlinear operators, Partial differential equations on manifolds; differential operators, Grassmannians, Schubert varieties, flag manifolds, General topics in partial differential equations, Geometric theory, characteristics, transformations in context of PDEs, Differential topology, Local submanifolds Partial extensions of jets and the polar distribution on Grassmannians of non-maximal integral 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 A stratification of the manifold of all square matrices is considered. One equivalence class consists of the matrices with the same sets of values of \(\operatorname{rank}(A - \lambda_i I)^j\). The stratification is consistent with a fibration on submanifolds of matrices similar to each other, i.e., with the adjoint orbits fibration. Internal structures of matrices from one equivalence class are very similar; among other factors, their (co)adjoint orbits are birationally symplectomorphic. The Young tableaux technique developed in the paper describes this stratification and the fibration of the strata on (co)adjoint orbits. stratification of the manifold of all square matrices Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Young tableaux and stratification of the space of square complex 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 Using the canonical isomorphism between the exterior power of solutions to generic linear Ordinary Differential Equations of order \(r<\infty\) and the polynomial ring with \(r\) indeterminates, we define and compute certain vertex operators whose expression for \(r=\infty\) is precisely that occurring in the classical treatment of the Boson-Fermion correspondence. linear ODEs; Boson-Fermion correspondence; vertex operators Grassmannians, Schubert varieties, flag manifolds, Vertex operators; vertex operator algebras and related structures, Ordinary differential equations of infinite order From solutions to linear ordinary differential equations to vertex 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 Let X be a smooth complex projective variety admitting the following properties:
(1) There exists an algebraic vector field V on X with precisely one zero \(s\in X\), and
(2) there is an algebraic \({\mathbb{C}}^*\)-action \(\lambda: {\mathbb{C}}^*X\to X\) and a positive integer p such that the induced tangent action \(d\lambda\) satisfies the relation \(d\lambda(t)\cdot V=t^ p\cdot V\) for any \(t\in {\mathbb{C}}^*.\)
Examples of such varieties are the projective spaces \({\mathbb{P}}^ n\), the flag varieties G/B defined by a complex semi-simple Lie group G and a Borel subgroup B, the Bott-Samuelson desingularizations of Schubert varieties, and certain Fano threefolds.
In the present paper, the authors prove a product formula for the Poincaré polynomial of varieties satisfying (1) and (2). This formula reads \(P(X,t^{p/2})= \prod^{n}_{i=1}(1-t^{-a_ i+p})/(1-t^{- a_ i}) \); where P denotes the Poincaré polynomial defined by the Betti numbers of X, and \(a_ 1,...,a_ n\) are the weights of \(\lambda\) at the zero s of V, \(n=\dim_{{\mathbb{C}}}X.\)
This explicit product formula for the Poincaré polynomial is then shown to have some amazing consequences: First, in the special case of X being a flag variety G/B of a semi-simple Lie group G, the product formula for the Poincaré polynomial turns out to coincide with the famous Kostant- Macdonald product identity for Lie algebras of maximal tori in B [cf. \textit{B. Kostant}, Am. J. Math. 81, 973-1033 (1959; Zbl 0099.256)] and \textit{I. G. Macdonald}, Math. Ann. 199, 161-174 (1972; Zbl 0286.20062)]. - Secondly, it is deduced that if \(\lambda\) and V arise from an algebraic \(SL_ 2({\mathbb{C}})\)-action on X, where X satisfies (1) and (2), then the second Betti number of X is just the multiplicity of the lagest weight of the induced linear \({\mathbb{C}}^*\)-action on the tangent space of X at the zero \(s\in X\). That means, in particular, \(b_ 2(X)\leq \dim_{{\mathbb{C}}}X=n.\)
