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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 In this note we study, for \(n=5,6,7\), the geometry of the full flag manifolds, \(F(n)={U(n)\over U(1)\times\cdots \times U(1)}\). By using tournaments we characterize all of the \((1,2)\)-symplectic invariant metrics on \(F(n)\), for \(n=5,6,7\) corresponding to different classes of non-integrable invariant almost complex structures. flag manifolds; \((1,2)\)-symplectic metrics; harmonic maps; Hermitian geometry; tournaments Differential geometric aspects of harmonic maps, Directed graphs (digraphs), tournaments, Grassmannians, Schubert varieties, flag manifolds, Harmonic maps, etc. Some results on the geometry of full flag manifolds and harmonic maps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 places itself within the rich framework of the literature regarding quantum cohomology of homogeneous varieties produced by the authors themselves. This new piece of mathematics regards certain \textsl{Quantum Giambelli} formulas for isotropic Grassmannians. If \(V\) is a vector space equipped with a non degenerate bilinear form \(\eta\), one can consider the Grassmannian \(X:=IG(k,V)\) of isotropic subspaces of \(V\) of a fixed dimension \(k\), namely the variety of those points \([W]\in G(k,V)\) such that the restriction of \(\eta\) to \(W\) is trivial (i.e. \(\eta_{|W\times W}=0\)).
If the bilinear form is skew-symmetric, the dimension of \(V\) is even and one is then concerned with symplectic vector spaces. As well known, the cohomology ring \(H^*(X,{\mathbb{Z}})\) is generated as a \({\mathbb{Z}}\)-algebra by certain special Schubert cycles and it is also a well known fact that such cycles generate the quantum cohomology of \(X\) as well. The latter is a deformation of the usual cohomology encoding the Gromov-Witten invariants which count, roughly speaking, numbers of maps of a given degree from the projective line to \(X\). The authors find and prove quantum Giambelli's formulas expressing an arbitrary Schubert class in the small quantum cohomology ring of \(X\) as a polynomial in the special Schubert classes alluded above.
The two main theorems of the article (concerning Giambelli's formulas) are analogous to those proven for the quantum cohomology of the orthogonal and Lagrangian Grassmannians in [J. Algebr. Geom. 12, No. 4, 777--810 (2003; Zbl 1051.53070)] and [Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083)], by \textit{A. Kresch} and \textit{H. Tamvakis}. The proof are however quite different, due to the fact that for non maximal isotropic Grassmannians, the explicit recursion used in the quoted references is no longer available. The latter is replaced, however, by another kind of recursion, neatly steted and proved in Proposition 3.
The reviewed paper is for all people interested in the combinatorial aspects of cohomology theories (quantum, equivariant, quantum-equivariant) of homogeneous varieties. quantum Schubert Calculus; isotropic Grassmannians; Giambelli's Formulas Buch, AS; Kresch, A; Tamvakis, H, Quantum Giambelli formulas for isotropic Grassmannians, Math. Ann., 354, 801-812, (2012) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Quantum Giambelli formulas for 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 establish one direction of a conjecture by \textit{N. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45-52 (1990; Zbl 0714.14033)] which describes combinatorially the singular locus of a Schubert variety. We prove that the conjectured singular locus is contained in the singular locus. singular locus of a Schubert variety Gasharov, Vesselin, Sufficiency of Lakshmibai-Sandhya singularity conditions for Schubert varieties, Compos. Math., 126, 1, 47-56, (2001), MR 1827861 Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Combinatorics of partially ordered sets Sufficiency of Lakshmibai-Sandhya singularity conditions for Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper proves a characterization of the Grassmann manifold \(G=G(n,m+n)\) which generalizes in a stronger form the former result of \textit{F. Hirzebruch} and \textit{K. Kodaira} [J. Math. Pures Appl., IX Sér. 36, 201-216 (1957; Zbl 0090.386)] about \({\mathbb{P}}^ n({\mathbb{C}})\), recognized by its cohomology ring and the existence of a positive line bundle L such that: \(c_ 1(L)\) generates \(H^ 2({\mathbb{Z}})\), and \(h^ 0(L^{\nu})=\left( \begin{matrix} n+\nu \\ n\end{matrix} \right).\)
The proof by Hirzebruch cannot be directly adapted; one needs the following lemma: Let \(\pi:\quad G\to {\mathbb{P}}^ N({\mathbb{C}})\) be the Plücker embedding; let \(V\varsubsetneq G\) be an irreducible subvariety not contained in any hyperplane. Then: \(\deg (V)>\deg (G)\). characterization of the Grassmann manifold; cohomology ring; positive line bundle A. Babakhanian and H. Hironaka, On the complex Grassmann manifold,Illinois J. Math 33 (1989), 170--179. Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry, (Co)homology theory in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On the complex Grassmann manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Curves in Lagrange Grassmannians naturally appear when one studies Jacobi equations for extremals, associated with geometric structures on manifolds. We fix integers \(d_i\) and consider curves \(\Lambda (t)\) for which at each \(t\) the derivatives of order \(\leqslant i\) of all curves of vectors \(\ell (t) \in \Lambda (t)\) span a subspace of dimension \(d_i\). We describe the construction of a complete system of symplectic invariants for such parametrized curves, satisfying a certain genericity assumption, and give applications to geometric structures, including sub-Riemannian and sub-Finslerian structures. Jacobi equations; symplectic invariants; sub-Finslerian structures Émery, M.: Stochastic calculus in manifolds. Universitext. Springer, Berlin (1989). 10.1007/978-3-642-75051-9. With an appendix by P.-A. Meyer Differential geometric aspects of harmonic maps, Differential geometry of homogeneous manifolds, Grassmannians, Schubert varieties, flag manifolds, Sub-Riemannian geometry Parametrized curves in Lagrange 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 Some years ago the authors developed the notion of uniformity and splitting for principal bundles, not just for vector bundles [\textit{R. Muñoz} et al., ``On uniform flag bundles on Fano manifolds'', Preprint, \url{arXiv:1610:05930}]. Here they raise several conjectures on the splitting of low rank principal bundles and prove them for principal bundles associated to classical group. It would be nice to know it in positive characteristic, but probably this must first done in more elementary cases. homogeneous spaces; flag and principal bundles; diagonalizability; split bundle Fano varieties, Minimal model program (Mori theory, extremal rays), Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Splitting conjectures for uniform 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 Given a framework determined by a system of geometric constraints, a fundamental question is to determine whether the framework is rigid or flexible. If the framework is a bar-and-joint or body-and-bar framework, the constraints are all given by fixing the distance between selected pairs of points with bars. The theorems by [\textit{G. Laman}, J. Eng. Math. 4, 331--340 (1970; Zbl 0213.51903)], (actually discovered first by \textit{H. Pollaczek-Geiringer}, Z. Angew. Math. Mech. 7, 58--72 (1927; JFM 53.0743.02), \textit{T.-S. Tay}, Adv. Appl. Math. 23, No. 1, 14--28 (1999; Zbl 0936.05027), and \textit{N. White} and \textit{W. Whiteley}, SIAM J. Algebraic Discrete Methods 8, 1--32 (1987; Zbl 0635.51014)] characterize rigidity combinatorially in various setting, and these combinatorial characterizations only determine the behavior of sufficiently generic realzations of a framework with given combinatorics.
The pure condition of a generically rigid framework, introduced by White, Whiteley [loc. cit.] and by \textit{N. L. White} and \textit{W. Whiteley}, SIAM J. Algebraic Discrete Methods 4, 481--511 (1983; Zbl 0542.51022)], is a polynomial in brackets. The goal of this chapter is to give an introduction to the Grassmann-Cayley algebra, which can be used to give a geometric interpretation of the vanishing of bracket polynomials in order to better understand when a generically rigid framework admits nontrival internal motions.
In order to motivate the development of the Grassmann-Cayley algebra we present the simplest example of a generically rigid structure in the plane, two rigid bodies connected by three bars, which we represent by a multigraph with a vertex for each body and an edge for each bar.
Following White, Whiteley [loc. cit.], we embed the framework in the projective plane \(\mathbb{P}^2\) and label each bar with vector corresponding to the line in the direction of the bar. This vector is obtained by taking the cross product of the vectors corresponding to the endpoints of the bars. Equivalently, the bars are labele by their Plücker coordinates in a projective plane that is dual to the plane in which the framework embedded. The resulting body-and-bar framework is generically infinitesimally rigid and has a nontrivial infinitesimal motion exactly when the three bars are parallel or meet at a point.
In \(\mathbb{P}^2\) lines are parallel exactly when they meet at a point on the line at infinity, hence the framework has an infinitesimal motion exactly when the lines along the bars are coincident. Three lines in \(\mathbb{P}^2\) are coincident when their Plücker coordinates in the dual projective plane are collinear. This is precisely the condition that the \(3\times 3\) matrix \([abc]\) has determinant zero.
In Section 4.2 we give an introduction to projective space, the Grassmannian, and Plücker coordinates. In Section 4.3 we discuss the bracket ring as the ring of invariant polynomials on \(\mathbb{P}^n\) and as a quotient ring with relations given by Plücker relations and Van der Waerden syzygies. We treat the Grassmann-Cayley algebra in Section 4.4 and Cayley factorization in Section 4.5, illustrating the theory with examples from the rigidity theory of body-and-bar frameworks.
In the final section we include a new result, Theorem 4.1 due to the first author, Cai, St. John, and Theran, characterizing which body-and-bar frameworks have pure conditions which may be represented by a bracket monomial. We use the notation and conventions by \textit{B. Sturmfels} [Algorithms in invariant theory. 2nd ed. Wien: Springer (2008; Zbl 1154.13003)] throughout, so that the reader who would like a fuller exposition may transition easily between this chapter and [Sturmfels, loc. cit.]. Exterior algebra, Grassmann algebras, Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) Special positions of frameworks and the Grassmann-Cayley algebra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For part I of this paper see J. Math. Kyoto Univ. 16, 201-207 (1976; Zbl 0326.14015); for part II see Bull. Kyoto Univ. Educ., Ser. B 73, 1-16 (1988; Zbl 0686.14048).]
In the previous papers we studied a 5-dimensional non-singular rational subvariety M of Gr(5,1) defined by seven equations. In this paper we show that M has an interesting structure. That is M is a blowing-down of a \({\mathbb{P}}^ 1\) bundle over a nonsingular quadric hypersurface of \({\mathbb{P}}^ 5\). Using this structure, we determine automorphisms of M which are induced by automorphisms of \({\mathbb{P}}^ 5\). 5-dimensional subvariety; Grassmann variety; automorphisms \(n\)-folds (\(n>4\)), Automorphisms of surfaces and higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds On a 5-dimensional non-singular rational subvariety of Grassmann variety Gr(5,1). III: Automorphisms of this 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 Concerns [Manuscr. Math. 106, No. 1, 101--116 (2001; Zbl 1066.14062)]. Projective techniques in algebraic geometry, \(n\)-folds (\(n>4\)), Grassmannians, Schubert varieties, flag manifolds, Questions of classical algebraic geometry Erratum: ''On first order congruences of lines of \({\mathbb{P}}^4\) with a fundamental curve'' | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The odd symplectic Grassmannian \(\text{IG}:=\text{IG}(k, 2n+1)\) parametrizes \(k\) dimensional subspaces of \({\mathbb {C}}^{2n+1}\) which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on \(\text{IG}\) with two orbits, and \(\text{IG}\) is itself a smooth Schubert variety in the submaximal isotropic Grassmannian \(\text{IG}(k, 2n+2)\). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of \(\text{IG}\), i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case \(k=2\), and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Equivariant quantum cohomology of the odd symplectic Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a homogeneous, projective variety, and let \(\overline M_{0,n}(X,\beta)\) be the space of genus zero, degree \(\beta\) stable curves to \(X\). \textit{B.~Kim} and \textit{R. Pandharipande} [in: Symplectic geometry and mirror symmetry, Proc. 4th KIAS Conf., Seoul 2000, 187--201 (World Scientific, Singapore) (2001; Zbl 1076.14517)] proved that \(\overline M_{0,n}(X,\beta)\) is a normal and irreducible, projective variety. Moreover they described its Bialynicki-Birula stratification corresponding to the maximal torus action. The virtual Poincaré polynomial of \(\overline M_{0,n}(X,\beta)\) has been computed by \textit{Yu.~Manin} [Topol. Methods Nonlinear Anal. 11, 207--217 (1998; Zbl 0987.14033)].
The tautological ring of \(\overline M_{0,n}(X,\beta)\) is the smallest subring of \(H^*(\overline M_{0,n}(X,\beta))\) having the following properties: it contains the pull-back by the evaluation morphisms of the cohomology ring of \(X\), and is closed under pull-backs and push-forwards by the forgetful morphisms, corresponding to various values of \(n\).
The goal of the article under review is to find generators, and to compute the dimensions of the cohomology (resp. Chow) groups of \(\overline M_{0,n}(X,\beta)\). The author succeeds in doing this for the Chow groups of codimension one and two.
At a first stage, the author describes the generators of the codimension one Chow group of \(\overline M_{0,n}(X,\beta)\); it turns out that all these are tautological classes. Assuming \(X\) is homogeneous for the special linear group, that is \(X\) is a flag variety, he computes moreover the dimension of \(A^1(\overline M_{0,n}(X,\beta))\).
Secondly, using a localization argument, the author describes a basis for the codimension two Chow group of \(\overline M_{0,n}(\mathbb P^r,d)\), for \(n=0,1,2\). One should compare his computations with those of \textit{K.~Behrend} and \textit{A.~O'Halloran} [Invent. Math. 154, 385--450 (2003; Zbl 1092.14019)].
Thirdly, a basis of \(A^2(\overline M_{0,n}(\text{Grass}(r,N),d))\), for \(n=0,1\), is determined. The computation is used for proving a reconstruction theorem, similar to that of Kontsevich-Manin, for genus zero Gromov-Witten invariants of \(\text{Grass}(r,N)\). stable maps; flag spaces; tautological ring; cohomology groups; Chow groups Oprea, D, Divisors on the moduli spaces of stable maps to flag varieties and reconstruction, J. Reine Angew. Math., 586, 169-205, (2005) Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory, (Equivariant) Chow groups and rings; motives, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Divisors on the moduli spaces of stable maps to flag varieties and reconstruction | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Via a natural degeneration of Grassmannian manifolds \(G(k, n)\) to Gorenstein toric Fano varieties \(P(k, n)\) with conifold singularities, we suggest an approach to study the relation between the tautological system on \(G(k, n)\) and the extended GKZ system on the small resolution \(\hat{P}(k, n)\) of \(P(k, n)\). We carry out the simplest case \((k, n) = (2, 4)\) to ensure its validity and show that the extended GKZ system can be regarded as a tautological system on \(\hat{P}(2, 4)\). tautological systems; extended GKZ systems; conifold transitions; toric degenerations of Grassmannians Mirror symmetry (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies Tautological systems under the conifold transition on \(G(2,4)\) | 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 We construct Koppelman formulas on Grassmannians for forms with values in any holomorphic line bundle as well as in the tautological vector bundle and its dual. As an application, we obtain new explicit proofs of some vanishing theorems of Bott-Borel-Weil type by solving the corresponding \(\bar\partial\)-equation. We also relate the projection part of our formulas to the Bergman kernels associated to the line bundles. Koppelman formulas; Grassmannian; holomorphic line bundle; vanishing theorem Götmark, E; Samuelsson, H; Seppänen, H, Koppelman formulas on Grassmannians, J. Reine Angew. Math., 640, 101-115, (2010) Integration on analytic sets and spaces, currents, Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels), Grassmannians, Schubert varieties, flag manifolds Koppelman formulas on Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in \(\text{GL}_n\) forms a real semi-algebraic cell of dimension \(n-1\). Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of \(\text{GL}_n(\mathbb{C})\) relying in particular on the positivity of the structure constants, which are enumerative Gromov-Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterson's which we explain with proofs in the type \(A\) case. flag varieties; quantum cohomology; total positivity; cell decomposition K. Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties, J. Amer. Math. Soc. 16 (2003), no. 2, 363--392 (electronic). Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over the reals, the complexes, the quaternions, Positive matrices and their generalizations; cones of matrices, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Totally positive Toeplitz matrices and quantum cohomology 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 We consider Deligne-Lusztig varieties and their analogs when the Frobenius endomorphism is replaced with conjugation by an element in a reductive group, especially a regular semisimple or regular unipotent one. We calculate their homology classes in the Chow group of the flag variety in terms of Schubert classes. Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over finite fields Homology class of a Deligne-Lusztig variety and its analogs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 is divided in two sections. In the first one the following theorem is proved:
Let \((R,{\mathfrak m}, K)\) be an excellent henselian Cohen-Macaulay local ring, with isolated singularity \(\text{Spec}(R)\) containing a field such that \(K\) is perfect or \([K:K^p] < \infty\) if \(\text{ch} (K)=p<0\). Let \(x=\{x_1, \dots, x_s\}\) be a system of parameters in \(R\) and \(R_t=R/(x^t_1, \dots, x_s^t)\), \(t\in\mathbb{N}\). Let \(L_{R_2}\) be the category of all finitely generated \(R_2\)-modules such that \(((x_1, \dots, x_{j-1}) P:x_j)_P=(x_1, \dots, x_j)P\), for all \(j=1, \dots, s\). Let \(\text{MCM} (R)\) be the category of maximal CM-modules of \(R\). Then the base change functor \(F_1: \text{MCM} (R)\to \text{Mod} (R_1)\), induces a bijection from the isomorphism classes of \(\text{MCM} (R)\) to the isomorphism classes of \(R_1\)-modules of the type \(x_1\dots x_sP\) for \(P\in L_{R_2}\).
