Vertices of Schubitopes

Fan, Neil J. Y.; Guo, Peter L.

doi:10.1016/j.jcta.2020.105311

Cited by 9 publications

(6 citation statements)

References 14 publications

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“…As a preliminary result (Theorem 4.7), we describe P w λ by its vertices, P w λ " convtvλ : v P W, v ď wu, generalizing the case for w " w 0 . We also note that when G " GL n and w is arbitrary, this vertex description follows from work of Fan and Guo on Schubitopes [9], answering a conjecture of Monical, Tokcan, and Yong [22,Conjecture 3.13] on key polynomials. With this in hand, we can now state the primary result of this paper (c.f.…”

Section: Introductionmentioning

confidence: 71%

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Weight polytopes and saturation of Demazure characters

Besson¹,

Jeralds²,

Kiers³

2022

Preprint

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For G a reductive group and T Ă B a maximal torus and Borel subgroup, Demazure modules are certain B-submodules, indexed by elements of the Weyl group, of the finite irreducible representations of G. In order to describe the T -weight spaces that appear in a Demazure module, we study the convex hull of these weights -the Demazure polytope. We characterize these polytopes both by vertices and by inequalities, and we use these results to prove that Demazure characters are saturated, in the case that G is simple of classical Lie type. Specializing to G " GL n , we recover results of Fink, Mészáros, and St. Dizier, and separately Fan and Guo, on key polynomials, originally conjectured by Monical, Tokcan, and Yong.

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Section: Introductionmentioning

confidence: 71%

“…As an immediate corollary of the agreement of P w λ and P ď w , we obtain that the vertices of P w λ are precisely tuλ : u ď wu. This was conjectured in type A r by Monical, Tokcan, and Yong [22, Conjecture 3.13], and first proven (in type A r ) by Fan and Guo [9].…”

Section: Convex Hullsmentioning

confidence: 76%

Weight polytopes and saturation of Demazure characters

Besson¹,

Jeralds²,

Kiers³

2022

Preprint

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“…Let rk S denote the rank function of a Schubert matroid SM n (S). Fan and Guo [10,Theorem 3.3] provided an efficient algorithm to compute rk S (T ) for any T ⊆ [n]. It is well known that the matroid polytope P(M) defined in (1.1) associated to a matroid M = ([n], B) is a generalized permutohedron perametrized by the rank function of M, see, for example, Fink, Mészáros and St. Dizier [15].…”

Section: The Rank Function Rkmentioning

confidence: 99%

“…We will use S, r or r(S) interchangeably with no further clarification. For example, let S = {2, 6, 7, 10} ⊆ [10], then I(S) = (0, 1, 0 3 , 1 2 , 0 2 , 1) and r(S) = (1, 1, 3, 2, 2, 1).…”

Section: Introductionmentioning

confidence: 99%

On the Ehrhart Polynomial of Schubert Matroids

Fan¹,

Li²

2021

Preprint

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In this paper, we give a formula for the number of lattice points in the dilations of Schubert matroid polytopes. As applications, we obtain the Ehrhart polynomials of uniform and minimal matroids as simple special cases, and give a recursive formula for the Ehrhart polynomials of (a, b)-Catalan matroids. Ferroni showed that uniform and minimal matroids are Ehrhart positive. We show that all sparse paving Schubert matroids are Ehrhart positive and their Ehrhart polynomials are coefficient-wisely bounded by those of minimal and uniform matroids. This confirms a conjecture of Ferroni for the case of sparse paving Schubert matroids. Furthermore, we express the Ehrhart polynomials of three families of Schubert matroids as positive combinations of the Ehrhart polynomials of uniform matroids, yielding Ehrhart positivity of these Schubert matroids.

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“…The lowest degree component of G w is the Schubert polynomial S w . Schubert polynomials have many combinatorial constructions and are well-understood [1,2,6,7,9,11,12,15,18,20,21,26,32]. However there is not nearly as much known combinatorially or discrete-geometrically about Grothendieck polynomials.…”

Section: Introductionmentioning

confidence: 99%

On the support of Grothendieck polynomials

Mészáros¹,

Setiabrata²,

Dizier³

2022

Preprint

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Grothendieck polynomials Gw of permutations w ∈ Sn were introduced by Lascoux and Schützenberger in 1982 as a set of distinguished representatives for the K-theoretic classes of Schubert cycles in the K-theory of the flag variety of C n . We conjecture that the exponents of nonzero terms of the Grothendieck polynomial Gw form a poset under componentwise comparison that is isomorphic to an induced subposet of Z n . When w ∈ Sn avoids a certain set of patterns, we conjecturally connect the coefficients of Gw with the Möbius function values of the aforementioned poset with 0 appended. We prove special cases of our conjectures for Grassmannian and fireworks permutations.

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Vertices of Schubitopes

Cited by 9 publications

References 14 publications

Weight polytopes and saturation of Demazure characters

Weight polytopes and saturation of Demazure characters

On the Ehrhart Polynomial of Schubert Matroids

On the support of Grothendieck polynomials

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