Besides these general consequences, the authors discuss, at the end of the paper, several concrete examples. product formula for the Poincaré polynomial; Kostant-Macdonald product; second Betti number Ersan Akyıldız and James B. Carrell, \(A generalization of the Kostant-Macdonald identity. \)Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 11, 3934--3937. Group actions on varieties or schemes (quotients), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) A generalization of the Kostant-Macdonald identity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An involutory birational transformation J: \(p\to p'\) in \(P_ 4\) is studied in this work, determined as follows: a random straight line \(p\in P_ 4\) and the straight line u define a hyperplane intersecting \(V^ 3_ 2\) in straight lines u, \(p_ 1, p_ 2\). In the general case, lines \(p_ 1\) and \(p_ 2\) are crossed and define J. The points of \(p'\) are harmonical conjugates of the points of p at J. The images of various straight lines in \(P_ 4\) are defined. An interpretation is made of J upon the Grassmann variety \(V^ 5_ 6\) in \(P_ 9\). involutory birational transformation Projective techniques in algebraic geometry, Projective analytic geometry, Grassmannians, Schubert varieties, flag manifolds, Rational and birational maps Linear inversion relative to a cubic norm-surface \(V^ 3_ 2\) in \(P_ 4\) | 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 classifies those smooth surfaces in \({\mathbb{P}}^ 5\) of degree less than or equal to 8 that are contained in the Grassmann variety of lines in \({\mathbb{P}}^ 3\). classification of smooth surface in projective 5-space; embedding in Grassmann variety; Chern class; canonical divisor; hyperplane section [P] Papantonopoulou, A.: Embeddings in G(1,3). Proc. Am. Math. Soc.89, 583-586 (1983); Corrigendum, Ibidem, Proc. Am. Math. Soc.95 (1985) Grassmannians, Schubert varieties, flag manifolds, Families, moduli, classification: algebraic theory, Embeddings in algebraic geometry Embeddings in G(1,3) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The straightening law of Doubilet-Rota-Stein tells that the standard bitableaux bounded by a pair \(m=(m(1),m(2))\) give a vector space basis of the polynomial algebra in \(m(1)m(2)\) variables. In an enumerative proof of the straightening law Abhyankar enumerated the set \(\text{stab}(2,m,p,a,V)\) of certain standard bitableaux. The Abhyankar formula gives also the Hilbert polynomial of a class of determinantal ideals \(I(p,a)\).
In the paper under review, the author outlines an alternate proof of the Abhyankar formula for the cardinality of \(\text{stab}(2,m,p,a,V)\) using a recent result on nonintersecting lattice paths obtained independently by Modak, Kulkarni and Krattenthaler. The lattice path approach leads also to some other known results on the numerators of the Hilbert-Poincaré series of \(I(p,a)\), the so-called \(h\)-vector of the associated simplicial complex, and gives better bounds for the degree of the numerator of the Hilbert series of \(I(p,a)\) (and in some cases the exact value of the degree). The author also discusses some related problems concerning possible generalizations to higher dimensions. He indicates as well connections between the Hilbert function for the Schubert varieties in Grassmannians and the Abhyankar formula. Stanley-Reisner ring; straightening law; standard bitableaux; Abhyankar formula; Hilbert polynomial; determinantal ideals; nonintersecting lattice paths; Hilbert series; Hilbert function; Schubert varieties DOI: 10.1016/0378-3758(95)00156-5 Combinatorial aspects of representation theory, Determinantal varieties, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Factorials, binomial coefficients, combinatorial functions, Exact enumeration problems, generating functions, Linkage, complete intersections and determinantal ideals, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Grassmannians, Schubert varieties, flag manifolds Young bitableaux, lattice paths and Hilbert functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the structure algebra \(\mathcal{Z}\) of the stable moment graph for the case of the affine root system \(A_1\). The structure algebra \(\mathcal{Z}\) is an algebra over a symmetric algebra and in particular, it is a module over a symmetric algebra. We study this module structure on \(\mathcal{Z}\) and we construct a basis. By ``setting \(c\) equal to zero'' in \(\mathcal{Z}\), we obtain the module \(\mathcal{Z}_{c = 0}\). This module can be described in terms of the finite root system \(A_1\) and we show that it is determined by a set of certain divisibility relations. These relations can be regarded as a generalization of ordinary moment graph relations that define sections