In the second section it is shown that the isomorphism classes of maximal Cohen-Macaulay \(R\)-modules and the isomorphism classes of modules of \(L_{R_2}\) can be seen as orbits of an affine group acting on a Grassmannian variety when the residue field of \(R\) is algebraically closed. isomorphism classes of finitely generated modules; isomorphism classes of maximal Cohen-Macaulay modules; isolated singularity; Grassmannian variety Popescu, D.: Maximal Cohen--Macaulay modules over isolated singularities. J. algebra 178, 710-732 (1995) Cohen-Macaulay modules, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Grassmannians, Schubert varieties, flag manifolds, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Maximal Cohen-Macaulay modules over isolated singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors, motivated by the applications of algebraic geometry techniques to linear control theory, investigate algorithmic approaches to describe Grassmann varieties of projective spaces. An algorithm for computing the reduced set of quadratic Plücker relations is presented. For this purpose, the authors prove that the set of homogeneous equations can be determined from the Plücker relations by using a criterion based on decomposable vectors and lexicographic orderings. Grassmann varieties; quadratic Plücker relations Computational aspects of higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds On the computation of a reduced set of quadratic Plücker relations describing the Grassmann variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a \((2n+1)\)-dimensional vector space over a field \(\mathbb F\), endowed with a non-singular quadratic form \(q\) of Witt index \(n\). Let \(\Delta\) be the building of type \(\mathsf B_n\) whose elements are the subspaces of \(V\) which are totally singular for \(q\), and for \(k \leq n\) denote \(\Delta_k \subset \Delta\) the \(k\)-Grassmannian of \(\Delta\), whose points are the totally singular \(k\)-subspaces of \(V\). Then \(\Delta_k\) is a point-line geometry.
The paper under review studies two classes of embeddings of \(\Delta_k\) into projective spaces, called the Grassmann embeddings and the Weyl embeddings. The Grassmann embedding is the natural embedding \(\varepsilon_k : \Delta_k \rightarrow \mathrm{PG}(W_k)\), where \(W_k = \bigwedge^k V\) and where \(\mathrm{PG}(W_k)\) denotes the associated projective space. To define the Weyl embedding, notice that \(\Delta_k\) has a natural action of the orthogonal group \(\mathrm{SO}(2n+1,\mathbb F)\). Given \(k < n\), denote \(\lambda_k\) the \(k^{th}\) fundamental weight of \(\mathrm{SO}(2n+1,\mathbb F)\), and set \(\lambda_n\) twice the last fundamental weight. Then \(\Delta_k\) is identified with the orbit of a highest weight line in the projective space of the Weyl module \(V(\lambda_k)\) of \(\mathrm{SO}(2n+1,\mathbb F)\), and the Weyl embedding is the associated embedding \(\widetilde\varepsilon_k : \Delta_k \rightarrow \mathrm{PG}(V(\lambda_k))\).
Set \(G = \mathrm{SO}(2n+1,\mathbb F)\) and, for \(k \leq n\), let \(\langle \varepsilon_k(\Delta_k) \rangle \subset W_k\) be the submodule generated by the image of \(\varepsilon_k\), which is a quotient of \(V(\lambda_k)\). As a consequence of the irreducibility of \(V(\lambda_k)\), the \(G\)-modules \(\langle \varepsilon_k(\Delta_k) \rangle\) and \(V(\lambda_k)\) are isomorphic if \(\mathrm{char}(\mathbb F) \neq 2\). An easy proof of this fact is given in the first theorem of the paper, without making use the irreducibility of \(V(\lambda_k)\) and relying only on elementary properties of quadratic forms in odd characteristic. If instead \(\mathrm{char}(\mathbb F) = 2\), then the authors show that \(\mathrm{dim}(V(\lambda_k)) - \mathrm{dim} \langle \varepsilon_k (\Delta_k) \rangle = {2n+1 \choose k-2}\).
In the second part of the paper the authors study the universality of Grassmann and Weyl embeddings, as defined by \textit{A. Kasikova} and \textit{E. Shult} [J. Algebra 238, No. 1, 265--291 (2001; Zbl 0988.51001)]. Given \(k < n\), the Weyl embedding \(\widetilde\varepsilon_k\) is called universal if all embeddings of \(\Delta_k\) defined over the same division ring as \(\widetilde\varepsilon_k\) are quotients of \(\widetilde\varepsilon_k\), it is a well known fact that \(\widetilde \varepsilon_1\) is universal for any field \(\mathbb{F}\). In the last theorem of the paper, assuming that \(\mathbb F\) is a perfect field of positive characteristic or a number field, the authors prove that \(\widetilde\varepsilon_2\) is universal provided that \(n > 2\), and that \(\widetilde\varepsilon_3\) is universal provided that \(n > 3\) and \(\mathbb F \neq \mathbb F_2\). Then they conjecture that the Weyl embedding \(\widetilde\varepsilon_k\) is always universal, for any \(k < n\) and for any field \(\mathbb F\). orthogonal Grassmannian; Grassmann embedding; Weyl embedding; universal embedding I. Cardinali, A. Pasini, Grassmann and Weyl embeddings of orthogonal grassmannians, J. Algebraic Combin., http://dx.doi.org/10.1007/s10801-013-0429-x, in press. Grassmannians, Schubert varieties, flag manifolds, Polar geometry, symplectic spaces, orthogonal spaces, Incidence structures embeddable into projective geometries, Buildings and the geometry of diagrams, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) Grassmann and Weyl embeddings of orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of \textit{V. Gorbounov} and \textit{C. Korff} [Adv. Math. 313, 282--356 (2017; Zbl 1386.14181)] in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of \textit{Y. Huang} and \textit{C. Li} [J. Algebra 441, 21--56 (2015; Zbl 1349.14173)] and known equivariant quantum Littlewood-Richardson rules, our formula does not require any calculations in a different Grassmannian or two-step flag variety. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds An equivariant quantum Pieri rule for the Grassmannian on cylindric shapes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In a paper of \textit{F. Burstall} [Minimal surfaces in quaternionic symmetric spaces, In: Geometry of low-dimensional manifolds, Proc. Symp. Durham/UK 1989, Lond. Math. Soc. Lect. Note Ser. 150, 231-235 (1990; Zbl 0726.53036)] the following was shown: between the twistor spaces over any two compact quaternionic Kähler symmetric spaces there exists a birational correspondence preserving the horizontal distributions.
The author studies, more generally, birational correspondences between flag manifolds. Here, a flag manifold \(F = G/P\) is a quotient of a complex (semi-)simple Lie group by a parabolic subgroup. Let \(\mathfrak g\) and \(\mathfrak p\) be the Lie algebras of \(G\) and \(P\) and \({\mathfrak n} = {\mathfrak p}^\perp\) the nilradical of \(\mathfrak p\), that is the polar of \(\mathfrak p\) with respect to the Killing form.
Important for the proof of Burstall was a paper of \textit{J. A. Wolf} [J. Math. Mech. 14, 1033-1047 (1965; Zbl 0141.382)] from which he deduced that the nilradical \(\mathfrak n\) is in this situation a generalized Heisenberg algebra. In the more general context of flag manifolds the horizontal distribution should be replaced by the so-called super- horizontal subdistribution, which coincides with the horizontal distribution in the symmetric quaternionic-Kähler case.
The main result of the paper under review states that flag manifolds for which nontrivial birational correspondences preserving the superhorizontal distribution exist must be quaternionic twistor spaces. The author obtains this by showing first that the existence of a birational equivalence between two flag manifolds \(F = G/P\) and \(F' = G'/P'\) which preserves the superhorizontal distribution implies that their nilradicals \(\mathfrak n\) and \({\mathfrak n}'\) are isomorphic. Using a result of \textit{A. L. Onishchik} and \textit{Y. B. Khakimdzhanov} [Mat. Zametki 18, 31-40 (1975; Zbl 0322.17003), Engl. transl.: Math. Notes 18(1975), 600-604 (1976)] he shows then the existence of Lie algebra isomorphisms \({\mathfrak g} \cong {\mathfrak g}'\) and \({\mathfrak p} \cong {\mathfrak p}'\) (if the nilradicals are neither Abelian nor Heisenberg), which implies the result. birational correspondences; flag manifolds; quaternionic twistor spaces Kobak, P. Z.: Birational correspondence between twistor spaces, Bull. London math. Soc. 26, 186-190 (1994) Differential geometry of homogeneous manifolds, Twistor theory, double fibrations (complex-analytic aspects), Grassmannians, Schubert varieties, flag manifolds, Homogeneous complex manifolds, Differential geometry of symmetric spaces Birational correspondences between twistor 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 Let \(X\subset {\mathbb P}^r\) be a complex integral non-degenerate variety of dimension \(n\). For \(k\leq r\), a general \((k+1)\)-uple of points in \(X\) spans a \(k\)-plane. Therefore we have the rational map: \(\Phi: X^{k+1} \to G(k,r)\) to the Grassmannian of \(k\)-planes in \({\mathbb P}^r\). By \(G_k(X)\) denote the closure of the image of \(\Phi\) and by \(G_{h,k}(X)\), \(h<k\), the closure of the subset
\[
\{(L,H) \mid L\in G(h,r), \;H\in G_k(X),\;L\subset H\;\}\subset G(h,r)\times G(k,r).
\]
\noindent Note that \(G_{0,k}\) is the secant variety \(S_k(X)\), which is the closure of the union of all \(k\)-planes, \((k+1)\)-secant to \(X\). The varieties \(G_{h,k}(X)\) are called Grassmannians of secant varieties. The present paper is devoted to a systematic study of these objects.
The expected dimension of \(G_{h,k}(X)\) is equal to
\[
\min\{(k-h)(h+1)+n(k+1), (r-h)(h+1)\}.
\]
In the first section it is shown that if this value is not attained, then the dimension of some secant variety is also less than the expected value.
In section 2 the case of irreducible surfaces is considered. It is proved that surfaces \(S\subset {\mathbb P}^r\), \(r\geq 5\) with \(\dim G_{1,2}(S)<8\) are either cones or rational normal surfaces of (minimal) degree \(4\) in \({\mathbb P}^5\), but not Veronese surfaces. secant varieties; expected dimension of Grassmannians of secant varieties; Veronese surfaces L. Chiantini and M. Coppens: ''Grassmannians of secant varieties'', Forum Math., Vol. 13, (2001), pp. 615--628. Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry Grassmannians of secant varieties. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Q\) be an acyclic quiver, and let \(\mathcal{S}\) be the associated Nakajima category associated to \(Q.\) Let \(\mathcal{M}_{0}\left( d\right) \) be the graded affine quiver variety associated with \(Q\) and the dimension vector \(d\), and let \(\mathcal{M}\left( d\right) \) be the variety of representations of the regular Nakajima category \(\mathcal{R}\). \textit{H. Nakajima} [J. Am. Math. Soc. 14, No. 1, 145--238 (2001; Zbl 0981.17016)] has constructed is a proper, surjective morphism (a pre-desingularization as the domain is smooth) \(\pi:\mathcal{M}\left( d\right) \rightarrow\mathcal{M}_{0}\left( d\right) \) which carries a representation in \(\mathcal{M}\left( d\right) \) to its restriction\ to \(\mathcal{S}.\) In the work being reviewed, the authors construct pre-desingularizations and desingularizations of quiver Grassmannians under certain conditions.
Let \(M\) be a module over \(\mathcal{S}\) with dimension vector \(d\); then \(M\) corresponds to a point in \(\mathcal{M}_{0}\left( d\right)\). Let \(w\) be a dimension vector, \(w\leq d.\) If the intermediate Kan extension \(K_{LR}\left( M\right) \) is rigid (i.e., its space of self-extensions vanishes), then the map on Grassmannians \(\pi_{\text{Gr}}:\coprod\)Gr\(_{\left( v,w\right) }\left( K_{LR}\left( M\right) \right) \rightarrow\)Gr\(_{w}\left( M\right) \)\ given by sending \(U\subset K_{LM}\left( M\right) \) to its restriction is shown to be a pre-desingularization. Here the coproduct is taken over certain vectors \(v\) which relate to the irreducible components of Gr\(_{w}\left( M\right) \).\ Assuming the rigidity condition, the authors consider the bistable Grassmannian Gr\(_{\left( v,w\right) }^{bs}\left( K_{LR}\left( M\right) \right) ,\) that is, the closure of the set of points which correspond to bistable submodules. Then it is shown that the map \(\pi ^{bs}:\coprod\)Gr\(_{\left( v,w\right) }^{bs}\left( K_{LR}\left( M\right) \right) \rightarrow\)Gr\(_{w}\left( M\right) \) sending \(U\subset K_{LM}\left( M\right) \) to its restriction is a desingularization, i.e., a pre-desingularization which induces an isomorphism between dense open subsets. It is unclear whether Gr\(_{\left( v,w\right) }^{bs}\left( K_{LR}\left( M\right) \right) \) is a proper restriction of Gr\(_{\left( v,w\right) }\left( K_{LR}\left( M\right) \right) \) as the authors conjecture that these two Grassmannians are equal.
It is not the case that \(K_{LR}\left( M\right) \) is always rigid -- in fact, an example example is provided to show this -- however some sufficient conditions are provided.
The work above is then applied to iterated algebras of Dynkin type, showing that for finite dimensional \(M\) the intermediate Kan extension is rigid, hence \(\pi^{bs}\) is a desingularization. The specific example of Dynkin type \(D_{4}\) is made explicit.
