Hilbert functions of points on Schubert varieties in the symplectic Grassmannian

Abstract. We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call "excited Young diagrams" and the second one is written in terms of factorial Schur Q-or P -functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety.

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“…Let w ≤ v ∈ W P , and λ ≤ µ ∈ SP ρn be the corresponding strict partitions. Then we have (1) ( [6], [14]…”

Section: Corollarymentioning

confidence: 99%

Excited Young diagrams and equivariant Schubert calculus

Ikeda

Naruse

2009

Trans. Amer. Math. Soc.

152

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“…As such, X w | v has a multigraded Hilbert series, and the K T -class S w | v is essentially this series times v·(the Weyl denominator); see [MS05,chapter 8.2]. This is the viewpoint of [GR06,KodR03,KrL04,RU#1,RU#2].…”

Section: (In Fact (3) Includes (2))mentioning

confidence: 99%

“…However one may change the game: rather than degenerating Schubert varieties, one may degenerate matrix Schubert varieties [KnM05,KnMY] or Schubert patches [GR06,KodR03,KrL04,RU#1,RU#2]. One difference when working with patches is that the choice of embedding becomes immaterial.…”

Section: There Is a Surjectionmentioning

confidence: 99%

Schubert Patches Degenerate to Subword Complexes

Knutson

2008

Transformation Groups

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ABSTRACT. We study the intersections of general Schubert varieties X w with permuted big cells, and give an inductive degeneration of each such "Schubert patch" to a StanleyReisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using the results of [Knutson 2007] on automatically reduced degenerations, or through more standard cohomology-vanishing arguments.The underlying simplicial complex of the Stanley-Reisner scheme is a subword complex, as introduced for slightly different purposes in [Knutson-Miller 2004], and is homeomorphic to a ball. This gives a new proof of the Andersen-Jantzen-Soergel/Billey and Graham/Willems formulae for restrictions of equivariant Schubert classes to fixed points. CONTENTS

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“…As a consequence of Lemma 6.3, we obtain Proposition 6. 4 The map BRSK is a degree-preserving bijection from the set of nonvanishing (resp. negative, positive) multisets on N 2 to the set of nonvanishing (resp.…”

Section: Lemma 63mentioning

confidence: 99%

“…3, 4, 5, 6, and 7, we define nonvanishing multisets on β × β bounded by T α , W γ , nonvanishing semistandard notched bitableaux on β × β bounded by T α , W γ , and the injection BRSK from the former to the latter. In Section 8, we prove that these two combinatorial objects are indeed indexing sets for the monomials of 4. In Sections 9 and 10, we show how using Corollary 2.4, Mult e β X γ α can be interpreted as counting certain families of nonintersecting paths in the lattice β × β.…”

mentioning

confidence: 97%

Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence

Kreiman

2007

J Algebr Comb

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The Richardson variety X γ α in the Grassmannian is defined to be the intersection of the Schubert variety X γ and opposite Schubert variety X α . We give an explicit Gröbner basis for the ideal of the tangent cone at any T -fixed point of X Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Gröbner basis result to deduce a formula which computes the multiplicity of X γ α at any T -fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattenthaler (Sém. Lothar. Comb. 45:B45c, 2000/2001 J. Algebr. Comb. 22:273-288, 2005).

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Hilbert functions of points on Schubert varieties in the symplectic Grassmannian

Abstract: Abstract. We give an explicit combinatorial description of the multiplicity as well as the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian.

Cited by 20 publications

References 21 publications

Excited Young diagrams and equivariant Schubert calculus

Excited Young diagrams and equivariant Schubert calculus

Schubert Patches Degenerate to Subword Complexes

Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence

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