of sheaves on moment graphs, and because of this we call them higher-order congruence relations. sheaves on moment graphs; affine Weyl group; alcoves Root systems, Grassmannians, Schubert varieties, flag manifolds Higher-order congruence relations on affine moment graphs: the subgeneric case | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{A. Skovsted Buch} and \textit{W. Fulton} [Invent. Math. 135, No. 3, 665--687 (1999; Zbl 0942.14027)] gave a formula for a general kind of degeneracy locus associated to an oriented Type A quiver. This formula involves Schur determinants and the quiver coefficients which generalize the classical Littlewood-Richardson coefficients. In the paper under review, the authors give a positive combinatorial formula for the quiver coefficients when the rank conditions defining the degeneracy locus are given by a permutation. As applications, one obtains new expansions for Fulton's universal Schubert polynomials, Schubert polynomials of Lascoux-Schützenberger, and explicit Giambelli formulas in the classical and quantum cohomology ring of any partial flag variety. quiver coefficients \beginbarticle \bauthor\binitsA. S. \bsnmBuch, \bauthor\binitsA. \bsnmKresch, \bauthor\binitsH. \bsnmTamvakis and \bauthor\binitsA. \bsnmYong, \batitleSchubert polynomials and quiver formulas, \bjtitleDuke Math. J. \bvolume122 (\byear2004), no. \bissue1, page 125-\blpage143. \endbarticle \OrigBibText Anders S. Buch, Andrew Kresch, Harry Tamvakis, and Alexander Yong, Schubert polynomials and quiver formulas , Duke Math. J. 122 (2004), no. 1, 125-143. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Schubert polynomials and quiver formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual to the Schubert VHS form a basis of the invariant characteristic cohomology associated with the infinitesimal period relation (a.k.a. Griffiths' transversality). Schubert variety; variation of Hodge structure; infinitesimal period relation; Griffiths' transversality; Hodge theory; Mumford-Tate group Robles, C., \textit{Schubert varieties as variations of Hodge structure}, Selecta Math. (N.S.), 20, 719-768, (2014) Period matrices, variation of Hodge structure; degenerations, Grassmannians, Schubert varieties, flag manifolds, Variation of Hodge structures (algebro-geometric aspects) Schubert varieties as variations of Hodge structure | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the adjacency diagrams and the local singularities of the stratification by the orbits of the group of symplectic linear transformations \(Sp(2n,\mathbb{R})\), which naturally acts on the flag manifolds \(F(2n;n_1, \dots, n_k)=\{V^{n_1} \subset\cdots \subset V^{n_k} \subset\mathbb{R}^{2n}\}\). A similar problem was considered in [\textit{M. Eh. Kazaryan}, Russ. Math. Surv. 46, No. 5, 91-136 (1991); translation from Usp. Mat. Nauk 46, No. 5(281), 79-119 (1991; Zbl 0783.32015)] in connection with the investigation of the adjacency of the Schubert cells on the manifold of complete flags. adjacency diagrams; local singularities; flag manifolds Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Grassmannians, Schubert varieties, flag manifolds, Topology of real algebraic varieties Stratifications by the orbits of symplectic groups on 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 We consider the action of a smooth, connected group scheme \(G\) on a scheme \(Y\), and discuss the problem of when the saturation map \(\Theta \colon G\times X\rightarrow Y\) is separable, where \(X\subset Y\) is an irreducible subscheme. We provide sufficient conditions for this in terms of the induced map on the fibres of the conormal bundles to the orbits. Using jet space calculations, one then obtains a criterion for when the scheme-theoretic image of \(\Theta\) is an irreducible component of \(Y\). We apply this result to Grassmannians of submodules and several other schemes arising from representations of algebras, thus obtaining a decomposition theorem for their irreducible components in the spirit of the result by \textit{W. Crawley-Boevey} and \textit{J. Schröer} for module varieties [J. Reine Angew. Math. 553, 201--220 (2002; Zbl 1062.16019)]. Grassmannians; decomposition theorem; module varieties Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds Irreducible components of quiver Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials By means of a combinatorial optimization approach, we reduce to a procedure of polynomial complexity the problem of computing the dimension of the projective algebraic variety of flags fixed by a nilpotent endomorphism. We also recover a necessary and sufficient condition to decide when this dimension attains a well-known upper bound. This last result is related to the celebrated Gale-Ryser theorem on the existence of \((0,1)\)-matrices. Díaz-Leal, H.; Martínez-Bernal, J.; Romero, D.: Dimension of the fixed point set of a nilpotent endomorphism on the flag variety. Bol. soc. Mat. mexicana (3) 7, 23-33 (2001) Computational aspects of higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory Dimension of the fixed point set of nilpotent endomorphism on the 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 For a complex semisimple algebraic group with parabolic subgroup \(P\) of \(G\), let \(m=\dim(H^2(G/P))\). For each \(t\in\mathbb C^m\), Belkale and Kumar defined a product \(\odot_t\). This product degenerates the usual cup product on \(H^\ast(G/P)\), and it gives applications to the eigenvalue problem and to the problem of finding \(G\)-invariants of the tensor products of representations. Previously, the authors have given a new construction of this product, and they have proved that \((H^\ast(G/P,\mathbb C),\odot_t)\) is isomorphic to a relative Lie algebra cohomology ring \(H^\ast(\mathfrak g_t,\mathfrak l_\Delta)\), where \(\mathfrak l_\Delta\) are subalgebras of \(\mathfrak g\times\mathfrak g\). The present article focuses on \((H^\ast(G/P),\odot_t)\). Let \(B\subset P\) be a Borel subgroup, let \(H\subset B\) be a Cartan subgroup, and let \(\alpha_1,\dots,\alpha_m\) be the simple roots with respect to \(B\). Let \(L\) be the Levi factor of \(P\) containing \(H\), let \(\mathfrak l\) be the Lie algebra of \(L\), and let the simple roots of \(\mathfrak l\) be \(I=\{\alpha_{m+1},\dots,\alpha_n\}\) where \(\alpha_1,\dots,\alpha_m\) are roots of the nilradical \(\mathfrak u\) of \(P\). For \(t=(t_1,\dots,t_m)\in\mathbb C^m\), \(J(t)=\{1\leq q\leq m:t_q\neq 0\}\), \(K=J(t)\cup I\). \(\mathfrak l_K\) denotes the Levi subalgebra generated by the Lie algebra \(\mathfrak h\) of \(H\) and the root spaces \(\mathfrak g_{\pm\alpha_i}\), \(i\in K\), \(L_K\) denotes the corresponding subgroup, and \(P_K=BL_K\) denotes the corresponding standard parabolic.
The main result is perfectly and precisely given as follows (direct quote): {Theorem 1.1.} For parabolic subgroups \(P\subset P_K\) of \(G\), with \(P_K\) determined by \(t\in\mathbb C^m\) as above,
(1) \(H^\ast(P_k/P)\) is isomorphic to a graded subalgebra \(A\) of \((H^\ast(G/P),\odot_t)\).
(2) The ring \((H^\ast(G/P_K),\odot_0)\cong (H^\ast(G/P),\odot_t)/I_+\), where \(I_+\) is the ideal of \((H^\ast(G/P),\odot_t)\) generated by positive degree elements of \(A\). \vskip0,2cm So \((H^\ast(G/P),\odot_t)\) has a classical part which is the usual cohomology ring, with the associated quotient given by the degenerate Belkale-Kumar product. The theorem is proved by applying the Hochschild-Serre spectral sequence in relative Lie algebra cohomology, that is, by using the relative Lie algebra cohomology description of the product. There is a need to show that the spectral sequence degenerates at the \(E_2\)-term, to compute the edge morphisms, and to determine the product structure on the \(E_2\)-term. Thus the main content of this article is to fulfill this need, that is carrying out these computations using the original approach of Hochschild and Serre. This approach is generalized to the the relative setting, with a main new point: A Lie algebra \(\mathfrak g\) is given, together with an ideal \(I\) and a subalgebra \(\mathfrak k\) which is reductive in \(\mathfrak g\). To construct the spectral sequence, there is a need for an action of \(\mathfrak g/I\) on the relative cohomology group \(H^\ast((I,I\cap\mathfrak k,M)\), \(M\) a \(\mathfrak g\)-module. If \(I\cap\mathfrak k\) is nonzero, the Lie algebra \(\mathfrak g/I\) acts in a special way on the space of cochains \(C^\ast(I,I\cap\mathfrak k)\). The authors construct a an action on the cohomology group by a formula involving cochains. Then they verify that this yields the \(d_1\) differential in the spectral sequence . The confirmation of the last fact is the main technical complication of the article.