The results obtained here are similar to [\textit{G. Cerulli Irelli} et al., Adv. Math. 245, 182--207 (2013; Zbl 1336.16015)], in fact the main results here are modeled after results from that paper. The inclusions of Kan extensions and bistable modules are new here -- the work of Cerulli, Feigin, and Reineke used objects from a certain category of contravariant functors, and the closure of a collection of subrepresentations, instead of these objects. A section of the paper under review is devoted to this connection, where it is shown that the other work is a specialization of this one. quiver Grassmannians; Nakajima quiver varieties; desingularizations; bistable Grassmannians; pre-desingularizations Keller, B; Scherotzke, S, Desingularizations of quiver Grassmannians via graded quiver varieties, Adv. Math., 256, 318-347, (2014) Grassmannians, Schubert varieties, flag manifolds, Representations of quivers and partially ordered sets Desingularizations of quiver Grassmannians via graded quiver varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors investigate some aspects of the Nichols-Woronowicz algebra associated to a particular kind of braided vector space called Yetter-Drinfeld module. Another focal point in this article is the study of relations among the Dunkl elements in the elliptic extension \(\tilde{\varepsilon}_n(\psi_{ij})\) of the Fomin-Kirillov algebra. quadratic algebra; elliptic function; Dunkl elements; Fomin-Kirillov quadratic algebra; Nichols-Woronowicz algebra Kirillov, Anatol N. and Maeno, Toshiaki, Braided differential structure on {W}eyl groups, quadratic algebras, and elliptic functions, International Mathematics Research Notices. IMRN, 2008, 14, no.~14, rnn046, 23~pages, (2008) Linear difference operators, Representation theory of linear operators, Hopf algebras and their applications, Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Braided differential structure on Weyl groups, quadratic algebras, and elliptic functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials {Let \(Fl_n\) be the manifold of complete flags in the \(n\)-dimensional vector space \(\mathbb C^n\). Inspired from ideas from string theory, recently the concept of quantum cohomology ring \(QH^*(X,\mathbb Z)\) of a Kähler algebraic manifold \(X\) has been defined. Then
\[
QH^*(Fl_n,\mathbb Z)\cong H^*(Fl_n,\mathbb Z) \otimes\mathbb Z[q_1,\dots,q_{n-1}],
\]
where \(H^*(X,\mathbb Z)\) is the usual cohomology ring of \(Fl_n\) and \(q_1,\dots,q_{n-1}\) are formal variables (deformation parameters). So, the additive structures of the two cohomology rings are essentially the same. The multiplicative structure of \(H^*(X,\mathbb Z)\) can be recuperated from the multiplicative structure of \(QH^*(X,\mathbb Z)\) by taking \(q_1=\cdots=q_{n-1}=0\). The structure constants for the quantum cohomology are the 3-point Gromov-Witten invariants of genus zero. Recently, Givental, Kim and Ciocan-Lafontaine found a canonical isomorphism
\[
QH^*(X,\mathbb Z)\cong\mathbb Z[q_1,\dots,q_{n-1}][x_1,\dots,x_n]/I_n^q,
\]
where \(x_1,\dots,x_n\) are variables and \(I_n^q\) is a certain ideal which can be explicitly described. This isomorphism extends an old isomorphism of Borel for the ordinary cohomology ring. The next problem naturally arising in the theory of quantum cohomology of the flag manifolds is to find an algebraic/combinatorial method for computing the structure constants of quantum multiplication in the basis of Schubert classes (the Gromov-Witten invariants). The aim of the paper under review is to solve this problem completely.} flag varieties; Schubert varieties; quantum cohomology ring; complete flags; Gromov-Witten invariants S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials , J. Amer. Math. Soc., 168 (1997), 565--596. JSTOR: Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Quantum 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 It is with the advent of Bose-Fermi supersymmetry that necessitated the creation of supermanifolds (superspaces), super Lie-groups. This notion has been further extended to many other geometrical objects and the present book is an outcome of investigations in this direction. The objective of this book is two fold: for mathematicians an access to understand supersymmetry whereas to physicists to learn the techniques of supersymmetry. The authors present the construction of Minkowski- and conformal superspaces as homogeneous spaces for the Poincaré- and conformal supergroups in mathematically rigorous way. This book contains five chapters, five appendices followed by bibliography and index. Chapter 1 gives a vivid account of supergeometry. The authors begin with the concepts of linear superalgebra, Lie-superalgebra; give an idea about sheaf and jump over to explain supermanifolds. From algebraic geometry, an introduction to supervarieties, functor of points, supergroups are discussed. Thereafter, first the notion of homogeneous superspaces is delineated. Next, revisiting Hopf algebra and its salient features briefly, the concept of Hopf superalgebra is developed in lucid way, thus incorporating all the tools required for studies in subsequent chapters of this book. Chapter 2 is devoted to revisit the ordinary Minkowski- and conformal spaces. The use of word `ordinary' looks rather abusive. The authors must have simply written Minkowski- and conformal spaces or might have used any other appropriate word. The authors reproduce a moderate but skilfully crafted description of Relativity with emphasis on physical motivation, discuss Galilean-, Lorentz transformations in detail which in turn give rise to the notions of Minkowski space, Lorentz-, Poincaré group. Thence the notion of conformal symmetry is introduced as a generalization of symmetry of special relativity, and the conformal transformations, corresponding Lie-group and their Lie-algebra are dealt with at length. Though the conformal symmetry is not global as Poincaré symmetry yet its role is important in the ensuing studies. Thereupon, after having explained the physical motivation and physical viewpoint, the concept of complex Minkowski-, complex conformal spaces and their real forms are studied. A mathematical construction of complex conformal space is made in terms of Grassmannian that may be viewed as an analytic manifold, as a projective variety in the language of algebraic geometry and as a homogeneous space. Later on, in order to study the geometry of conformal space, the functor of points of Grassmannian manifold is discussed. Chapter 3 deals with the supersymmetry in physics. In fact, supersymmetry can be viewed as generalization of Lie-groups, the Poincaré- and conformal groups from mathematical point of view, while from stand point of Physics, it is not possible to get a clear understanding of supersymmetry without having an insight to quantum physics. So first a succinct introduction to quantum physics is presented as a smooth transition from classical to quantum keeping an eye and emphasizing on what is required in due course of further discussion. Next, an elaborate answer to `what is spin in the context of special relativity' is tried to obtain. Moreover, the concept of particles and spin in the relativistic setting intimately related to unitary representations of Poincaré group is also undertaken for in depth study. Quantum Field Theory (QFT) came into being for merging of relativity and quantum physics. So some fundamental ideas of QFT are given here. Now, in order to understand the concept of particles and their spin statistics connection in the context of QFT, there arose the need of superalgebra to describe two types of particles namely Bosons and Fermions, and supersymmetry came here whose role in parcticle physics is fairly described. In the end of this chapter, the authors give a clear picture of the development and the philosophy that led to the introduction of supersymmetry in physics. They further go on to delineate its need and aspect in the framework of string field theories unifying all the fundamental forces of nature. Chapter 4 takes up the main theme of this book: Minkowski- and conformal superspaces, wherein generalization of spacetime leading to Poincaré- and conformal superspace fairly used in supersymmetric theories, is made. In first two sections, spinors, spin representations, and then the complex Clifford algebra and the spin groups with their properties and important results are discussed. In section three, real Clifford algebra and the spin group are studied. Subsequently, in section four complex Wess-Zumino or conformal superalgebra is defined and its real form is described with significant results and proofs thereof. The Poincaré algebra plus dilations is known as parabolic subalgebra of the conformal algebra. In next two sections, Borel and parabolic subalgebra and the Poincaré superalgebra are studied. The authors now go on to describing complex Grassmannian supermanifold \(\mathrm{Gr}_{ch}\) and super Plücker embedding on it to projective superspace. It is then they look into the relationship of super Grassmannian with super invariant theory. Now, first the chiral version of conformal superspace is taken for discussion in terms of super Grassmannian, then the full version as super flag and the properties of superspaces are described. A point worthy to note is that the Minkowski superspaces are considered as big cells inside the super Grassmannian and super flag. This study is carried out first to complex and then to real form in terms of the functor of points as well as the sheaf. In Chapter 5, the authors devote a great deal to the non-commutative deformation of the Minkowski-, conformal spaces and superspaces. As well known, General Relativity as such and Quantum Physics are inconsistent. Since gravity being extremely weak at very short distances in the realm of quantum world, it is not accessible in the lab. so we are left with only Mathematics to search for a consistent theory of gravity and quantum physics. However, in the world of theoretical physics, there are events such as black holes wherein and about, one can visualize the consistency of gravity and quantum physics. Through such an event, their spectacular behaviour is largely explored and examined. This in turn gives us the deformations of algebra of functions on spacetime to a non-commutative algebra. It is in the same way as the quantization of a physical system regarded as a deformation of the algebra of the functions on phase space. The parameter of the deformation could be related to Planck's scale. It is worth mentioning here that Minkowski spacetime is flat so gravity is absent. On account of this, the deformation of Minkowski space is far off from view point of quantum gravity consideration. Nevertheless, there are several versions of deformations of Minkowski space. Most interesting ones are those somehow preserving the action of Poincare group or some quantum version of it. The deformation proposed in this chapter goes a bit further ahead for it also quantizes the conformal space. This is done using the concept of quantum group and quantum homogeneous space. In order to accomplish their task,the authors first discuss quantum groups and the quantum complex conformal- and Minkowski spaces in detail and then take up real quantum Minkowski space and quantum supergroups for study. Finally, the quantum deformation of the chiral conformal superspace and chiral Minkowski superspace are studied. In the end, readers may find it useful the brief notes on some topics namely categories, representability criterion, Lie-superalgebra and Lie-supergroups, super Harish Chandra pairs and quantum supergroups in the appendices. The collection of references in the bibliography is as good as to be. There is an error on p. 158: A. Salam, S. Glashow and S. Weinberg received the Physics Nobel Prize not in 1972 (as mentioned there) but in 1979. However, in the reviewer's opinion, third chapter is the gem of the book. The presentation of the subject matter is very clear, concise, cohesive and easy to understand, neat explanation given at every technical point is the beauty of the book. The authors incorporated important results and furnished examples thereof wherever required. On the one hand, they tried to maintain the mathematical rigour as far as possible, on the other hand, their efforts to build the physical intuition throughout the text is praiseworthy. The authors deserve a word of appreciation for their industrious work. This book is useful for graduate students of both Mathematics and Physics as well as for researchers. generalized spaces; supersymmetry; supergroups; quantum groups; Minkowsky; conformal superspaces; Minkowski and conformal geometry R. Fioresi and M. A. Lledó, \textit{The Minkowski and Conformal Superspaces} (World Scientific, 2015). Research exposition (monographs, survey articles) pertaining to quantum theory, Supersymmetric field theories in quantum mechanics, Analysis on supermanifolds or graded manifolds, Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics, Supervarieties, Lie (super)algebras associated with other structures (associative, Jordan, etc.), Structure and representation of the Lorentz group, Special relativity, Clifford algebras, spinors, Grassmannians, Schubert varieties, flag manifolds, Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Physics The Minkowski and conformal superspaces. The classical and quantum descriptions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 dedicated to developing a positive characteristic version of the Kazhdan-Lusztig cells. In [Contemp. Math. 683, 333--361 (2017; Zbl 1390.20001)], the author and \textit{G. Williamson} defined the \(p\)-canonical basis for the Hecke algebra of a crystallographic Coxeter system. It can be thought of as a positive characteristic analogue of the Kazhdan-Lusztig basis. The \(p\)-canonical basis shares strong positivity properties with the Kazhdan-Lusztig basis (similar to the ones described by the Kazhdan-Lusztig positivity conjectures), but it loses many of its combinatorial properties. Replacing the Kazhdan-Lusztig basis by the \(p\)-canonical basis in the definition of the left (resp. right or two-sided) cells leads to the notion of left (resp. right or two-sided) p-cells. These \(p\)-cells are the main subject of the present paper.
The first properties of \(p\)-cells are proved in Section 3. Left and right \(p\)-cells are related by taking inverses (see Lemma 3.6), just like for Kazhdan-Lusztig cells. The set of elements with a fixed left descent set decomposes into right \(p\)-cells (see Lemma 3.4). The most important result of this section is a certain compatibility of \(p\)-cells with parabolic subgroups that shows that any right \(p\)-cell preorder relation in a finite standard parabolic subgroup \(W_I\) induces right \(p\)-cell preorder relations in each right \(W_I\)-coset (see Theorem 3.9). It is also shown that unfortunately Kazhdan-Lusztig cells do not always decompose into \(p\)-cells, but the author expects that this may still be the case when the prime \(p\) is good for the corresponding algebraic group. Indeed, in Section 4 it is proved that Kazhdan-Lusztig left cells decompose into left \(p\)-cells in finite types B and C for \(p > 2\). This is done by studying the consequences of the Kazhdan-Lusztig star-operations for the \(p\)-canonical basis, these operations appear as the main technical tool of the present work. Kazhdan-Lusztig cells; Hecke algebra; \(p\)-canonical basis Other geometric groups, including crystallographic groups, Representation theory for linear algebraic groups, Combinatorial aspects of representation theory, Hecke algebras and their representations, Representations of finite symmetric groups, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Modular representations and characters, Reflection and Coxeter groups (group-theoretic aspects) The ABC of \(p\)-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 Given a projective variety \(X\subset\mathbb{P}^n\) of codimension \(k+1\), the Chow hypersurface \(Z_X\) is the hypersurface of the Grassmannian \(\mathrm{Gr}(k,n)\) parametrizing projective linear spaces that intersect \(X\). We introduce the tropical Chow hypersurface \(\mathrm{Trop}(Z_X)\). This object only depends on the tropical variety \( \mathrm{Trop}(X)\) and we provide an explicit way to obtain \(\mathrm{Trop}(Z_X)\) from \(\mathrm{Trop}(X)\). We also give a geometric description of \(\mathrm{Trop}(Z_X)\). We conjecture that, as in the classical case, \(\mathrm{Trop}(X)\) can be reconstructed from \(\mathrm{Trop}(Z_X)\) and prove it for the case when \(X\) is a curve in \(\mathbb{P}^3\). This suggests that tropical Chow hypersurfaces could be the key to construct a tropical Chow variety. Geometric aspects of tropical varieties, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of tropical varieties Tropical Chow 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 \textit{G.Higman} [Proc. internat. Conf. Theory Groups, Canberra 1965, 167--173 (1967; Zbl 0242.20040)] has constructed functors of a vectorspace \(E\), related to the irreducible representations of the general linear group; these functors
will here be denoted by \(\wedge^\alpha E\) (where \(\alpha\) is a partition). A different construction
is given for vectorspaces over fields of characteristic O by \textit{D. B. A. Epstein} [Am. J. Math. 91, 395--414 (1969; Zbl 0201.02501 )], and Epstein's construction has been generalized for \(E\) a module over any commutative ring by \textit{R. W. Carter} and \textit{G.Lusztig} [Math. Z. 136, 193--242 (1974; Zbl 0298.20009)]; this second class of functors will here be denoted by \(\vee_\alpha E\). The present paper concerns
the following new constructions and results involving these functors \(\wedge^\alpha E\) and \(\vee_\alpha E: \oplus \vee_\alpha E\) is given the natural structure of a graded anti-commutative associative algebra. (Note that \(\oplus\wedge^\alpha E\) automatically has the structure of a commutative associative algebra, as is implicit in Higman's construction.) Generators and relations are given for the two rings \(\oplus\wedge^\alpha E\), \(\oplus \vee_\alpha E\) which thus make sense for \(E\) a module over a commutative ring. Note 1: In the present paper, \(\wedge^\alpha E\) and \(\vee_\alpha E\) are defined via these generators and relations. The resulting functors of modules \(\wedge^\alpha E\) are naturally isomorphic to those of Higman when \(E\) is a vectorspace, but appear to be new in general. The resulting functors \(\vee_\alpha E\) are naturally Isomorphic to those of Carter-Lusztig when \(E\) is finitely generated projective, but not in general. Note 2: At the end of the paper being reviewed, the author mentioned that these relations for \(\wedge^\alpha E\) may have been already published by Higman and/or his students; Higman has since written the author that this is not the case. If \(\alpha\) and \(\alpha\ast\) are dual partitions, a natural pairing \(\wedge^\alpha E^\oplus \vee_{\alpha\ast} E^\ast \to k\) is constructed, which exhibits \(\wedge^\alpha E\) and \(\vee_{\alpha\ast} E^\ast\) a dual modules if \(E\) is finitely generated projective. If \(E\) is free of rank \(r\), then \(\wedge^\alpha E\), \(\vee_\alpha E\) are free of rank depending only on \(\alpha\) and \(r\) (this is proved by Carter and Lusztig for \(\vee_\alpha E\)). If \(k\) is a field of characteristic 0, an explicit isomorphism is constructed between \(\wedge^\alpha E\) and \(\vee_{\alpha^\ast} E\); this yields new structure on the affine coordinate ring of the flag manifold (in characteristic 0 only), namely a natural graded anti-commutative associative wedge-product operation. One final concept in this paper which appears to be new is that of Young symmetry, with respect to a partition of \(n\), of a function of \(n\) variables taking values in a \(\mathbb{Z}\)-module; this coincides with Young's original concept if the function takes values in \(\mathbb{Q}\)-module. In this connection it should be mentioned that the citation on p. 100 for a certain combinatorial lemma, now appears not to exist in the literature; the author will be glad to mail a proof of this lemma on request. Towber, J.: Two new functors from modules to algebras. J. Alg.47, 80--104 (1977) Vector and tensor algebra, theory of invariants, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Representations of finite symmetric groups, Theory of modules and ideals in commutative rings, Grassmannians, Schubert varieties, flag manifolds Two new functors from modules to algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors give a description of the intersection cohomology groups of the Schubert varieties in partial flag manifolds over symmetrizable Kac-Moody Lie algebras in terms of parabolic Kazhdan-Lusztig polynomials introduced by \textit{V. Deodhar} [J. Algebra 111, 483-506 (1987; Zbl 0656.22007)]. Grassmanians; Schubert varieties; flag manifolds; intersection cohomology groups; symmetrizable Kac-Moody Lie algebras Kashiwara, M., \& Tanisaki, T. (2002). Parabolic Kazhdan-Lusztig polynomials and Schubert varieties. J. Algebra, 249, 306--325. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Parabolic Kazhdan-Lusztig polynomials and Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Each cohomology ring of a Grassmannian or flag variety has a basis of Schubert classes indexed by the elements of the corresponding Weyl group. Classical Schubert calculus computes the cohomology rings of Grassmannians and flag varieties in terms of the Schubert classes. This paper is ``doing Schubert calculus'' in the equivariant cohomology rings of Peterson varieties.
The Peterson variety is a subvariety of the flag variety \(G/B\) parameterized by a linear subspace \(H_{\mathrm{Pet}} \subseteq \mathfrak g\) and a regular nilpotent operator \(N_0 \in \mathfrak g\). We can define the Peterson variety as
\[
\mathrm{Pet}=\{gB\in G\backslash B:\mathrm{Ad}(g^{-1})N_0 \in H_{\mathrm{Pet}}\}.
\]
Peterson varieties were introduced by Peterson in the 1990s. Peterson constructed the small quantum cohomology of partial flag varieties from what are now Peterson varieties. \textit{B. Kostant} [Sel. Math., New Ser. 2, No. 1, 43--91 (1996; Zbl 0868.14024)] used Peterson varieties to describe the quantum cohomology of the flag manifold and \textit{K. Rietsch} [Nagoya Math. J. 183, 105--142 (2006; Zbl 1111.14048)] gave the totally non-negative part of type A Peterson varieties. \textit{E. Insko} and \textit{A. Yong} [Transform. Groups 17, No. 4, 1011--1036 (2012; Zbl 1267.14066)] explicitly identified the singular locus of type A Peterson varieties and intersected them with Schubert varieties. \textit{M. Harada} and \textit{J. Tymoczko} [Proc. Lond. Math. Soc. (3) 103, No. 1, 40--72 (2011; Zbl 1219.14065)] proved that there is a circle action \(\mathbb S^1\) which preserves Peterson varieties. In this paper the authors study the equivariant cohomology of the Peterson variety with respect to this action and also they use GKM theory as a model for studying equivariant cohomology, but Peterson varieties are not GKM spaces under the action of \(\mathbb S^1\). Using work by Harada and Tymoczko [Zbl 1219.14065] and \textit{M. Precup} [Sel. Math., New Ser. 19, No. 4, 903--922 (2013; Zbl 1292.14032)], they construct a basis for the \(\mathbb S^1\)-equivariant cohomology of Peterson varieties in all Lie types. This construction gives a set of classes which we call Peterson Schubert classes. The name indicates that the classes are projections of Schubert classes, they do not satisfy all the classical properties of Schubert classes.
Classical Schubert calculus asks how to multiply Schubert classes; here, the authors asks how to multiply in the basis of Peterson Schubert classes. She gives a Monk's formula for multiplying a ring generator and a module generator, and a Giambelli's formula for expressing any Peterson Schubert class in the basis in terms of the ring generators. Peterson variety; equivariant cohomology; Monk's rule; Giambelli's formula; Schubert calculus Drellich, Elizabeth, Monk's rule and Giambelli's formula for Peterson varieties of all Lie types, J. Algebraic Combin., 41, 2, 539-575, (2015) Classical problems, Schubert calculus, Equivariant homology and cohomology in algebraic topology, Grassmannians, Schubert varieties, flag manifolds Monk's rule and Giambelli's formula for Peterson varieties of all Lie 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 This is the first of a series of papers which review the geometric construction of the double affine Hecke algebra via affine flag manifolds and explain the main results of the authors on its representation theory. There are also some simplifications of the original arguments and proofs for some well-known results for which there exists no reference.
This paper concerns the most basic facts of the theory: the geometric construction of the double affine Hecke algebra via the equivariant, algebraic K-theory and the classification of the simple modules of the category \(\mathcal O\) of the double affine Hecke algebra. It is our hope that by providing a detailed explanation of some of the difficult aspects of the foundations, this theory will be better understood by a wider audience.
