Excited Young diagrams, equivariant K -theory, and Schubert varieties

By W. Graham, V. Kreiman. 2013

61 citations recorded by the source corpus.

We give combinatorial descriptions of the restrictions to T-fixed points of the classes of structure sheaves of Schubert varieties in the T-equivariant K-theory of Grassmannians and of maximal isotropic Grassmannians of orthogonal and symplectic types. We also give formulas, based on these descriptions, for the Hilbert series and Hilbert polynomials at T-fixed points of the corresponding Schubert varieties. These descriptions and formulas are given in terms of two equivalent combinatorial models: excited Young diagrams and set-valued tableaux. The restriction fomulas are positive, in that for a Schubert variety of codimension d, the formula equals (-1)^d times a sum, with nonnegative coefficients, of monomials in the expressions (e^{-\alpha}-1), as \alpha runs over the positive roots. In types A_n and C_n the restriction formulas had been proved earlier by [Kreiman 05], [Kreiman 06] by a different method. In type A_n, the formula for the Hilbert series had been proved earlier by [Li-Yong 12]. The method of this paper, which relies on a restriction formula of [Graham 02] and [Willems 06], is based on the method used by [Ikeda-Naruse 09] to obtain the analogous formulas in equivariant cohomology. The formulas we give differ from the K-theoretic restriction formulas given by [Ikeda-Naruse 11], which use different versions of excited Young diagrams and set-valued tableaux. We also give Hilbert series and Hilbert polynomial formulas which are valid for Schubert varieties in any cominuscule flag variety, in terms of the 0-Hecke algebra.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eb5dd601271f4b736b4216aa82e9171b25f8f81b

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Citations in the corpus

K-theory of minuscule varieties (2013, background): “Formulas for the restrictions of Schubert classes to T -fixed points can be found in [13, 30, 16].”
K-theoretic analogues of factorial Schur P- and Q-functions (2011, background): “ents. Bumping procedure for ‘orthogonal’ type, which is relevant for GPλ(x1,...,xn), is not known now. After the ﬁrst version of this paper was completed, a closely related work by Graham and Kreiman [8] has appeared. They give combinatorial descriptions (using excited Young diagrams and set-valued ”