The authors give a nice introduction to Lie algebra cohomology, and they generalize this to the relative setting. This involves explicit formulas for the differentials, needed in the relative computations. Then they give a rather complete module on spectral sequences, including the computation of general edge morphisms. Finally they apply the spectral sequences on the relative Hochscild Serre cohomology. This means that they give explicit conditions of convergence and descriptions of the edge morphisms. Finally, the article apply the Hochschild-Serre spectral sequence to prove the main theorem, and this proves that the cohomology of a generalized flag variety, equipped with the Belkale-Kumar product, has a structure analogous to the cohomology of a fibre bundle.
The article also includes some interesting theorems on the structure of cohomology rings in the relative case. The article is detailed and precise, and in addition to giving nice results, it is also a nice introduction to the techniques of Lie algebra cohomology rings. Belkale-Kumar product; relative Lie cohomology; Hochschild-Serre spectral sequence; Lie cohomology ring Evens, S; Graham, W, The relative Hochschild-Serre spectral sequence and the belkale-kumar product, Trans. Amer. Math. Soc., 365, 5833-5857, (2013) Cohomology of Lie (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups The relative Hochschild-Serre spectral sequence and the Belkale-Kumar product | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(X_w\) be a Schubert subvariety of a cominuscule Grassmannian \(X\), and let \(\mu :T^*X\rightarrow\mathcal{N}\) be the Springer map from the cotangent bundle of \(X\) to the nilpotent cone \(\mathcal{N}\). In this paper, we construct a resolution of singularities for the conormal variety \(T^*_XX_w\) of \(X_w\) in \(X\). Further, for \(X\) the usual or symplectic Grassmannian, we compute a system of equations defining \(T^*_XX_w\) as a subvariety of the cotangent bundle \(T^*X\) set-theoretically. This also yields a system of defining equations for the corresponding orbital varieties \(\mu (T^*_XX_w)\). Inspired by the system of defining equations, we conjecture a type-independent equality, namely \(T^*_XX_w=\pi^{-1}(X_w)\cap\mu^{-1}(\mu (T^*_XX_w))\). The set-theoretic version of this conjecture follows from this work and previous work for any cominuscule Grassmannian of type A, B, or C.
For Part I, see [the author and \textit{V. Lakshmibai}, ``Conormal varieties on the cominuscule Grassmannian'', Preprint, \url{arXiv:1712.06737}]. Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Coadjoint orbits; nilpotent varieties Conormal varieties on the cominuscule Grassmannian. 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 A surface in the Grassmannian of lines of the complex projective space \(\mathbb{P}^3\) is classically called a congruence (of lines). The paper under review is devoted to the study of the family \(B(Y)\) of lines which are tangent to a hypersurface \(Y\) in \(\mathbb{P}^3\) of degree \(d\) at two points. For \(d=4\) it is known that \(B(Y)\) is a smooth irreducible surface, provided that \(Y\) is smooth and contains no line. For \(d \geq 5\), if \(Y\) has at worst isolated singularities and its dual variety is a hypersurface, then \(B(Y)\) is in fact a congruence, named the Fano congruence of bitangents to \(Y\). However, it is not smooth, even when \(Y\) is general. The authors consider its normalization \(S(Y)\) and show that, for \(Y\) general and \(d \geq 5\), \(S(Y)\) is a smooth irreducible projective surface. Moreover, they prove that for an arbitrary \(Y\), \(S(Y)\) is smooth at any point representing a line \(\ell \subset\mathbb{P}^3\) such that \(\ell \not\subset Y\), \(Y\) is smooth at the two tangency points and they are disjoint from the further intersections. Furthermore, the authors describe the singularities of the Fano congruence of bitangents occurring in a general Lefschetz pencil in the space of hypersurfaces of degree \(d\). Fano congruence; bitangents; Lefschetz pencil Special surfaces, Pencils, nets, webs in algebraic geometry, Fibrations, degenerations in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Normalization of the congruence of bitangents to a hypersurface in \(\mathbb{P}^3\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simple algebraic group over \(\mathbb{C}\). Fix a maximal torus \(T \subset G\). Let \(\Delta =\{ \alpha_1,\dots, \alpha_n \}\) be the system of simple roots of \(G\) and \(P=P_k\) denote the maximal parabolic subgroup associated to a simple root \(\alpha_k\).