This paper contains three chapters. The first one is a reminder on \(\mathcal O\)-modules over non Noetherian schemes and over ind-schemes. The second one deals with affine flag manifolds. The last chapter concerns the classification of simple modules in the category \(\mathcal O\) of the double affine Hecke algebra. double affine Hecke algebras; degenerate Hecke algebras; simple modules in category \(\mathcal O\); simple spherical representations; induced modules; rational Cherednik algebras; spherical finite-dimensional modules M. Varagnolo and E. Vasserot, Double affine Hecke algebras and affine flag manifolds. I, Affine flag manifolds and principal bundles, 233--289, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2010. Hecke algebras and their representations, Grassmannians, Schubert varieties, flag manifolds, Grothendieck groups, \(K\)-theory, etc., Quantum groups (quantized enveloping algebras) and related deformations Double affine Hecke algebras and affine flag manifolds. 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 These notes are based upon a series of lectures that both authors had given in a summer school at Thurnau, Germany, held from June 19 to June 23, 1995. The lectures were designed to provide an introduction to the theory of Schubert varieties, at its contemporary state of knowledge, and to the related theory of degeneracy loci of vector bundle morphisms in algebraic geometry. The text under review follows closely the lectures delivered at Thurnau, the notes of which had been circulating, since then, among the community of algebraic geometers, but it has been enhanced, in its present published from, by ten additional appendices and a few up-dating remarks or footnotes. As the authors emphasize in the preface to the book, this text is neither intended to be a textbook, nor a research monograph, nor a survey on the subject. Instead, they have tried to describe what they, in their capacity of being two of the most active and competent researches in this area of algebraic geometry, regard as essential features of the whole complex of topics, each from his own point of view. The outcome is a great, huge panorama of a fascinating subject in both classical and contemporary algebraic geometry. The present text consists of nine chapters, ten appendices to them, and an utmost rich bibliography.
Chapter I starts with the classical origin of the whole subject, that is, with the description of loci of matrices of various ranks. This is followed by discussing classical and modern solutions of these old problems, including the combinatorial framework of Schur functions and Schubert polynomials. Chapter II turns to the modern generalization of the classical background, namely to morphisms of vector bundles over algebraic varieties, their degeneracy loci, and the cohomological invariants of these degeneracy loci. The fundamental case of Grassmannians and flag manifolds, together with the Schubert subvarieties associated with them, is the central topic of this chapter. Chapter III is devoted to the crucial combinatorial tools: the various kinds of symmetric functions such as Schur \(S\)-polynomials, Schur \(Q\)-polynomials, supersymmetric polynomials, and others, together with their fundamental properties and identities. Chapter IV discusses symmetric polynomials supported on degeneracy loci of vector bundle maps. The powerful general technique of Gysin maps is also explained in this chapter, and that for the important special case of Grassmannians and flag manifolds. In addition, chapters III and IV touch upon the problem of determining those polynomials that are universally supported on degeneracy loci with an explicit description of their defining ideals. Chapter V gives an application of the technique described in chapter IV to the problem of computing topological Euler characteristics of degeneracy loci and Brill-Noether loci in Jacobians of curves. The geometry of flag manifolds and determinantal formulas for Schubert varieties in the case of general homogeneous spaces associated with various classical groups are treated in chapters VI-VII. Following the correspondence method described in chapter III, degeneracy loci for generalized vector bundles (over homogeneous spaces) are investigated, too. Chapter VIII provides a particularly important application of the general theory developed in chapter VII, namely the computation of cohomology classes of some Brill-Noether loci in Prym varieties.
Although several further applications and open problems are pointed out in the course of chapters I-VIII, the concluding chapter IX is exclusively devoted to the discussion of a huge variety of other applications, related questions, and more open problems.
The following ten appendices A-J serve the purpose of making the text as self-contained as possible, on the one hand, and of indicating some closely related work that has been done since 1995, on the other hand.
Appendix A provides some background material from general intersection theory and the representation theory of degeneracy loci by symmetric polynomials. Appendix B gives a recent improvement of Fulton's theorem on the characterization of vexillary permutations in terms of degeneracy loci. Appendix C points to the relation between degeneracy loci, Demazure's resolution scheme for singularities, and the so-called Bott-Samelson schemes, just so for the sake of completeness. Appendix D compiles the definition and basic properties of Pfaffians, while appendix \(E\) sketches the relevant background material from the group-theoretic approach to Schubert varieties. Appendix F explains a useful Gysin formula for Grassmannian bundles, and appendix G discusses a general criterion for computing the classes of relative diagonals. A special construction for vector bundles, which is well-known and due to D. Mumford, is explained in appendix H (and used in chapter VIII). Appendix I provides a little bit of the relevant representation theory of groups and the combinatorics of Young tableaux, though this is not needed anywhere in the text. Finally, appendix J points to the very recent developments in quantum cohomology, in particular to the significance of the so-called ``quantum double Schubert polynomials'' introduced by \textit{I. Ciocan-Fontanine} and \textit{W. Fulton} (cf.: ``Quantum double Schubert polynomials'', Inst. Mittag-Leffler Report No. 6 (1996-97). Throughout the entire, highly enlightening and inspiring text, the authors have focused on careful explanations of the treated material, with lots of included examples and hints to the original papers. Proofs are mostly just indicated, but always come with precise references to the original papers. The omittance of technical details is to the benefit of the non-expert reader, because this makes the beauty of the entire panorama drawn here more transparent and enjoyable. It should be mentioned that another beautiful, recent introduction to the topic of Schubert varieties and symmetric polynomials is given by the lecture notes ``Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence'' by \textit{L. Manivel} [Cours Spécialisés, No. 3, Paris (1998; Zbl 0911.14023)]. Schubert varieties; quantum double Schubert polynomials; Schur functions; Schubert polynomials; morphisms of vector bundles; degeneracy loci; Grassmannians; flag manifolds; symmetric functions Fulton, W., Pragacz, P.: Schubert Varieties and Degeneracy Loci. Lecture Notes in Mathematics, vol. 1689. Springer, Berlin (1998) Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Determinantal varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], (Equivariant) Chow groups and rings; motives Schubert varieties and degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce the notion of a cominuscule point in a Schubert variety in a generalized flag variety for a semisimple group. We derive formulas expressing the Hilbert series and multiplicity of a Schubert variety at a cominuscule point in terms of the restrictions of classes in torus-equivariant \(K\)-theory and cohomology to that point, generalizing previously known formulas for flag varieties of cominuscule type. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. The formulas for Schubert varieties are special cases of more general formulas valid at generalized cominuscule points of schemes with torus actions. flag variety; Schubert variety; cominuscule; minuscule; equivariant; \(K\)-theory Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Equivariant \(K\)-theory Cominuscule points and Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The jeu-de-taquin-based Littlewood-Richardson rule of \textit{H. Thomas} and \textit{A. Yong} [Algebra Number Theory 3, No. 2, 121--148 (2009; Zbl 1229.05285)] for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, \textit{A. Skovsted Buch} and \textit{M. J. Samuel} [J. Reine Angew. Math. 719, 133--171 (2016; Zbl 1431.19001)] developed a combinatorial theory of `unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to \(K\)-theory. Separately, \textit{P.-E. Chaput} and \textit{N. Perrin} [J. Lie Theory 22, No. 1, 17--80 (2012; Zbl 1244.14036)] used the combinatorics of R. Proctor's `\(d\)-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of \(d\)-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain \(K\)-theoretic Schubert structure constants in the Kac-Moody setting. unique rectification target; jeu de taquin; \(d\)-complete poset; Schubert calculus; Kac-Moody group Rahul Ilango, Oliver Pechenik, Michael Zlatin, Unique rectification in \(d\)-complete posets: towards the \(K\)-theory of Kac-Moody flag varieties, preprint 2018, 34 pages, arXiv:1805.02287. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorics of partially ordered sets, Grassmannians, Schubert varieties, flag manifolds Unique rectification in \(d\)-complete posets: towards the \(K\)-theory of Kac-Moody flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a conjecture by \textit{A. Kuznetsov} and \textit{A. Polishchuk} [J. Eur. Math. Soc. (JEMS) 18, No. 3, 507--574 (2016; Zbl 1338.14021)] on the existence of some particular full exceptional collections in bounded derived categories of coherent sheaves on Grassmann varieties. A. V. Fonarev, On the Kuznetsov--Polishchuk conjecture, Proc. Steklov Inst. Math. 290 (2015), no. 1, 11--25. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds On the Kuznetsov-Polishchuk conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to group theory, Groups with a \(BN\)-pair; buildings, Reflection and Coxeter groups (group-theoretic aspects), Buildings and the geometry of diagrams, Reflection groups, reflection geometries, Geometry of classical groups, Grassmannians, Schubert varieties, flag manifolds, Lie (super)algebras associated with other structures (associative, Jordan, etc.), Proceedings of conferences of miscellaneous specific interest Groups of exceptional type, Coxeter groups and related geometries. Invited articles based on the presentations at the international conference on ``Groups and geometries'', Bangalore, India, December 10--21, 2012. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 space \({\mathcal M}\) of holomorphic mappings of fixed degree \(d\) from a compact Riemannian surface of genus \(g\) to the Grassmannian of complex \(r\)-planes in \(\mathbb{C}^k\) are considered under the assumptions \(g\geq 1\) and \(d>2r(g-1)\). The Uhlenbeck compactification [see \textit{J. Sacks} and \textit{K. Uhlenbeck}, Ann. Math., II. Ser. 113, 1-24 (1981; Zbl 0462.58014) and \textit{J. E. Wolfson}, J. Differ. Geom. 28, No. 3, 383-405 (1988; Zbl 0661.53024)] of \({\mathcal M}\) is shown to have the structure of a projective variety. Also an algebraic compactification of \({\mathcal M}\) is considered, which is obtained as particular case of the Grothendieck Quot Scheme [\textit{A. Grothendieck}, ``Techniques de construction et théorèmes d'existence en géometrie algébrique. IV: Les schémas de Hilbert'', Sémin. Bourbaki 13 (1960/61), No. 221 (1961; Zbl 0236.14003)]. It is shown that there is an algebraic surjection from the latter compactification to the first, which is an isomorphism in case of projective space (i.e. \(r=1)\). This latter compactification is shown to embed into the moduli space of solutions of a certain nonlinear partial differential equation, i.e. a generalized version of the vortex equations [see \textit{S. B. Bradlow} and \textit{G. D. Daskalopoulos}, Int. J. Math. 2, No. 5, 477-513 (1991; Zbl 0759.32013)]. That this can be used to calculate the Gromov invariants [certain intersection numbers, see \textit{M. Gromov} in: Proc. Int. Congr. Math., Berkeley 1986, Vol. 1, 81-98 (1987; Zbl 0664.53016)] is demonstrated for a Riemannian surface of genus one. holomorphic mappings; algebraic compactification; Gromov invariants; intersection numbers A. Bertram, G. Daskalopoulos, and R. Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, J. Amer. Math. Soc., to appear. Riemann surfaces; Weierstrass points; gap sequences, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Algebraic moduli problems, moduli of vector bundles, Complex-analytic moduli problems, Moduli problems for differential geometric structures Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathcal{O}_\lambda\) be a generic coadjoint orbit of a compact semisimple Lie group \(K\). Weight varieties are the symplectic reductions of \(\mathcal{O}_\lambda\) by the maximal torus \(T\) in \(K\). We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of \(\mathcal{O}_\lambda\), and it makes explicit the dependence on \(\lambda\) and a parameter in Lie\((T)^*=:\mathfrak{t}^*\). compact semisimple Lie group; weight varieties; symplectic reductions; equivariant cohomology Momentum maps; symplectic reduction, Grassmannians, Schubert varieties, flag manifolds Cohomology of symplectic reductions of generic coadjoint orbits | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 positroid is a special case of a realizable matroid, that arose from the study of totally nonnegative part of the Grassmannian by \textit{A. Postnikov} [``Total positivity, Grassmannians, and networks'', Preprint, \url{arXiv:math/0609764}]. Postnikov demonstrated that positroids are in bijection with certain interesting classes of combinatorial objects, such as Grassmann necklaces and decorated permutations [\textit{S. Oh}, J. Comb. Theory, Ser. A 118, No. 8, 2426--2435 (2011; Zbl 1231.05061)]. The bases of a positroid can be described directly in terms of the Grassmann necklace and decorated permutation. In this paper, we show that the rank of an arbitrary set in a positroid can be computed directly from the associated decorated permutation using non-crossing partitions. Grassmannian; Grassmann necklace; decorated permutations Combinatorial aspects of matroids and geometric lattices, Combinatorial aspects of partitions of integers, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Grassmannians, Schubert varieties, flag manifolds The rank function of a positroid and non-crossing partitions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 present paper we introduce and study the push pull operators on the formal affine Demazure algebra and its dual. As an application we provide a non-degenerate pairing on the dual of the formal affine Demazure algebra which serves as an algebraic counterpart of the natural pairing on the equivariant oriented cohomology of the complete flag variety induced by multiplication and push-forward to a point. Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Homology and cohomology of homogeneous spaces of Lie groups, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Push-pull operators on the formal affine Demazure algebra and its dual | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 involution Stanley symmetric functions \( \hat{F}_y\) are the stable limits of the analogs of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words and are indexed by the involutions in the symmetric group. By construction, each \( \hat{F}_y\) is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur \(P\)-positive. We give an algorithm to efficiently compute the decomposition of \( \hat{F}_y\) into Schur \(P\)-summands and prove that this decomposition is triangular with respect to the dominance order on partitions. As an application, we derive pattern avoidance conditions which characterize the involution Stanley symmetric functions which are equal to Schur \(P\)-functions. We deduce as a corollary that the involution Stanley symmetric function of the reverse permutation is a Schur \(P\)-function indexed by a shifted staircase shape. These results lead to alternate proofs of theorems of Ardila-Serrano and DeWitt on skew Schur functions which are Schur \(P\)-functions. We also prove new Pfaffian formulas for certain related involution Schubert polynomials. flag variety; Schur \(P\)-function Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Schur \(P\)-positivity and involution Stanley symmetric functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A torus action on a symplectic variety allows one to construct solutions to the quantum Yang-Baxter equations (\(R\)-matrices). For a torus action on cotangent bundles over flag varieties the resulting \(R\)-matrices are the standard rational solutions of the Yang-Baxter equation, well known in the theory of quantum integrable systems. The torus action on the instanton moduli space leads to more complicated \(R\)-matrices, depending additionally on two equivariant parameters \(t_{1}\) and \(t_{2}\). In this paper we derive an explicit expression for the \(R\)-matrix associated with the instanton moduli space. We study its matrix elements and its Taylor expansion in the powers of the spectral parameter. Certain matrix elements of this \(R\)-matrix give a generating function for the characteristic classes of tautological bundles over the Hilbert schemes in terms of the bosonic cut-and-join operators. In particular we rederive from the \(R\)-matrix the well known Lehn's formula for the first Chern class. We
explicitly compute the first several coefficients for the power series expansion of the \(R\)-matrix in the spectral parameter. These coefficients are represented by simple contour integrals of some symmetrized bosonic fields. J.-E. Bourgine and D. Fioravanti, \textit{Omega-deformed Seiberg-Witten relations}, to appear. Groups and algebras in quantum theory and relations with integrable systems, Exactly solvable models; Bethe ansatz, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Yang-Baxter equations, Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory, Grassmannians, Schubert varieties, flag manifolds, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) On the instanton \(R\)-matrix | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Deligne-Lusztig variety \(\overline X (w)\) arising from one of the classical (possibly twisted) groups, we show that the Picard group of \(\overline X(w)\) is generated by the finitely many Deligne-Lusztig subvarieties of \(\overline X (w)\). It is conjectured that this more generally should hold in any codimension, and a good deal of supporting evidence for this claim is presented.
This article is based on Chapter 3 of the author's PhD thesis ``The geometry of Deligne-Lusztig varieties. Higher-dimensional AG codes`` [University of Aarhus (1999)]. Hansen S.H.: Picard groups of Deligne-Lusztig varieties--with a view toward higher codimensions. Beiträge Algebra Geom. 43(1), 9--26 (2002) Grassmannians, Schubert varieties, flag manifolds, Algebraic cycles, (Equivariant) Chow groups and rings; motives, Picard groups Picard groups of Deligne-Lusztig varieties -- with a view toward higher codimensions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 given flag variety, we characterize the primes \(p\) for which there exists a weight \(\lambda\) such that the hard Lefschetz theorem holds for multiplication by \(\lambda\) on the cohomology of the flag variety with coefficients in an infinite field of characteristic \(p\). Grassmannians, Schubert varieties, flag manifolds, Positive characteristic ground fields in algebraic geometry, Reflection and Coxeter groups (group-theoretic aspects) The hard Lefschetz theorem in positive characteristic for the 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 give a complete list of smooth and rationally smooth normalized Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat-Tits building. The particular case of the quasi-minuscule Schubert variety in the quasi-split but non-split form of \(\text{Spin}_8 \) (ramified triality) provides an input needed in the article by He-Pappas-Rapoport classifying Shimura varieties with good or semi-stable reduction. affine Grassmannians; Schubert varieties; intersection cohomology; local models of Shimura varieties Grassmannians, Schubert varieties, flag manifolds, Modular and Shimura varieties, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Smoothness of Schubert varieties in twisted affine Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper continues part I [Transform. Groups 17, No. 4, 953-987 (2012; Zbl 1318.20005)], where the authors introduced nil-DAHA as a certain limit of type \(A_1\) double affine Hecke algebra (DAHA). In the present paper, the authors define a so-called core subalgebra of nil-DAHA and study its properties and representation theory. This subalgebra can be defined either explicitly by generators and relations, or as an intersection of two other subalgebras in nil-DAHA. It is bigraded (in contrast to nil-DAHA and these subalgebras), and invariant under the natural anti-involution. The main results of the paper compare the induced representations of the core subalgebra with certain limits of the corresponding DAHA representations, studied by the first author and \textit{X. Ma} [in Sel. Math., New Ser. 19, No. 3, 737-817 (2013; Zbl 1293.20005) and ibid. 19, No. 3, 819-864 (2013; Zbl 1293.20006)]. double affine Hecke algebras; nil-DAHAs; nonsymmetric Macdonald polynomials; Whittaker functions; core subalgebras; induced representations I. Cherednik and D. Orr. ''One-dimensional nil-DAHA and Whittaker functions II''. Trans form. Groups 18 (2013), pp. 23--59.DOI. Hecke algebras and their representations, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Filtered associative rings; filtrational and graded techniques One-dimensional nil-DAHA and Whittaker functions. 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 Let \(\sigma,\theta\) be commuting involutions of a connected reductive group \(G\) over an algebraically closed field of characteristic zero. Let \(H,K\) be the fixed point groups. The group \(H\times K\) acts on \(G\) by \(((h,k),g)\mapsto hgk^{-1}\). It is shown that the categorical quotient \(G//(H\times K)\) is smooth in the case when \(G\) is semisimple and simply connected. Some results are obtained for more general \(G\) when the quotient is not necessarily smooth. categorical quotients; connected reductive algebraic groups; commuting involutions A. G. Helminck, G. W. Schwarz, Smoothness of quotients associated with a pair of commuting involutions, Canad. J. Math. 56 (2004), no. 5, 945--962. Linear algebraic groups over arbitrary fields, Linear algebraic groups over the reals, the complexes, the quaternions, Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Semisimple Lie groups and their representations Smoothness of quotients associated with a pair of commuting convolutions. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(G(d,n)\) denote the Grassmannian of \(d\)-planes in \(\mathbb{C}^n\) and let \(T\) be the torus \((\mathbb{C}^*)^n/\text{diag}(\mathbb{C}^*)\) which acts on \(G(d,n)\). Let \(x\) be a point of \(G(d,n)\) and let \(\overline{T_X}\) be the closure of the \(T\)-orbit through \(x\). Then the class of the structure sheaf of \(\overline{T_X}\) in the \(K\)-theory of \(G(d,n)\) depends only on which Plücker coordinates of \(x\) are nonzero-combinatorial data known as the matroid of \(x\).
