Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, and Sphericity Consequences

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

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Polynomial identities related to special Schubert varieties

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-021-00496-6 ◽

2021 ◽

Author(s):

Francesca Cioffi ◽

Davide Franco ◽

Carmine Sessa

Keyword(s):

Arbitrary Number ◽

Schubert Variety ◽

Decomposition Theorem ◽

Point Of View ◽

Explicit Description ◽

Intersection Cohomology ◽

Schubert Varieties ◽

Polynomial Identities ◽

Poincaré Polynomial

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

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Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

Forum of Mathematics Sigma ◽

10.1017/fms.2020.60 ◽

2021 ◽

Vol 9 ◽

Author(s):

Alex Chirvasitu ◽

Ryo Kanda ◽

S. Paul Smith

Keyword(s):

Elliptic Curve ◽

Symmetric Power ◽

Canonical Homomorphism ◽

Coordinate Ring ◽

Characteristic Variety ◽

Closed Subvariety ◽

Coordinate Rings ◽

Cubic Hypersurfaces ◽

Elliptic Algebras ◽

Twisted Homogeneous Coordinate Ring

Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

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On a conjecture of Pappas and Rapoport about the standard local model for GL_d

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2020-0030 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Dinakar Muthiah ◽

Alex Weekes ◽

Oded Yacobi

Keyword(s):

Type A ◽

Positive Answer ◽

Local Model ◽

Schubert Varieties ◽

Shimura Varieties ◽

Local Models ◽

Full Generality ◽

Frobenius Splitting ◽

Affine Grassmannians ◽

Main Ideas

AbstractIn their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

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A decomposition formula for representations

Nagoya Mathematical Journal ◽

10.1017/s0027763000002543 ◽

1987 ◽

Vol 107 ◽

pp. 63-68 ◽

Cited By ~ 2

Author(s):

George Kempf

Keyword(s):

Irreducible Representation ◽

General Formula ◽

Parabolic Subgroup ◽

Characteristic Zero ◽

Maximal Torus ◽

Reductive Group ◽

Irreducible Representations ◽

Levi Subgroup ◽

Know How ◽

Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

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