It is known that the category of \(G\)-equivariant vector bundles on \(G/P\) is equivalent to the category of finite-dimensional representations of \(P\).
Let \(X \subset \mathbb{P}^N\) be a smooth projective variety of dimension \(d\) over \(\mathbb{C}\). A vector bundle \(E\) on \(X\) is called Ulrich if the cohomology groups \(H^i(X, E(-t)) = 0\) for all \(0 \leq i \leq d\) and \(1\leq t \leq d\). The twisted bundle \(E\otimes O_X(-t)\) is denoted by \(E(-t)\).
For an integral weight \(\omega\) dominant with respect to \(P\), one has an irreducible representation \(V(\omega)\) of \(P\) with highest weight \(\omega\), and let us denote by \(E_{\omega}\) the corresponding irreducible equivariant vector bundle \(G \times_{P} V(\omega)^{*} \) on \( G/P:E_{\omega}:=G\times_{P}V(\omega)^{*}= (G \times V(\omega)^{*})/P\), where the equivalence relation is given by \((g, v) \sim (gp, p^{-1}. v)\) for \(p \in P\).
The cohomology of such bundles is computed by the famous the Borel-Bott-Weil theorem, see [\textit{J. M. Weyman}, Cohomology of vector bundles and syzygies. Cambridge: Cambridge University Press (2003; Zbl 1075.13007)] and [\textit{D. M. Snow}, CMS Conf. Proc. 10, 193--205 (1989; Zbl 0701.14017)].
If \(L\) be a Levi factor of a maximal parabolic subgroup \(P_k \subset G\), then for an \(L\)-dominant integral weight \(\omega \in \Lambda_{+}\) , define a set \(\mathrm{Sing}(\omega) :=\{t \in Z:\omega + \rho -t\omega_k\) is singular\}, where \(\rho\) is the sum of fundamental weights of \(G\).
The main tool used in this paper is Fonarev's criterion (Lemma 2.4 of [\textit{A. Fonarev}, Mosc. Math. J. 16, No. 4, 711--726 (2016; Zbl 1386.14180)]) which says that for \( \omega \in \Lambda^{+}_{L} \) an irreducible equivariant vector bundle \(E_{\omega}\) on a rational homogeneous variety \(G/P_k\) with Picard number 1 and dimension \(d\) is Ulrich if and only if \(\mathrm{Sing}(\omega) =\{1,2,\dots, d-1, d\}\).
Further, using this criterion the authors check whether the rational homogeneous varieties admit Ulrich bundles or not.
To check it properly they use the following helpful arguments: Assume that \(G/P_k\) is a Hermitian symmetric space of compact type. If an irreducible equivariant vector bundle \(E_{\omega}\) on \(G/P_k\) is Ulrich, then the maximum of \(\mathrm{Sing}(\omega)\) is a singular value attained at the highest root \(\theta\) of the Lie algebra \(g\).
and
Let \(G/P_k\) be an adjoint variety such that \(\mathrm{rank}(G)\geq 2\) and \(G\) is not of type A. If an irreducible equivariant vector bundle \(E_{\omega}\) on \(G/P_k\) is Ulrich, then the maximum of \(\mathrm{Sing}(\omega)\) is a singular value attained at the root \(\theta - \alpha_{k}\), where \(\theta\) is the highest root of \(g\).
More concretely, for \(G_2\)-homogeneous varieties:
\begin{itemize}
\item \(G_2/P_1\) does not admit Ulrich bundles;
\item \(G_2/P_2\) does not admit Ulrich bundles.
\end{itemize}
\(E_6\)-homogeneous varieties:
\begin{itemize}
\item Cayley plane \(E_6/P_2\subset \mathbb{P}(V_{E_6})(\omega_2)=\mathbb{P}^{77}\) has dimension 21 and Fano index 11, does admit the only one irreducible Ulrich bundle \(E_{\omega_5+3\omega_6}\);
\item \(E_6/P_3\subset \mathbb{P}^{350}\) does not admit Ulrich bundles;
\item \(E_6/P_4 \subset \mathbb{P}^{2924}\) has dimension 29 and Fano index 7 and does not admit Ulrich bundles.