In this paper, we will define a certain map of additive groups from \(K^0(G(d,n))\) to \(\mathbb{Z}[t]\). Letting \(g_x(t)\) denote the image of \((-1)^{n-\dim T_X}[{\mathcal O}_{\overline T_X}]\), \(g_x\) behaves nicely under the standard constructions of matroid theory, such as direct sum, two-sum, duality and series and parallel extensions. We use this invariant to prove bounds on the complexity of Kapranov's Lie complexes [\textit{M. Kapranov}, Chow quotients of Grassmannians. I: Gelfand, Sergej (ed.) et al., I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in functional analysis held at Moscow University, Russia, September 1993. Providence, RI: American Mathematical Society. Adv. Sov. Math. 16(2), 29--110 (1993; Zbl 0811.14043)]. \textit{P. Hacking}, \textit{S. Keel} and \textit{J. Tevelev's} very stable pairs [J. Algebraic Geom. 15, No. 4, 657--680 (2006; Zbl 1117.14036)] and the author's tropical linear spaces when they are realizable in characteristic zero [(*) SIAM J. Discrete Math. 22, No. 4, 1527--1558 (2008; Zbl 1191.14076)]. Namely, in characteristic zero, a Lie complex or the underlying \((d-1)\)-dimensional scheme of a very stable pair can have at most \({(n-i-1)!\over (d-i)!(n- d- i)!(i- 1)!}\) strata of dimensions \(n-i\) and \(d-i\), respectively. This prove the authors \(f\)-vector conjecture, from [(*)], in the case of a tropical linear space realizable in characteristic 0. \(K\)-theory; Grassmannian; matroid; tropical linear space; polytope; valuation Speyer, D, A matroid invariant via the \(K\)-theory of the Grassmannian, Adv. Math., 221, 882-913, (2009) , Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) A matroid invariant via the \(K\)-theory of the Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author studies holomorphic immersions \(f:\widetilde{X}\to G\) of the universal cover \(\widetilde X\) of a compact Riemann surface \(X\) into the Grassmannian \(G = G(n,\mathbb{C}^{2n})\). The mapping \(f\) is assumed to satisfy a nondegeneracy condition, and to be equivariant with respect to a homomorphism \(\rho:\Gamma\to \text{GL}(\mathbb{C}^{2n})\), where \(\Gamma\) is the decktransformation group acting on \(\widetilde{X}\), \(X = \widetilde{X}/\Gamma\), and \(\rho\Gamma\) acts on \(G\). The author proves that such immersions \(f\) (up to equivalence) are in bijective correspondence with the holomorphic differential operators (up to equivalence) of order 2 on a rank \(n\) vector bundle over \(X\) such that the symbol of the operator is an isomorphism. holomorphic equivariant immersions; differential operators; Grassmannians; holomorphic differential operators Global submanifolds, Vector bundles on curves and their moduli, Grassmannians, Schubert varieties, flag manifolds Differential operators and immersions of a Riemann surface into a Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper is organized as follows. Three equivalent definitions of the combinatorial flag variety \(\Omega_n\) are given in Section 1, and properties of \(\Omega_n\) are discussed in Section 2. Section 3 contains all necessary definitions concerning (ordinary) matroids, flag matroids, and vocabulary for translation to the language of Coxeter matroids. Equivalence of the first two definitions of \(\Omega_n\) becomes clear at this stage. Section 4 discusses Coxeter metrics on chamber complexes and Gaussian schemes. Section 5 contains the proofs of main results about \(\Omega_n\). combinatorial flag variety; matroids; flag matroids; Coxeter matroids; Coxeter metrics; chamber complexes; Gaussian schemes Borovik, A., Gelfand, I., White, N.: Combinatorial flag varieties. J. Comb. Theory (A) 91, 111--136 (2000) Combinatorial aspects of matroids and geometric lattices, Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Combinatorial flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author identifies certain multiplicity spaces corresponding to the cohomology ring of a \(\text{GL}_N\) flag variety with multiplicity spaces of irreducible \(sl_n\)- modules in a fusion product. More specifically, if the \(m\)-tuple \(\mu=(\mu_1,\dots ,\mu _m)\) represents the highest weight of a symmetric power representation and \(\sum \mu _i =N\), then the cohomology ring \(R\) of the flag variety \(F_\mu\) of \(\text{GL}_N\) can be decomposed into irreducible \(S_N\) (symmetric group) modules as \(R=\bigoplus _{| \lambda| \leq n} W_\lambda\otimes M_{\lambda ,\mu }\), where \(W_\lambda\) is a Specht module and \(M_{\lambda ,\mu}\) is a graded multiplicity space. The author identifies the \(M_{\lambda ,\mu}\) with multiplicity spaces of irreducible \(sl _n\)-modules in a fusion produce under the special condition that the evaluation modules in the fusion product are symmetric power representations. fusion product; flag manifold; Kostka polynomial; cohomology ring Rinat Kedem, Fusion products of \?\?_{\?} symmetric power representations and Kostka polynomials, Quantum theory and symmetries, World Sci. Publ., Hackensack, NJ, 2004, pp. 88 -- 93. Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Fusion products, cohomology of \(\text{GL}_N\) flag manifolds, and Kostka 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 This paper introduces a program for a general theory of hypergeometric functions of several variables based on the view that the natural domain of the general hypergeometric function is a certain line bundle over a Grassmann manifold. In this approach the general hypergeometric function appears as a Radon transform.
The paper surveys basic definitions and propositions, as well as problems to be considered in a series of future publications devoted to the program. hypergeometric functions of several variables; Grassmann manifold; Radon transform I. M. Gel'fand,The general theory of hypergeometric functions, Sov. Math. Dokl.,33, n. 3 (1986), pp. 573--577. Classical hypergeometric functions, \({}_2F_1\), Grassmannians, Schubert varieties, flag manifolds General theory of hypergeometric 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 There is a remarkable formula for the principal specialization of a type A Schubert polynomial as a weighted sum over reduced words. Taking appropriate limits transforms this to an identity for the backstable Schubert polynomials recently introduced by \textit{T. Lam} et al. [``Back stable Schubert calculus'', Preprint, \url{arXiv:1806.11233}]. This note identifies some analogues of the latter formula for principal specializations of Schubert polynomials in classical types B, C, and D. We also describe some more general identities for Grothendieck polynomials. As a related application, we derive a simple proof of a pipe dream formula for involution Grothendieck polynomials. Schubert polynomials; Grothendieck polynomials; Coxeter systems; reduced words Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Principal specializations of Schubert polynomials in classical types | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [Duke Math. J. 168, No. 18, 3437--3527 (2019; Zbl 1439.14142)], \textit{K. Rietsch} and \textit{L. Williams} relate cluster structures and mirror symmetry for type A Grassmannians \(\operatorname{Gr}(k, n)\), and use this interaction to construct Newton-Okounkov bodies and associated toric degenerations. In this article we define a cluster seed for the Lagrangian Grassmannian, and prove that the associated Newton-Okounkov body agrees up to unimodular equivalence with a polytope obtained from the superpotential defined by \textit{C. Pech} and \textit{K. Rietsch} on the mirror Orthogonal Grassmannian in [``A Landau-Ginzburg model for Lagrangian Grassmannians, Langlands duality and relations in quantum cohomology'', Preprint, \url{arXiv:1304.4958}]. cluster algebra; Newton-Okounkov body; plabic graph Grassmannians, Schubert varieties, flag manifolds, Mirror symmetry (algebro-geometric aspects), Cluster algebras, Toric varieties, Newton polyhedra, Okounkov bodies Towards cluster duality for Lagrangian and orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials After giving an explicit description of all the non vanishing Dolbeault cohomology groups of ample line bundles on Grassmannians, the author gives two series of vanishing theorems for ample vector bundles on a smooth projective variety. They imply a part of a conjecture by \textit{W. Fulton} and \textit{R. Lazarsfeld} [in: Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math. 862, 26--92 (1981; Zbl 0484.14005 )] about the connectivity of some degeneracy loci. Schur diagrams; partitions Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Vanishing theorems in algebraic geometry, Projective techniques in algebraic geometry Vanishing theorems and degeneracy loci in small corang | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We apply down operators in the affine nil Coxeter algebra to yield explicit combinatorial expansions for certain families of noncommutative \(k\)-Schur functions. This yields a combinatorial interpretation for a new family of \(k\)-Littlewood-Richardson coefficients. symmetric function; \(k\)-Schur function; Young tableaux; \(k\)-core; Coxeter group; Littlewood-Richardson coefficient; affine Grassmannian Berg, C.; Saliola, F.; Serrano, L., Combinatorial expansions for families of non-commutative \textit{k}-Schur functions Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Combinatorial expansions for families of noncommutative \(k\)-Schur functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In his fundamental paper [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV: Les schemes de Hilbert'', Sémin. Bourbaki, exp. 221 (1961; Zbl 0236.14003)], \textit{A. Grothendieck} introduced the so called Hilbert scheme, which is an object which parametrizes all projective subschemes of the projective space with fixed Hilbert polynomial.
In the paper under review, the authors present a more general object, called the multigraded Hilbert scheme, parametrizing all homogeneous ideals with fixed Hilbert function in a graded polynomial ring \(S\).
As in the case of Hilbert schemes, the multigraded Hilbert scheme is a projective scheme (quasi-projective if the grading of \(S\) is not positive), and, when the ground ring is a field, its tangent space at a point corresponding to an ideal \(I\) has a simple description: it is canonically isomorphic to the degree \(0\) piece of \(\Hom(I,S/I)\).
The construction of the multigraded Hilbert scheme is obtained in a great generality, and it enables the authors to prove a conjecture from \textit{D. Bayer}'s thesis [The division algorithm and the Hilbert scheme, Ph.D. thesis, Harvard University (1982)] on equations defining the Hilbert scheme, and to construct a natural morphism from the toric Hilbert scheme to the toric Chow variety, resolving Problem 6.4 appearing in the paper of \textit{B. Sturmfels} [The geometry of A-graded algebras, preprint, \texttt{http://arXiv.org/abs/math.AG/9410032}]. graded polynomial ring; Hilbert function; Chow morphism M. Haiman - B. Sturmfels, Multigraded Hilbert schemes. J. Algebraic Geom., 13 (4) (2004, pp. 725-769. Zbl1072.14007 MR2073194 Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Grassmannians, Schubert varieties, flag manifolds Multigraded 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 In this article, orbits of real reductive groups G acting on real affine varieties X are investigated. The main result is a generalization of previous results of Kempf-Ness in the complex case [cf. \textit{G. Kempf} and \textit{L. Ness} in Algebraic geometry, Proc. Summer. Meet., Copenhagen 1978, Lect. Notes Math. 732, 233-243 (1979; Zbl 0407.22012)]. For example, it is proved that the points of a closed G-orbit with minimal distance to zero in a real linear representation X form an orbit under a maximal compact subgroup K of G.
Several applications are given, among them a new proof of the G- properness of the algebraic quotient \(X\to X/G\) in the real Hausdorff- topology (i.e. the image of a closed G-stable subset of X is closed in X/G) and a reduction of G-conjugacy of reductive subgroups of G to K- conjugacy. orbits of real reductive groups acting on real affine varieties R. W. Richardson and P. J. Slodowy, ''Minimum vectors for real reductive algebraic groups,'' J. London Math. Soc., vol. 42, iss. 3, pp. 409-429, 1990. Group actions on varieties or schemes (quotients), Linear algebraic groups over the reals, the complexes, the quaternions, Real algebraic and real-analytic geometry, Grassmannians, Schubert varieties, flag manifolds Minimum vectors for real reductive algebraic 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 Let \(G\) be a connected semisimple algebraic group over an algebraically closed field of characteristic zero, let \(B\) be a Borel subgroups of \(G\), and let \(T\) be a maximal torus of \(B\). Consider the action of \(T\) on \(X=G/B\) by left translations. Let \(\chi\) be a strictly dominant character of \(B\) and let \(X_{L_\chi}^{\text{ss}}\) be the semistable locus of \(X\) with respect to the sheaf \(L_\chi\) on \(X\) corresponding to \(\chi\). Then there is a categorical quotient \(X_{L_\chi}^{\text{ss}}/\!\!/T\).
In [\textit{V. S. Zhgoon}, Izv. Math. 71, No. 6, 1105--1122 (2007); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 71, No. 6, 29--46 (2007; Zbl 1146.14025)], the rank of \({\text{Pic}}(X_{L_\chi}^{\text{ss}}/\!\!/T)\) has been computed in the case where \(G\) does not contain simple factors of type \({\text A}_{\ell}\). In the present paper this rank is computed in the case where \(G={\text{SL}}(\ell+1)\). It is also shown that \({\text{Pic}}(X_{L_\chi}^{\text{ss}}/\!\!/T)\) is torsion free.
[For part I, cf. Izv. Math. 71, No. 6, 1105--1122 (2007); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 71, No. 6, 29--46 (2007; Zbl 1146.14025).] semisimple algebraic group; quotient; torus; action Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds Variation of Mumford quotients by torus actions on full flag varieties. II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over \(\mathbb{C}\) and point counts over \(\mathbb{F}_q)\) to Khovanov-Rozansky homology of the associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational \(q,t\)-Catalan numbers. Via the curious Lefschetz property, this implies the \(q,t\)-symmetry and unimodality properties of rational \(q,t\)-Catalan numbers. We show that the \(q,t\)-symmetry phenomenon is a manifestation of Koszul duality for category \(\mathcal{O}\), and discuss relations with equivariant derived categories of flag varieties, and open Richardson varieties. positroid varieties; \(q,t\)-Catalan numbers; HOMFLY polynomial; Khovanov-Rozansky homology; mixed Hodge structure; equivariant cohomology; Koszul duality Grassmannians, Schubert varieties, flag manifolds, Polynomial rings and ideals; rings of integer-valued polynomials Positroids, knots, and \(q,t\)-Catalan numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study those subvarieties of the variety \(X[M]\) of planar normal sections on a natural embedding of a flag manifold \(M\), that are projective spaces. When \(M=G/T\) is the manifold of complete flags of a compact simple Lie group \(G\), those subspaces of the tangent space \(T[T](M)\), invariant by the torus action, give rise to real projective spaces of \(X[M]\). planar normal section; flag manifold; projective space; variety Dal Lago, W., García, A.N., Sánchez, C.U.: Maximal projective subspaces in the variety of planar normal sections of a flag manifold. Geom. Dedicata 75, 219--233 (1999) Differential geometry of homogeneous manifolds, Grassmannians, Schubert varieties, flag manifolds Maximal projective subspaces in the variety of planar normal sections of a flag manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(X\), \(Y\) be finite sets and \(T\) a set of functions \(X \rightarrow Y\) which we will call ``tableaux''. We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such tableau complexes have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions [BP79]. In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are labeled by Buch's set-valued semistandard tableaux. One vertex decomposition of this ``Young tableau complex'' parallels Lascoux's transition formula for vexillary double Grothendieck polynomials [La01, La03]. Consequently, we obtain formulae (both old and new) for these polynomials. In particular, we present a common generalization of the formulae of Wachs [Wa85] and Buch [Bu02], each of which implies the classical tableau formula for Schur polynomials. Young tableaux; simplicial complex; semistandard Young tableaux; Lascoux transition formula A. Knutson, E. Miller, and A. Yong. ''Tableau complexes''. Isr. J. Math. 163.1 (2008), pp. 317-- 343.DOI. 12 Cara Monical Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Tableau complexes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 projective variety \(M\), denote by \(\text{Hol}(M)\) the space of all holomorphic maps from the Riemann sphere \(\mathbb{P}^1\) into \(M\). It is known that \(\text{Hol}(M)\) can be given the structure of a quasiprojective variety and hence it has the homotopy type of a finite CW complex. In general \(\text{Hol}(M)\) is not connected. Computing the exact homotopy type of various components in \(\text{Hol}(M)\) in terms of well known spaces therefore presents itself as an interesting problem. In the present paper, the authors study the case of \(M =\text{Gr}(n, m)\), the Grassmann manifold of \(n\)-dimensional complex linear subspaces in \(\mathbb{C}^{n+m}\). For a holomorphic map \(f:\mathbb{P}^1 \to\text{Gr}(n, m)\), the degree \(d\geq 0\) is the multiplicative factor for the induced map between the second homology groups (\(\mathbb{Z}\) for both spaces). It turns out that two maps in \(\text{Hol}(\text{Gr}(n, m))\) are in the same connected component if and only if they have the same degree. Hence the components of \(\text{Hol}(\text{Gr}(n, m))\) are determined by the degree \(d\) of maps. Denote by \(\text{Hol}_d(\text{Gr}(n, m))\) the component of maps of degree \(d\). Clearly \(\text{Hol}_0(\text{Gr}(n, m))=\text{Gr}(n, m)\) are the constant maps. The space \(\text{Hol}_1(\text{Gr}(n, m))\), which is the main object of a study in the paper, is the component of linear maps. In one of the main theorems, a very complete description is given of \(\text{Hol}_1(\text{Gr}(n, m))\) as a sphere bundle over a flag manifold. For the quadric Grassmann manifold \(\text{Gr}(2, 2)\) the paper contains a detailed study of the homology of \(\text{Hol}_1(\text{Gr}(2, 2))\). The paper is well structured and adequately referenced. holomorphic maps; Grassmannians; characteristic classes Loop spaces, Grassmannians, Schubert varieties, flag manifolds, Real rational functions, Sphere bundles and vector bundles in algebraic topology The space of linear maps into a Grassmann manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 an explicit closed formula, written as a sum of Pfaffians, which describes each equivariant Schubert class for the Grassmannian of isotropic subspaces in a symplectic vector space. symplectic Grassmannian; Schubert calculus; Giambelli formula; Pfaffian; signed permutation; \(k\)-strict partition Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Equivariant Giambelli formula for the symplectic Grassmannians-Pfaffian sum formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\subset {\mathbb{P}}^ n\) be a k-dimensional complex manifold whose osculating spaces \(S_ p\) have everywhere maximal dimension \(r(p)=\binom {p+k}{k}-1\). These spaces define Gauss maps \(G_ p: V\to\) Grass\((r(p),n).\) Let \(\Omega_ p\) be the pull back, via \(G_ p\), of the standard Kähler form of the grassmannian. - The author finds some links among the \(\Omega_ p's\); they express \(\Omega_ 1\) in terms of \(\Omega_ 0\) and the \({\bar \partial}\)-cohomology classes of \(\Omega_ p\) in terms of those of \(\Omega_ 0\) and \(\Omega_ 1\). In case \(k=n-1\) the author obtains a stronger result, holding under local maximality hypotheses for r(p).
