Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry

Abe, Hiraku; Billey, Sara

doi:10.2969/aspm/07110001

Cited by 20 publications

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“…is zero-dimensional, consisting of exactly c B λ (1) ,...,λ (r) points of Gr(n, k), where B is the ambient rectangle and c B λ (1) ,...,λ (r) is a certain Littlewood-Richardson coefficient, defined in Section 4.6. When we refer to a "generic" choice of flags, we mean that we are choosing from an open dense subset of the flag variety.…”

Section: Variation 1: Intersections Of Schubert Varieties In the Grasmentioning

confidence: 99%

“…This is the cohomology class of the single point Ω B (F • ) for some flag F • . Thus the intersection of the Schubert varieties Ω λ (1)…”

Section: Cellular Homology and Cohomologymentioning

confidence: 99%

“…• ) is rationally equivalent to a finite union of points, the number of which is the coefficient c B λ (1) ,...,λ (r) in the expansion…”

Section: Cellular Homology and Cohomologymentioning

confidence: 99%

“…For a sufficiently general choice of flags, the c B λ (1) ,...,λ (r) points in the intersection are distinct with no multiplicity.…”

Section: Cellular Homology and Cohomologymentioning

confidence: 99%

“…Theorem 4.26. We have s λ (1) · · · · · s λ (m) = ν c ν λ (1) ,...,λ (m) s ν where c ν λ (1) ,...,λ (m) is the number of chains of Littlewood-Richardson tableaux of contents λ (i) with total shape ν.…”

Section: The Littlewood-richardson Rulementioning

confidence: 99%

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Variations on a Theme of Schubert Calculus

Gillespie

2019

Association for Women in Mathematics Series

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In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear spaces. It is intended as a guide for readers with a combinatorial bent to understand and appreciate the geometric and topological aspects of Schubert calculus, and conversely for geometric-minded readers to gain familiarity with the relevant combinatorial tools in this area.

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Section: Variation 1: Intersections Of Schubert Varieties In the Grasmentioning

confidence: 99%

“…This is the cohomology class of the single point Ω B (F • ) for some flag F • . Thus the intersection of the Schubert varieties Ω λ (1)…”

Section: Cellular Homology and Cohomologymentioning

confidence: 99%

“…• ) is rationally equivalent to a finite union of points, the number of which is the coefficient c B λ (1) ,...,λ (r) in the expansion…”

Section: Cellular Homology and Cohomologymentioning

confidence: 99%

“…For a sufficiently general choice of flags, the c B λ (1) ,...,λ (r) points in the intersection are distinct with no multiplicity.…”

Section: Cellular Homology and Cohomologymentioning

confidence: 99%

Section: The Littlewood-richardson Rulementioning

confidence: 99%

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Variations on a Theme of Schubert Calculus

Gillespie

2019

Association for Women in Mathematics Series

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Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

Abe

DeDieu

Galetto

et al. 2018

Sel. Math. New Ser.

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In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton-Okounkov bodies of the two-dimensional Peterson variety.

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Transition formulas for involution Schubert polynomials

Hamaker

Marberg

Pawlowski

2018

Sel. Math. New Ser.

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The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group S n . Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote asŜ y (to be called involution Schubert polynomials) andŜ FPF y (to be called fixed-point-free involution Schubert polynomials). Our main results are explicit formulas decomposing the product ofŜ y (respectively,Ŝ FPF y ) with any y-invariant linear polynomial as a linear combination of other involution Schubert polynomials. These identities serve as analogues of Lascoux and Schützenberger's transition formula for Schubert polynomials, and lead to a self-contained algebraic proof of the nontrivial equivalence of several definitions ofŜ y andŜ FPF y appearing in the literature. Our formulas also imply combinatorial identities about involution words, certain variations of reduced words for involutions in S n . We construct operators on involution words based on the Little map to prove these identities bijectively. The proofs of our main theorems depend on some new technical results, extending work of Incitti, about covering relations in the Bruhat order of S n restricted to involutions.Let S Z denote the group of permutations of Z which fix all but finitely many points, and write S ∞ for the subgroup of elements in S Z with support contained in P = {1, 2, 3, . . . }. Define I ∞ (respectively, I Z ) as the subset of involutions in S ∞ (respectively, S Z ). We also write S n and I n for the subsets of S ∞ and I ∞ which fix all numbers outside [n] = {1, 2, . . . , n}, and F n ⊂ I n for the subset of fixed-point-free involutions. The Schubert polynomials are a family of homogeneous polynomials S w ∈ Z[x 1 , x 2 , . . .] indexed by w ∈ S ∞ . Write B for the subgroup of lower triangular matrices in GL n (C). It is well-known that the right B-orbits in the flag variety Fl(n) = B\GL n (C) are in bijection with S n , that the integral cohomology ring of Fl(n) is isomorphic to a quotient of Z[x 1 , x 2 , . . . , x n ], and that under this isomorphism, the Schubert polynomials {S w : w ∈ S n } correspond to the cohomology classes Poincaré dual to the closures of the aforementioned B-orbits; see [27] for details.The involution Schubert polynomials are homogeneous polynomialsŜ y indexed by y ∈ I ∞ serving a similar geometric purpose: the right orbits of O n (C) on Fl(n) are in bijection with I n , and the cohomology classes of their orbit closures are (up to a constant factor) represented by the involution Schubert polynomials {Ŝ y : y ∈ I n }. The family of fixed-point-free involution Schubert polynomials {Ŝ FPF z : z ∈ F n } plays an analogous role when n is even and O n (C) is replaced by Sp n (C). The precise definitions ofŜ y andŜ FPF z appear in Sections 3 and 4. We attribute the definitions of these polynomials to Wyser and Yong [32], although they occur as special cases of the cohomology representatives des...

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Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry

Cited by 20 publications

References 0 publications

Variations on a Theme of Schubert Calculus

Variations on a Theme of Schubert Calculus

Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

Transition formulas for involution Schubert polynomials

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