\end{itemize}
\(F_4\)-homogeneous varieties:
\begin{itemize}
\item \(F_4/P_4\) is a general hyperplane section of a Cayley plane \(E_6/P_1 \subset \mathbb{P}^{26}\) does not admit Ulrich bundles; if one will restrict an Ulrich bundle from the Cayley plane to \(F_4/P_4\) one can get an equivariant Ulrich bundle on \(F_4/P_4\);
\item \(F_4/P_3 \subset \mathbb{P}^{272}\) is the closed \(F_4\)-orbit in the space of lines on the rational homogeneous variety \(F_4/P_4 \subset \mathbb{P}^{25}\) has dimension 20 and Fano index 7. It does not admit Ulrich bundles;
\item \(F_4/P_2 \subset \mathbb{P}^{1273}\) has dimension \(20\) and Fano index \(7\) and does not admit Ulrich bundles;
\item \(F_4/P_1 \subset \mathbb{P}^{51}\) has dimension \(15\) and Fano index \(8\) and does not admit Ulrich bundles;
\end{itemize}
\(E_7\)-homogeneous varieties:
\begin{itemize}
\item \(E_7/P_1 \subset \mathbb{P}^{132}\) has dimension \(33\) and Fano index \(17\) and does admit the only one irreducible equivariant Ulrich bundle \(E_{\omega_5+3\omega_6+8\omega_7}\);
\item The odd symplectic grassmanians of planes \(Gr_{\omega}(2, 2n+1)\) admit Ulrich bundles (see [ Zbl 1386.14180]);
\item \(E_7/P_3 \subset \mathbb{P}^{8644}\) has dimension \(47\) and Fano index \(11\) and does not admit Ulrich bundles;
\item \(E_7/P_4 \subset \mathbb{P}^{365749}\) has dimension \(53\) and Fano index \(8\) and does not admit Ulrich bundles;
\item \(E_7/P_5 \subset \mathbb{P}^{27663}\) has dimension \(50\) and Fano index \(10\) and does not admit Ulrich bundles;
\item \(E_7/P_6 \subset \mathbb{P}^{1538}\) has dimension \(42\) and Fano index \(13\) and does not admit Ulrich bundles;
\item \(E_7/P_7 \subset \mathbb{P}^{55}\) has dimension \(27\) and Fano index \(18\) (Freundenthal variety, one of two exceptional Hermitian symmetric spaces of compact type) and does not admit Ulrich bundles;
\end{itemize}
\(E_8\)-homogeneous varieties: none of them admit equivariant Ulrich bundles. equivariant Ulrich bundles; exceptional homogeneous varieties; Borel-Weil-Bott theorem; Cayley plane; Dynkin diagram Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Grassmannians, Schubert varieties, flag manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results Equivariant Ulrich bundles on exceptional homogeneous varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The article under review continues the study of the algebras \(B_{Q}\) from previous work of the authors [\textit{G. Cerulli Irelli} et al., Adv. Math. 245, 182--207 (2013; Zbl 1336.16015)], in which the algebra is used to construct desingularizations for quiver Grassmannians.
Given a quiver \(Q\), it is constructed the Gabriel quiver \(\hat{Q}\) whose vertices are the original vertices of \(Q\) plus the non-projective indecomposable modules over \(kQ\). The algebra \(B_{Q}\) is defined as \(k\hat{Q}/I\) where \(I\) is certain ideal.
On the other hand, \textit{D. Hernandez} and \textit{B. Leclerc} [``Quantum Grothendieck rings and derived Hall algebras'', \url{arXiv:1109.0862}] defined an algebra \(\tilde{\Lambda}_{Q}\) related to graded Nakajima quiver varieties which are isomorphic to representation varieties \(R_{d}(Q)\) of the quiver \(Q\). This paper shows that the algebras \(B_{Q}\) and \(\tilde{\Lambda}_{Q}\) isomorphic which allows to explicitly describe the representation varieties as affine quotients of \(B_{Q}\)-representations. quiver varieties; Hernandez-Leclerc construction; quiver Grassmannians; Gabriel quiver; Nakajima varieties Cerulli Irelli, G.; Feigin, E.; Reineke, M., Homological approach to the Hernandez-Leclerc construction and quiver varieties, Representation Theory, 18, 1-14, (2014) Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Categories in geometry and topology Homological approach to the Hernandez-Leclerc construction and quiver varieties | 0 |
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