Finally a characterization of the Veronese surface is given as the only one smooth surface in \({\mathbb{P}}^ 5\) such that the 2-osculating space at every point has dimension 5. This result, which seems to go back to Terracini, has been generalized by \textit{W. Fulton}, \textit{S. Kleiman}, \textit{R. Piene}, and \textit{H. Tai} [Bull. Soc. Math. Fr. 113, 205-210 (1985; Zbl 0581.14037)] to the case of the p-th Veronese embedding of \({\mathbb{P}}^ k\). osculating spaces; Veronese surface Projective techniques in algebraic geometry, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Projective differential geometry, Grassmannians, Schubert varieties, flag manifolds On Gauß mappings defined by osculating spaces at a complex variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a case of a positivity conjecture of
\textit{L. C. Mihalcea} and \textit{R. Singh} [``Mather classes and conormal spaces of Schubert varieties in cominuscule spaces'', Preprint, \url{arXiv:2006.04842}],
concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassmannian \(LG(n,2n)\). Combined with work of
\textit{P. Aluffi} et al. [``Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells'', Preprint, \url{arXiv:1709.08697}],
this further implies the positivity of the Mather classes for Schubert varieties in \(LG(n,2n)\), which Mihalcea-Singh had verified for the other cominuscule spaces of classical Lie type. Building on the work of Boe and Fu, we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan-Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, for \(LG(n,2n)\) the Euler obstructions \(e_{y,w}\) may vanish for certain pairs \((y,w)\) with \(y\le w\) in the Bruhat order. Our combinatorial description allows us to classify all the pairs \((y,w)\) for which \(e_{y,w}=0\). Restricting to the big opposite cell in \(LG(n,2n)\), which is naturally identified with the space of \(n\times n\) symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification. local Euler obstructions; Schubert stratification; Lagrangian Grassmannian; tree labelings Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Trees, Local complex singularities, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) Euler obstructions for the Lagrangian Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a complex vector space equipped with a nondegenerate symmetric bilinear form. Let \(X\) denote the flag variety for the even orthogonal group, which parameterizes flags of isotropic subspaces in \(V\). In the paper under review, the author develops a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of \(X\). These polynomials are applied to understand the structure of the Gillet-Soulé arithmetic Chow ring of \(X\). Actually, the author studies an extra relation which comes from the vanishing of the top Chern class of the maximal isotropic subbundle of the trivial vector bundle over \(X\) and he computes the natural arithmetic Chern numbers on \(X\). Finally, he shows that these arithmetic Chern numbers are all rational. Schubert polynomial; orthogonal flag variety; arithmetic Chow ring; arithmetic Chern number Harry Tamvakis, Schubert polynomials and Arakelov theory of orthogonal flag varieties, Math. Z. 268 (2011), no. 1-2, 355 -- 370. Grassmannians, Schubert varieties, flag manifolds, Arithmetic varieties and schemes; Arakelov theory; heights, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Schubert polynomials and Arakelov theory of orthogonal flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The article under review studies quiver Grassmannians associated with finite dimensional indecomposable representations of a Kronecker quiver. A finite dimensional representation of a Kronecker quiver on the category of vector spaces is given by two finite dimensional complex vector spaces, \(M_{1}\) and \(M_{2}\) and linear maps \(m_{a}\), \(m_{b}\) between them. Given \(e_{1}\) and \(e_{2}\), define
\[
Gr_{(e_{1},e_{2})}=\{(N_{1},N_{2})\in Gr_{e_{1}}(M_{1})\times Gr_{e_{2}}(M_{2}):m_{a}(N_{1})\subset M_{2},m_{b}(N_{1})\subset N_{2}\}
\]
which is a projective variety, in general non smooth, where \(Gr_{e}(V)\) denotes the Grassmannian of \(e\)-dimensional vector subspaces of \(V\). We call these varieties quiver Grassmannians.
For the case \(M_{1}=M_{2}=\mathbb{C}^{n}\), \(m_{a}=Id\) and \(m_{b}\) an indecomposable nilponent Jordan block, the representation is denoted by \(R_{n}\) and is called regular indecomposable. The article concentrates on this particular case, \(X=Gr_{(e_{1},e_{2})}(R_{n})\), because all the rest of indecomposable representations either have the same quiver, or are easier cases previously studied in the literature.
There is an action of a \(1\)-dimensional torus on \(X\) which provides a stratification
\[
X_{s}\subseteq\cdots\subseteq X_{1}\subseteq X_{0}=X
\]
into closed subvarieties \(X_{k}\simeq Gr_{(e_{1}-k,e_{2}-k)}(R_{n-2k})\), where each one is the singular locus of the previous one and the difference between two consecutive ones is smooth quasiprojective. Bialynicki-Birula's theorem is applied to show that \(X\) has a cellular decomposition and, relations between the dimensions of the cells through the Euler form associated to the Kronecker quiver, are used to compute the Betti numbers and the Poincare polynomials of the quiver Grassmannians, from which the Euler characteristics are recovered.
As an application, the authors give a new geometric realization for the atomic basis of a cluster algebra \(\mathcal{A}_{Q}\) of type \(A_{1}^{(1)}\) (which are \(\mathbb{Z}\)-subalgebras of the field of rational functions associated to quivers \(Q\) without loops and \(2\)-cycles) by using the Caldero-Chapoton map (cf. [\textit{P. Caldero} and \textit{F. Chapoton}, Comment. Math. Helv. 81, No. 3, 595--616 (2006; Zbl 1119.16013)]) associating cluster monomials of the cluster algebra to rigid \(Q\)-representations. The authors propose a truncation of the Caldero-Chapoton map to realize the extra elements of the atomic basis. quiver Grassmannians; cluster algebras; Caldero-Chapoton map; quiver representations Cerulli Irelli, G., Esposito, F.: Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras. Algebra \& Number Theory. arXiv:1003.3037v2 (2010) Cluster algebras, Representation theory of lattices, Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Projective techniques in algebraic geometry Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A Riemannian manifold \((M,\rho )\) is called Einstein if the metric \(\rho \) satisfies the condition \(\text{Ric}(\rho)= c\cdot \rho\) for some constant \(c\). This paper is devoted to the investigation of \(G\)-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces \(G/H\) of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds \(\text{SO}(n)/\text{SO}(l)\). Furthermore, we show that for any positive integer \(p\) there exists a Stiefel manifold \(\text{SO}(n)/\text{SO}(l)\) that admits at least \(p\) \(\text{SO}(n)\)-invariant Einstein metrics. Riemannian manifolds; homogeneous spaces; Einstein metrics; Stiefel manifolds Arvanitoyeorgos, A., Dzhepko, V.V., and Nikonorov, Yu.G., Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups, Canad. J. Math., 2009, vol. 61, no. 6, pp. 1201--1213. Special Riemannian manifolds (Einstein, Sasakian, etc.), Differential geometry of homogeneous manifolds, Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Homogeneous complex manifolds, Homology and cohomology of homogeneous spaces of Lie groups Invariant Einstein metrics on some homogeneous spaces of classical Lie 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 Landau problem on the flag manifold \(\mathbb F_2= \text{SU}(3)/U(1)\times U(1)\) is analyzed from an algebraic point of view. The involved magnetic background is induced by two \(U(1)\) Abelian connections. In quantizing the theory, we show that the wave functions, of a nonrelativistic particle living on \(\mathbb F_2\), are the SU(3) Wigner -functions satisfying two constraints. Using the \(\mathbb F_2\) algebraic and geometrical structures, we derive the Landau Hamiltonian as well as its energy levels. The lowest Landau level (LLL) wave functions coincide with the coherent states for the mixed \(SU(3)\) representations. We discuss the quantum Hall effect for a filling factor \(\nu=1\), where the obtained particle density is constant and finite for a strong magnetic field. In this limit, we also show that the system behaves like an incompressible fluid. We study the semiclassical properties of the system confined in LLL. These will be used to discuss the edge excitations and construct the corresponding Wess-Zumino-Witten action. flag manifold; geometry; quantization; quantum Hall effect Daoud, Mohammed; Jellal, Ahmed, Quantum Hall effect on the flag manifold \(F_2\), Int. J. Mod. Phys. A, 23, 3129-3154, (2008) Many-body theory; quantum Hall effect, Geometry and quantization, symplectic methods, Grassmannians, Schubert varieties, flag manifolds Quantum Hall effect on the flag manifold \(\mathbb F_2\) | 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 Using the method of degenerating a Grassmannian into a toric variety, we calculate formulas for the dimensions of the eigenspaces of the action of an \(n\)-dimensional torus on a Grassmannian of planes in an \(n\)-dimensional space. toric variety; Grassmannian; Poincare-Hilbert series Witaszek, J.: The Degeneration of the Grassmannian into a Toric Variety and the Calculation of the Eigenspaces of a Torus Action. arXiv:1209.3689 [math.AG] (2012) Toric varieties, Newton polyhedra, Okounkov bodies, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves The degeneration of the Grassmannian into a toric variety and the calculation of the eigenspaces of a torus action | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 [Algebr. Geom. Topol. 2, 665--741 (2002; Zbl 1002.57006)], the author defined for any positive integer \(n\) a ring \(H^n\) so that one can associate to a tangle a special complex of \(H^n-H^m\) bimodules as invariant. Let \({\mathcal B}_{n,n}\) be the variety of complete flags in \({\mathbb C}^{2n}\) fixed by a special nilpotent operator given by two Jordan blocks of size \(n\). A special case of a result by \textit{C. de Concini} and \textit{C. Procesi} [Invent. Math. 64, 203--219 (1981; Zbl 0475.14041)] is the explicit computation of the integral cohomology ring of \({\mathcal B}_{n,n}\) by generators and relations.
The main result of the present paper is the computation of the centre of \(H^n\), and the observation that the centre of \(H^n\) and the cohomology ring of \({\mathcal B}_{n,n}\) coincide. In the proof the author realizes \(H^n\) as a product of certain homology spaces, and the centre as an equalizer of certain natural maps between these. Then, the author identifies these homology spaces and corresponding homology classes of the variety \({\mathcal B}_{n,n}\). In a second part the author mentions that there is a weak action of the braid group on the derived category of \(H^n\)-modules analogous to the one defined by the author and Seidel, as well as the one defined by Rouquier and the reviewer in similar situations. The author then shows that this action gives a natural permutation action of the symmetric group on the cohomology of \({\mathcal B}_{n,n}\). Springer variety; tangles; braid group action Khovanov, M., Crossingless matchings and the cohomology of \((n, n)\) Springer varieties, Commun. Contemp. Math., 6, 2, 561-577, (2004) Invariants of knots and \(3\)-manifolds, Grassmannians, Schubert varieties, flag manifolds Crossingless matchings and the cohomology of \((n,n)\) Springer 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 article surveys recent developments on Hessenberg varieties, emphasizing some of the rich connections of their cohomology and combinatorics. In particular, we will see how hyperplane arrangements, representations of symmetric groups, and Stanley's chromatic symmetric functions are related to the cohomology rings of Hessenberg varieties. We also include several other topics on Hessenberg varieties to cover recent developments. Hessenberg varieties; flag varieties; cohomology; hyperplane arrangements; representations of symmetric groups; chromatic symmetric functions. Grassmannians, Schubert varieties, flag manifolds, Configurations and arrangements of linear subspaces, Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry A survey of recent developments on Hessenberg varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Y\) be a Noetherian scheme over a field of characteristic zero, \(\mathcal E\) a vector bundle of rank \(n\) over \(Y\), and \(G=\text{Gr}(k,\mathcal E)\xrightarrow {p} Y\) the bundle of Grassmannians associated with \(\mathcal E\). The author shows that the natural homomorphism \(K_0(G)\otimes K_i(Y)\to K_i(G)\) is an isomorphism, and describes two bases of \(K_0(G)\) over \(K_0(Y)\). A similar result is obtained for the bundle of Grassmannians \(p: G\to Y\) determined by an element \(\phi \in H^1_{\text{et}}(Y, \mathrm{PGL}(n))\).
These results represent a generalization of results by \textit{D. Quillen} [in: Algebraic K-theory I, Proc. Conf. Battelle Inst. 1972, Lecture Notes Math. 341, 85--147 (1973; Zbl 0292.18004)] for \(k=1\). The methods used by the author differ substantially from those of D. Quillen. \(K_0\); bundle of Grassmannians I. A. Panin, Algebraic \?-theory of Grassmannian manifolds and their twisted forms, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 71 -- 72 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 2, 143 -- 144. Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory in geometry Algebraic K-theory of Grassmann varieties and their convoluted forms | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a companion paper of [\textit{A. Braverman} et al., Compos. Math. 157, No. 8, 1724--1765 (2021; Zbl 1481.14025)]. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of \(\text{SO}(N-1,\mathbb{C} [\![t]\!])\)-equivariant perverse sheaves on the affine Grassmannian of \(\text{SO}_N\). We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh. Satake equivalence; affine Grassmannian; supergroups Geometric Langlands program (algebro-geometric aspects), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras Orthosymplectic Satake equivalence | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 obtain upper and lower bounds for the number of \((n,k)\)-MDS linear codes over \({\mathbb F}_q\), and an asymptotic formula for this number. \textit{A. N. Skorobogatov} [Linear codes, strata of Grassmannians, and the problems of Segre. In: Coding theory and algebraic geometry, Proc. Int. Workshop, Luminy/Fr. 1991, Lect. Notes Math. 1518, 210-223 (1992; Zbl 0809.94022)] showed that this number is related to the number of \({\mathbb F}_q\)-rational points of the Grassmannian of all \(k\)-dimensional linear subspaces of \({\mathbb F}_q^n\) for which all the Plücker coordinates are nonzero. He also showed that this number is the number of inequivalent representations of the uniform matroid \(U_{k,n}\) over \({\mathbb F}_q\), and it is related to three classical questions of B. Segre on \(n\)-arcs and the rational normal curve in \({\mathbb P}^{k-1}\). The authors obtain bounds on the number of rational points on hyperplane sections of the Grassmannian by using results on the parameters of Grassmann codes obtained by \textit{D. Yu. Nogin} [Codes associated to Grassmannians. In: Pellikaan, R. (ed.) et al., Arithmetic, geometry, and coding theory. Proc. Int. Conf. CIRM, Luminy, France, 1993, Walter de Gruyter, 145-154 (1996; Zbl 0865.94032)]. The authors also obtain results on geometric properties of these hyperplane sections of the Grassmannian. In the final section, they give some tables containing numerical values for the lower and upper bounds on the number of MDS codes for certain small values of \(n, k\), and \(q\). Grassmannian; MDS code; Grassmann code; hyperplane section; uniform matroid; arcs in projective space Ghorpade S.R., Lachaud G.: Hyperplane sections of Grassmannians and the number of MDS linear codes. Finite Fields Appl. 7, 468--506 (2001) Geometric methods (including applications of algebraic geometry) applied to coding theory, Grassmannians, Schubert varieties, flag manifolds, Finite ground fields in algebraic geometry Hyperplane sections of Grassmannians and the number of MDS linear codes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 define Schubert Eisenstein series as sums like usual Eisenstein series but with the summation restricted to elements of a particular Schubert cell, indexed by an element of the Weyl group. They are generally not fully automorphic. We will develop some results and methods for \(\mathrm{GL}_3\) that may be suggestive about the general case. The six Schubert Eisenstein series are shown to have meromorphic continuation and some functional equations. The Schubert Eisenstein series \(E_{s_1s_2}\) and \(E_{s_2s_1}\) corresponding to the Weyl group elements of order three are particularly interesting: at the point where the full Eisenstein series is maximally polar, they unexpectedly become (with minor correction terms added) fully automorphic and related to each other. Schubert Eisenstein series; Schubert cell; meromorphic continuation; Weyl group Other groups and their modular and automorphic forms (several variables), Grassmannians, Schubert varieties, flag manifolds Schubert Eisenstein series | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 ``Soergel theory'' for Bruhat-constructible perverse sheaves on the flag variety \(G /B\) of a complex reductive group \(G\), with coefficients in an arbitrary field \(\Bbbk\). Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a ``multiplicative'' coinvariant algebra and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of ``Soergel modules'' over this algebra. We also obtain a description of the derived category of unipotently \(T\) monodromic \(\Bbbk\) sheaves on \(G /U\) (where \(U\), \(T \subset B\) are the unipotent radical and the maximal torus), as a monoidal category, in terms of coherent sheaves on the formal neighborhood of the base point in \(T_\Bbbk^\vee \times_{(T_\Bbbk^\vee)^W} T_\Bbbk^\vee\), where \(T_\Bbbk^\vee\) is the \(\Bbbk\)-torus dual to \(T\). Derived categories, triangulated categories, Representation theory for linear algebraic groups, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Hecke algebras and their representations A topological approach to Soergel 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 We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to \textit{N. Bergeron} and \textit{F. Sottile} [Duke Math. J. 95, 373--423 (1998; Zbl 0939.05084)] in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive the construction of Bergeron and Sottile in a purely combinatorial way, relating it to the construction of Schubert polynomials in terms of rc-graphs. Schubert polynomial; Bruhat order; Littlewood-Richardson coefficient Lenart, C., Sottile, F.: Skew Schubert polynomials. Proceedings of the American Mathematical Society 131(11), 3319--3328 (2003) Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Combinatorics of partially ordered sets Skew 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 For the original paperback edition see [Zbl 1297.14002]. Seshadri C S, Introduction to the theory of standard monomials, Brandeis Lecture Notes 4 (1985) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial), Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) Introduction to the theory of standard monomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 class of \(F\)-regular varieties consists of projective algebraic varieties defined over a field in positive characteristic such that all the ideals in their homogeneous coordinates are tightly closed. In this note, the authors obtain the global (resp. strong regularity) \(F\)-regularity for the class of varieties referred to as ``large Schubert varieties'' (see section four) proved in theorem 4.3 (resp. corollary 4.1) which are the main results of the paper. They extend the previous results to the case that the field has characteristic zero, thereby introducing the notion of strong (resp. global) \(F\)-regular type to affine (resp. projective) varieties. As a consequence of his result, the large Schubert varieties in any equivariant embedding of \(G\) are normal and Cohen-Macaulay as has been proved by e.g. \textit{M. Brion} and \textit{P. Polo} [Represent. Theory 4, 97--126 (2000; Zbl 0947.14026)] and more recently by \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, 1245--1318 (2005; Zbl 1089.14007)].
The paper is organized as follows. In section two, they recall the notion of strong \(F\)-regularity defined over an algebraically closed field of positive characteristic. In section three, they recall the notion of global \(F\)-regularity for projective varieties and in section 3.1 the notion of Frobenius splitting for a scheme of finite type is introduced. In particular, globally \(F\)-regular varieties are Frobenius split. Also the notion of being stably Frobenius split along an effective cartier divisor is introduced. They collect a number of standard facts in lemma 3.1. In section 3.2, they state a well known result connecting global \(F\)-regularity, Frobenius splitting and strong \(F\)-regularity and in corollary 3.2 state sufficient conditions for a projective variety to be globally \(F\)-regular. In section four, the notion of an equivariant embedding of a connected reductive algebraic group \(G\) over an algebraically closed field is introduced.
The authors define for an equivariant embedding \(X\) the concept of large Schubert varieties parametrized by the Weyl group of \(G\). In section 4.2, they prove proposition 4.2 which states that if \(X\) is a non-singular equivariant embedding of \(G\) over a field of characteristic \(p\) , then \(X\) is Frobenius split along \((p-1) \partial X\), where \(\partial X\) is the boundary of the equivariant embedding of \(X\) . In section 4.3, they prove the main results, namely theorem 4.3 and corollary 4.1. For the proof of the last they use lemma 3.1 and proposition 4.2 for the case that \(X\) is non-singular. For the case of singular \(X\) they use lemma 4.1, lemma 3.2 and corollary 3.2. For the proof of corollary 4.1, they use theorem 4.3 and prove that the local rings of the large Schubert varieties are strongly \(F\)-regular. In section 4.4, they assume from the beginning that the field is of characterisitic zero and prove in theorem 4.4, the analogous version of theorem 4.3 and of corollaries 4.1, 4.2 by using the well known notions of strongly (resp. globally) \(F\)-regular type. The paper ends with corollary 4.3 showing that the large Schubert varieties have rational singularities. group actions on varieties or schemes (quotients); Grassmannians; flag manifolds Michel Brion and Jesper Funch Thomsen, \?-regularity of large Schubert varieties, Amer. J. Math. 128 (2006), no. 4, 949 -- 962. Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure \(F\)-regularity of large Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\widetilde G_{n,k}\) be the Grassmann manifold of oriented \(k\)-dimensional subspaces in \(\mathbb R^n\) (\(1\leq k\leq n-k\)). It is well known that \(\widetilde G_{n,k}\) is a closed orientable \(k(n-k)\)-dimensional manifold. It was shown in [\textit{V. Ramani} and \textit{P. Sankaran}, Proc. Indian Acad. Sci., Math. Sci. 107, No. 1, 13--19 (1997; Zbl 0884.55002)] that if \((n,k)\neq(m,l)\), \(l\geq2\) and \(k(n-k)=l(m-l)\), then every map \(f:\widetilde G_{n,k}\rightarrow\widetilde G_{m,l}\) has degree zero.
In the paper under review, the authors consider the Grassmann manifold \(\widetilde I_{2n,k}\) (where \(1\leq k\leq n\)) of oriented isotropic \(k\)-dimensional subspaces of \(\mathbb R^{2n}\) (which is equipped with the standard symplectic form) and obtain an analogous result: if \(\widetilde I_{2n,k}\) and \(\widetilde I_{2m,l}\) are two distinct ``oriented isotropic Grassmannians'' of the same dimension, and if \(k,l\geq2\), then \(\deg f=0\) for all maps \(f:\widetilde I_{2n,k}\rightarrow\widetilde I_{2m,l}\). Moreover, they establish that the same conclusion holds for all maps of the form \(\widetilde I_{2n,k}\rightarrow\widetilde G_{m,l}\) and \(\widetilde G_{m,l}\rightarrow\widetilde I_{2n,k}\), provided that \(\dim\widetilde I_{2n,k}=\dim\widetilde G_{m,l}\) and \(k,l\geq2\). The authors actually prove that there is no ring monomorphism between rational cohomology rings of the manifolds in question and use the following fact (easily obtained from Poincaré duality): if \(f:M\rightarrow N\) is a nonzero degree map between oriented closed connected manifolds of the same dimension, then \(f^*:H^*(N;\mathbb Q)\rightarrow H^*(M;\mathbb Q)\) is a monomorphism. isotropic Grassmann manifolds; Brouwer degree; characteristic classes Degree, winding number, Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Homology and cohomology of homogeneous spaces of Lie groups, Homology of classifying spaces and characteristic classes in algebraic topology Degrees of maps between isotropic Grassmann 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 [For the entire collection see Zbl 0663.00011.]
Let \(\phi: x\to {\mathbb{C}}{\mathbb{P}}^ N\) be a smooth projective variety of dimension n embedded in \({\mathbb{C}}{\mathbb{P}}^ N\), T be the holomorphic tangent bundle of X and \(H=\phi^*{\mathcal O}_{{\mathbb{P}}^ N}(1)\) the hyperplane line bundle. \(\phi\) defines a Gauss map of X into the Grassmannian \(Gr(n+1,N+1)\). Let E the pull-back of the universal bundle of \(Gr(n+1,N+1)\) by the Gauss map. Then there is the Euler sequence \(0\to 1\to E\otimes H\to T\to 0.\) The author gets from this exact sequence some information about Chern classes of T and E: in particular some inequalities between Chern numbers of complete intersections. tangent bundle; hyperplane line bundle; Gauss map; Grassmannian; Chern classes; Chern numbers of complete intersections Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry, Characteristic classes and numbers in differential topology, Complete intersections A class of symmetric functions and Chern classes of projective 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 associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's program on ``freeness'' for rational points of bounded height on Fano varieties. Manin conjecture; freeness; Grassmannian; equidistribution; lattice Rational points, Asymptotic results on counting functions for algebraic and topological structures, Lattice points in specified regions, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over the reals, the complexes, the quaternions Equidistribution and freeness on Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A combinatorial mutation, introduced in [\textit{M. Akhtar} et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 8, Paper 094, 17 p. (2012; Zbl 1280.52014)], is a transformation of the Newton polytope of a Laurent polynomial undergoing mutation; which can be considered as a local transformation for lattice polytopes. The theory of combinatorial mutations was further developed by Higashitani from a combinatorial viewpoint, and later has been used to study combinatorial mutation equivalence classes of Newton-Okounkov bodies of flag varieties [\textit{N. Fujita} and \textit{A. Higashitani}, Int. Math. Res. Not. 2021, No. 12, 9567--9607 (2021; Zbl 07503553)].
For the Grassmannian \(\text{Gr}(k, n)\), a matching field is a map taking each Plücker variable to a permutation, and can be interpreted as a choice of initial term for the corresponding Plücker form. They were introduced by \textit{B. Sturmfels} and \textit{A. V. Zelevinsky} [Adv. Math. 98, No. 1, 65--112 (1993; Zbl 0776.13009)] to study the Newton polytope of a product of maximal minors of a generic matrix and have proved to be a useful tool in many contexts.
In this paper, the authors use combinatorial mutations to find relations between matching field polytopes. In fact, each matching field \(\Lambda\) admits a toric ideal \(J_\Lambda\) with associated polytope \(P_\Lambda\), and they show that understanding the polytope associated to a matching field is equivalent to finding toric degenerations of the Grassmannian as the following:
Theorem 1. Let \(\Lambda\) be a coherent matching field for the Grassmannian \(\text{Gr}(k, n)\) with polytope \(P_\Lambda\). If \(P_\Lambda\) is obtained from the Gelfand-Tsetlin polytope by a sequence of combinatorial mutations, then \(\Lambda\) gives rise to a toric degeneration of \(\text{Gr}(k, n)\).
In particular, this paper investigate the block diagonal matching fields which are examples of coherent matching fields with particularly simple description. It shows that all block diagonal matching field polytopes are related by a sequence of combinatorial mutations:
Theorem 2. Any pair of block diagonal matching field polytopes can be obtained from one another by a sequence of combinatorial mutations such that all intermediate polytopes are matching field polytopes.
The matching fields associated to the intermediate polytopes can be thought of as interpolating between the block diagonal matching fields. As a result, the authors obtain a large family of toric degenerations for the Grassmannian given by matching fields. toric degeneration; combinatorial mutation; matching field; polytope; Grassmannian Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Toric varieties, Newton polyhedra, Okounkov bodies, Tropical geometry, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Combinatorial mutations and block diagonal 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 We extend the group-theoretic construction of local models of Pappas and Zhu [\textit{G. Pappas} and \textit{X. Zhu}, Invent. Math. 194, No. 1, 147--254 (2013; Zbl 1294.14012)] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when \(p\geqslant 5\). We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group. algebraic groups; affine Grassmannians; nearby cycles; Shimura varieties Levin, B., \textit{local models for Weil-restricted groups}, Compos. Math., 152, 2563-2601, (2016) Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, Grassmannians, Schubert varieties, flag manifolds Local models for Weil-restricted 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 We provide a construction of Saito primitive forms for Gepner singularity by studying the relation between Saito primitive forms for Gepner singularities and primitive forms for singularities of the form \(F_{k, n} = \sum_{i = 1}^n x_i^k\) invariant under the natural \(S_n\)-action. Frobenius manifolds; Saito primitive form; Saito structures; singularity theory Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Deformation quantization, star products, Topological field theories in quantum mechanics, Grassmannians, Schubert varieties, flag manifolds Primitive forms for Gepner singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathbb{C}}\) be the field of complex numbers (or any algebraically closed field of zero characteristic), V an n-dimensional \({\mathbb{C}}\)-space and \(G=G(k,V)\) the Grassmannian of k-dimensional subspaces of V. The author describes the derived category \(D^ b_{coh}(G)\) of coherent sheaves on G. The main results asserts that \(D^ b_{coh}(G)\) is equivalent with the triangulated category of homotopical category of bounded complexes of graduated \(\bigwedge (V^*)\)-modules, consisting of finite coproducts of modules \(M_{\alpha}\). The modules \(M_{\alpha}\) are canonical associated to Young's diagrams \(\alpha\) with at most k-rows and most n-k-columns. derived category of coherent sheaves on Grassmannian; Young diagram Kapranov M.M., On the derived category of coherent sheaves on Grassmann manifolds, Math. USSR-Izv., 1985, 24(1), 183--192 Grassmannians, Schubert varieties, flag manifolds, Derived categories, triangulated categories, Exterior algebra, Grassmann algebras, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Holomorphic bundles and generalizations On the derived category of coherent sheaves on Grassmann 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 As following from the celebrated results of \textit{A. A. Beilinson} [Funct. Anal. Appl. 12, 214-216 (1979; Zbl 0424.14003)] and \textit{M. M. Kapranov} [Math. USSR, Izv. 24, 183-192 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No.1, 192--202 (1984; Zbl 0564.14023)], the Grassmannian \(Gr(n,r)\), and more generally the partial flag varieties of type A have a remarkable property that their bounded derived categories of coherent sheaves have a tilting object, given by direct sum of certain Schur powers of the tautological bundles. Such a tilting object induces an equivalence from the bounded derived category of coherent sheaves to the derived category of representations of a finite directed quiver. This provides a powerful tool to study the moduli spaces of sheaves on such varieties \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No.180, 515--530 (1994; Zbl 0837.16005)].
In this paper, the author constructs a class of smooth projective varieties, called quiver flag varieties, as a generalization of the partial flag varieties of type A. A quiver flag variety \(\mathcal{M}_\theta(Q,r)\) is constructed as the fine moduli space of \(\theta\)-stable representations with dimension vector \(r\) of certain finite directed quiver \(Q\). By considering the filtration of \(Q\) by directed subquivers, the author shows that \(\mathcal{M}_\theta(Q,r)\) arises as a tower of Grassmann bundles. This description leads to two interesting results. First, combined with a result of \textit{Yi Hu} and \textit{S. Keel} [Mich. Math. J. 48, Spec. Vol., 331-348 (2000; Zbl 1077.14554)], the author shows that \(\mathcal{M}_\theta(Q,r)\) is a Mori dream space. Secondly, using certain vanishing theorems, the author shows that the derived category of \(\mathcal{M}_\theta(Q,r)\) has a tilting object. At the end, the author introduces the Plucker embedding of a quiver flag variety by using multigraded linear series. Craw, A, Quiver flag varieties and multigraded linear series, Duke Math. J., 156, 469-500, (2011) Fine and coarse moduli spaces, Representations of quivers and partially ordered sets, Derived categories, triangulated categories, Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies Quiver flag varieties and multigraded linear series | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 nilpotent orbit \({\mathcal O}_ u\) in a complex semisimple Lie algebra gives rise to a collection of cone bundles on the flag variety, by taking the closed components of its preimage under Springer's resolution of singularities. Using the generalization of inverse Chern classes of vector bundles to Segre classes of cone bundles due to Fulton and the third author, we attach to each such cone bundle a characteristic class in the cohomology of the flag variety, which is interpreted as a harmonic polynomial on the Cartan subalgebra. Using the intersection homology approach to the study of nilpotent varieties as by \textit{W. Borho} and \textit{R. MacPherson} in Astérisque 101-102, 23-74 (1983; Zbl 0576.14046) and C. R. Acad. Sci., Paris, Sér. I 292, 707-710 (1981; Zbl 0467.20036) we show that this collection of polynomials transforms under the action of the Weyl group according to Springer's irreducible representation \(\rho_ u\) which is usually constructed from \({\mathcal O}_ u\) by quite different means. Weyl group representations; cone bundles on the flag variety; characteristic class; cohomology of the flag variety; intersection homology; action of the Weyl group Borho, W.; Brylinksky, J. -L; Macpherson, J. -L; Macpherson, R.: Springer's Weyl group representations through characteristic classes of cone bundles. (1986) (Co)homology theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Characteristic classes and numbers in differential topology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Springer's Weyl group representations through characteristic classes of cone 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 The author gives an analogue of a theorem of \textit{W.-L. Chow} in [Ann. Math. (2) 50, 32--67 (1949; Zbl 0040.22901)] for infinite dimensional vector spaces. He considers vector spaces \(X\) over division rings, the lattice \({\mathcal G}(X)\) of vector subspaces, and the graph \(\Gamma(X)\), whose set of vertices is \({\mathcal G}(X)\) with pairs \((U,V)\) of adjacent subspaces \(U\), \(V\) as edges; \(U\) is called adjacent to \(V\) (\(U\sim V\)) iff \(\dim U /(U\cap V) = \dim V / (U\cap V)\).
\({\mathcal C} \subset {\mathcal G}(X)\) is called a component iff \({\mathcal C}\) induces a connected component in the graph \(\Gamma(X)\).
For components \({\mathcal C} \subset {\mathcal G}(X)\) and \({\mathcal D} \subset {\mathcal G}(Y)\), where \(X\) and \(Y\) are vector spaces, a bijective map \(\Phi : {\mathcal C} \longrightarrow {\mathcal D}\) which preserves \(\sim\) in both directions is called isomorphism.
First, the author gives a careful description of components \({\mathcal C}\) and the sublattices \({\mathcal C}_\pm\) generated by them in \({\mathcal G}(X)\).
The main result of the paper describes isomorphisms in the case of infinite diameter of the connected components. First, it is shown that for diameters \(\geq 2\) any isomorphism \(\Phi : {\mathcal C} \longrightarrow {\mathcal D}\) of components may be uniquely extended to an isomorphism or antiisomorphism \(\rho : {\mathcal C}_\pm \longrightarrow {\mathcal D}_\pm\) of lattices. Furthermore, any such \(\rho\) which maps \({\mathcal C}\) bijectively onto \({\mathcal D}\) restricts to an isomorphism of components. Second, for components of infinite diameter such lattice isomorphisms or antiisomorphisms are characterized by two bijective semilinear maps between certain subspaces of \(X\) and \(Y\).
Let \({\mathcal G}_{\alpha,\beta}(X)\) denote the set of subspaces of dimension \(\alpha\) and of codimension \(\beta\) such that \(\alpha + \beta = \dim X\). Chow's theorem shows in the finite dimensional case that \(\alpha < \beta\) makes \({\mathcal G}_{\alpha,\beta}(X)\) a component, whose automorphisms may be described by bijective semilinear maps on \(X\). From his main result the author derives that an analogous description for infinite dimensional \(X\) and infinite \(\alpha\) and \(\beta\) is not possible.
Continuing the work of \textit{A. Blunck} and \textit{H. Havlicek} [Discrete Math. 301, No. 1, 46--56 (2005; Zbl 1083.51001)], the author applies his results to infinite dimensional vector spaces \(X\) and \(Y\), to characterize complementarity preserving bijections of \({\mathcal G}_{\alpha,\alpha}(X)\longrightarrow {\mathcal G}_{\gamma,\gamma}(Y)\) by certain bijective semilinear maps \(X\longrightarrow Y\). adjacent subspaces; Grassmann graph; complementary subspaces; distant graph; semilinear mapping Plevnik, L., Top stars and isomorphisms of Grassmann graphs, Beitr. algebra geom., 56, 703-728, (2015) Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Distance in graphs, Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.), Grassmannians, Schubert varieties, flag manifolds, Linear transformations, semilinear transformations, Linear preserver problems, Homomorphism, automorphism and dualities in linear incidence geometry, Incidence structures embeddable into projective geometries Top stars and isomorphisms of Grassmann graphs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 different degeneration processes appearing in algebraic Lie theory. They study toric degenerations of flag varieties \(G/B\) of complex semisimple Lie algebras, the totality encoded in tropical flag varieties with respect to Plücker embeddings. The authors also investigate the PBW-type degenerations of quantized enveloping algebras \(U_q(\mathfrak{g})\) associated with complex semisimple Lie algebras \(\mathfrak{g}=\text{Lie}(G)\).
One link between these two degenerations is via a class of polyhedral cones called quantum degree cones, which depend on the complex simple Lie algebra \(\mathfrak{g}\) and a choice of a reduced decomposition of the longest element in the Weyl group of \(\mathfrak{g}\). In fact, the negative part of the quantized enveloping algebra is generated by the quantum PBW root vectors with respect to some noncommutative straightening relations. The quantum degree cone is a certain Gröbner fan, where the monomial ordering is encoded in the reduced decomposition of the longest element in the Weyl group of \(\mathfrak{g}\).
The first main result is an embedding of the quantum degree cone into the negative tight monomial cone of the Langlands dual Lie algebra (Corollary 1, page 824). The second main result is a proof for reduced decompositions adapted to a Dynkin quiver (Theorem 4, page 830). quiver representations; canonical bases; quantized enveloping algebras; quantum degree cones, PBW degenerations; Plucker embeddings; \(K\)-theoretic cones; tropical flag variety; maximal prime cones Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Foundations of tropical geometry and relations with algebra Cones from quantum groups to tropical 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 Numerical algebraic geometry has a close relationship to intersection theory from algebraic geometry. We deepen this relationship, explaining how rational or algebraic equivalence gives a homotopy. We present a general notion of witness set for subvarieties of a smooth complete complex algebraic variety using ideas from intersection theory. Under appropriate assumptions, general witness sets enable numerical algorithms such as sampling and membership. These assumptions hold for products of flag manifolds. We introduce Schubert witness sets, which provide general witness sets for Grassmannians and flag manifolds. intersection theory; numerical algebraic geometry; Schubert variety; witness set Geometric aspects of numerical algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds General witness sets for numerical algebraic geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(K\) be a number field, \({\mathbf e}_1,\ldots ,{\mathbf e}_n\) the standard basis of \(K^n\), and \(K^m\) the linear subspace of \(K^n\) generated by \({\mathbf e}_1,\ldots , {\mathbf e}_m\). Further, let \(\alpha =(\alpha_1,\ldots ,\alpha_d)\) be a tuple of integers with \(1\leq\alpha_1<\cdots <\alpha_d\leq n\). For \(B>0\), denote by \(M(\alpha ,B)\) the number of \(d\)-dimensional linear subspaces \(S\) of \(K^n\) of height at most \(B\) satisfying the conditions \(\dim_K(S\cap K^{\alpha_i})=i\) for \(i=1,\ldots , d\). Here the height of \(S\) is some sort of twisted multiplicative height of the exterior product of the vectors from a basis of \(S\). In other words, \(M(\alpha ,B)\) is the number of points of heights at most \(B\) in a Schubert cell. Under the hypothesis \(\alpha_1>1\), the author proves that
\[
M(\alpha ,B)\gg\ll B^{c_1(\alpha )}(\log B)^{c_2(\alpha )-1}\quad\text{for } B\gg 1
\]
where \(c_1(\alpha )=\max\{ \alpha_i-2i+d+1:\, 1\leq i\leq d\}\), \(c_2(\alpha )\) is the number of \(i\) for which the maximum is assumed, and the constants implied by the Vinogradov symbols depend on \(n\) and \(K\). It is conjectured that there is an asymptotic formula of the shape
\[
M(\alpha ,B)= a(\alpha ,K)B^{c_1(\alpha )}(\log B)^{c_2(\alpha )-1}+ O(B^{c_1(\alpha )}(\log B)^{c_2(\alpha )-2})\quad \text{as \(B\to\infty\).}
\]
The author obtains such a formula with slightly larger error term in the special case \(c_1(\alpha )=\alpha_d+1-d\) and \(c_2(\alpha )=1\).
The author derives these results from a counting result of his for rational points of bounde height on flag varieties [Compos. Math. 88, No. 2, 155--186 (1993; Zbl 0806.11030)], using partial summation techniques. heights; Schubert varieties J.L. Thunder, Points of bounded height on Schubert varieties , Inter. J. Number Theory, Heights, Varieties over global fields, Grassmannians, Schubert varieties, flag manifolds Points of bounded height on Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Given a projective variety \(X\subseteq \mathbb{P}V\) we define its dual projective variety as the Zariski closure \(X^\vee \subset \mathbb{P}V^*\) of the set of hyperplanes \(H\subset \mathbb{P}V\) containing the tangent space of \(X\) at some smooth point. When \(X^\vee\) is a hypersurface we denote its defining polynomial by \(\Delta_X\) and call it a hyperdeterminant or discriminant. With a decomposition \(V=A\oplus B\) of the vector space \(V\) we can define the ``restriction'' of \(\Delta_X\) to \(A^*\) and denote it as Res\((\Delta_X,A^*).\)
The first result in this paper gives sufficient conditions on projective varieties \(X\subseteq \mathbb{P}V\) and \(Y\subseteq \mathbb{P}A\) such that if \(X^\vee\) and \(Y^\vee\) are hypersurfaces then \(\Delta_Y^m|\)Res\((\Delta_X,A^*).\) They use it then to establish division conditions involving discriminants of varieties constructed from Lie algebras.
For a semi-simple Lie algebra \(\mathfrak{g}\) the adjoint variety \(X^{ad}_G\) is the projectivization of the highest weight orbit in \(\mathfrak{g}\) for the adjoint action of the Lie group \(G.\) It is known that dual varieties of adjoint varieties are hypersurfaces and its discriminant is denoted \(\Delta_G\) instead of \(\Delta_{X_G^{ad}}.\) Therefore, given a \(\mathbb{Z}_k\)-grading \(\mathfrak{g}=\mathfrak{g_0}\oplus\mathfrak{g_1}\oplus \cdots \oplus \mathfrak{g}_{k-1}\) when can consider Res\((\Delta_G,\mathfrak{g}_s^*)\) for \(s=0,\dots,k-1.\)
In this context the main result of the present paper is to describe the discriminants of the Grassmanians \(\mathrm{Gr}(3,9)\) and \(\mathrm{Gr}(4,8)\) in terms of restrictions of the discriminants from exceptional Lie algebras: \([\Delta_{\mathrm{Gr}(3,9)}^2]=[\mbox{Res}(\Delta_{E_8}\bigwedge^3\mathbb{C}^9)]\) and \([\Delta_{\mathrm{Gr}(4,8)}]=[\mbox{Res}(\Delta_{E_7}\bigwedge^4\mathbb{C}^8)].\) The authors also express \(\Delta_{\mathrm{Gr}(3,9)}\) and \(\Delta_{\mathrm{Gr}(4,8)}\) as polynomials in the fundamental invariants of \(\bigwedge^3\mathbb{C}^9\) and \(\bigwedge^4\mathbb{C}^8.\) hyperdeterminants; dual varieties; sparse resultants; interpolation; Lie algebra; Grassmannians; adjoint varieties; quantum information Grassmannians, Schubert varieties, flag manifolds, Lie algebras and Lie superalgebras, Vector and tensor algebra, theory of invariants, Toric varieties, Newton polyhedra, Okounkov bodies, Multilinear algebra, tensor calculus Hyperdeterminants from the \(E_8\) discriminant | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If \(H\) is a finite-dimensional vector space, it is well-known that the irreducible representations of the unitary group \(U(H)\) can be realized geometrically as the natural action of \(U(H)\) on the holomorphic sections of a holomorphic line bundle over a space of flags. In this paper, the authors consider a separable Hilbert space \(H\) and construct a manifold of flags in \(H\) over which there exist holomorphic line bundles that are similar to the finite-dimensional ones. As an application they show how such constructions occur in quantum field theory and in the framework of integrable systems. finite-dimensional vector space; irreducible representations; unitary group; holomorphic line bundle; flags; Hilbert space; manifold of flags; quantum field theory; integrable systems A. G. Helminck and G. F. Helminck, \(H_k\)-fixed distributionvectors for representations related to \(\mathfrak p\)-adic symmetric varieties, To appear. Infinite-dimensional Lie groups and their Lie algebras: general properties, Analysis on \(p\)-adic Lie groups, Grassmannians, Schubert varieties, flag manifolds, Other completely integrable equations [See also 58F07], Analysis on other specific Lie groups, Infinite-dimensional Lie (super)algebras Holomorphic line bundles over Hilbert flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(S_n\) be the symmetric group on \([n] := \{1,\dots,n\}\). An inversion of \(\pi \in S_n\) is a pair \(i, j\in [n]\) such that \(i < j\) and \(\pi (i) > \pi (j\). The number of inversions of \(\pi\) is its length, denoted \(\mathcal l(\pi)\). The Bruhat order on \(S_n\) is a partial ordering on \(S_n\), graded by length. It arises in geometry as the face poset for the Schubert decomposition of the variety of complete flags in \(\mathbb C^n\).
Its cover relations have the form \(\pi s_{ij}< \pi\) where \(s_{ij} := (i, j)\) is a transposition such that \((\pi) = (\pi s_{ij} ) + 1\). The maximal element of the Bruhat order, written in row notation, is \(\pi_{\mathrm{top}} = [n, n - 1, \dots, 1]\) of length \(r := \binom{n}{2}\) and the smallest element is the identity permutation \(id = [1, 2, \dots, n]\) of length \(0\). In the Bruhat order, each maximal chain has the form \(id = \pi_0 < \pi_1< \dots < \pi_r = \pi_{\mathrm{top}}\). Let \({\alpha}_1, \dots, {\alpha}_n\) be indeterminates. Define the weight of a covering \(\pi s_{ij} < \pi\) with \(i < j\) to be \({\alpha}_i + {\alpha}_{i+1} +\cdots + {\alpha}_{j-1}\), and then define the weight of a maximal chain to be the product of the weights of its cover relations. In a result that extends to all Weyl groups, Stembridge shows that the sum of the weights of the maximal chains is \[ \frac{\binom{n}{2}!}{1^{n-1} 2^{n-2}\dots (n-1)^1}\prod_{1\leq i<j\leq n}{\alpha}_i + {\alpha}_{i+1} +\cdots + {\alpha}_{j-1}. \] The totally nonnegative Grassmannian \(Gr(k, n)_{\geq 0}\) is a subset of points in the real Grassmannian \(Gr(k, n)\) which have all nonnegative Plücker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of Postnikov's positroid cell decomposition of\(Gr(k, n)_{\geq 0}\).
In this paper the authors provide a Stembridge-like formula for the circular Bruhat orderusing the hook formula and Young tableoux for rectangles. circular Bruhat order; \(k\)-Bruhat order; positroid; totally nonnegative Grassmannians Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Factorials, binomial coefficients, combinatorial functions Counting weighted maximal chains in the circular Bruhat order | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article relates the Grassmannian complexes of geometric configurations to the tangent to the Bloch-Suslin complex and to the tangent to Goncharov's motivic complex. By means of morphisms, we bring the geometry of configurations in tangent groups, \(T\mathcal{B}_2 (F)\) and \(T\mathcal{B}_3 (F)\) that produce commutative diagrams. To show the commutativity of diagrams, we use combinatorial techniques that include permutations in symmetric group \(S_6\). We also create analogues of the Siegel's cross-ratio identity for the truncated polynomial ring \(F[\epsilon]_{\nu}.V\) affine spaces; cross-ratio; tangent complex; odd permutation; symmetric group Grassmannians, Schubert varieties, flag manifolds, Exterior algebra, Grassmann algebras, Higher algebraic \(K\)-theory Geometry of configurations in tangent 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 In this paper we show that there are explicit Yang-Baxter (YB) maps with Darboux-Lax representation between Grassman extensions of algebraic varieties. Motivated by some recent results on noncommutative extensions of Darboux transformations, we first derive a Darboux matrix associated with the Grassmann-extended derivative nonlinear Schrödinger (DNLS) equation, and then we deduce novel endomorphisms of Grassmann varieties, which possess the YB property. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional YB maps on invariant leaves, related to the Grassmann-extended NLS and DNLS equations. We consider their vector generalisations. Yang-Baxter maps; Grassmann algebraic varieties; Grassmann extensions of Yang-Baxter maps; Grassmann extensions of Darboux transformations; noncommutative extensions of Yang-Baxter maps Grahovski, G.G.; Konstantinou-Rizos, S.; Mikhailov, A.V., Grassmann extension of Yang-Baxter maps, J. phys. A, math. theor., 49, (2016) Groups and algebras in quantum theory and relations with integrable systems, Grassmannians, Schubert varieties, flag manifolds, Yang-Baxter equations, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems, NLS equations (nonlinear Schrödinger equations), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Grassmann extensions of Yang-Baxter maps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(X\subset\mathbb{P}(V)\) be a complex projective spherical \(G\)-variety, where \(G\) is a classical group and \(V\) a finite-dimensional \(G\)-module. Generalizing the case of toric varieties, one can associate to \(X\) an integral convex polytope \(\Delta(X)\) such that the Hilbert polynomial of \(X\) is the Ehrhardt polynomial of \(\Delta(X)\). The polytope \(\Delta(X)\) is fibred over the moment polytope of \(X\) with the Gelfand-Tsetlin polytopes as fibres. This polytope was defined by Okounkov based on results of Brion.
It is proved that, for \(G=\text{Sp}(2n)\), the variety \(X\) can be degenerated, by a flat deformation, to the toric variety corresponding to the polytope \(\Delta(X)\). Firstly, it is known that any spherical variety can be degenerated to a horospherical variety, i.e., to a variety where the stabilizer of a point in the dense \(G\)-orbit contains a maximal unipotent subgroup. Secondly, the main result of the paper states that the homogeneous coordinate ring of a horospherical variety has a (finite) SAGBI basis. Here the homogeneous ring of \(X\) is embedded into a Laurent polynomial algebra and the SAGBI basis is considered with respect to a natural term order. Moreover, the author shows that the semigroup of initial monomials is the semigroup of integral points in the cone over the polytope \(\Delta(X)\). The proof is based on the result of \textit{A.~Okounkov} [in: Kirillov's seminar on representation theory, Am. Math. Transl., Ser.~2, 181, 231--244 (1998; Zbl 0920.20032)]. More general results on degenerations of spherical varieties to toric varieties may be found in [\textit{V.~Alexeev} and \textit{M.~Brion}, Sel. Math., New Ser. 10, 453--478 (2004; Zbl 1078.14075)]. Gelfand-Tsetlin polytope Kaveh, K, SAGBI bases and degeneration of spherical varieties to toric varieties, Michigan Math. J., 53, 109-121, (2005) Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies SAGBI bases and degeneration of spherical varieties to toric varieties | 0 |
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