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Excited Young diagrams, equivariant \(K\)-theory, and Schubert varieties. (English) Zbl 1317.05187
Summary: 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 formulas 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 V. Kreiman [“Schubert classes in the equivariant \(K\)-theory and equivariant cohomology of the Grassmannian”, Preprint, arXiv:math/0512204; “Schubert classes in the equivariant \(K\)-theory and equivariant cohomology of the Lagrangian Grassmannian”, Preprint, arXiv:math/0602245] using a different method. In type \( A_n\), the formula for the Hilbert series had been proved earlier by L. Li and A.Yong [Adv. Math. 229, No. 1, 633–667 (2012; Zbl 1232.14033)].
The method of this paper, which relies on a restriction formula of W. Graham [“Equivariant \(K\)-theory and Schubert varieties”, Preprint] and M. Willems [Duke Math. J. 132, No. 2, 271–309 (2006; Zbl 1118.19002)], is based on the method used by T. Ikeda and H. Naruse [Trans. Am. Math. Soc. 361, No. 10, 5193–5221 (2009; Zbl 1229.05287)] to obtain the analogous formulas in equivariant cohomology. The formulas we give differ from the \( K\)-theoretic restriction formulas given by T. Ikeda and H. Naruse [Adv. Math. 243, 22–66 (2013; Zbl 1278.05240)], 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.

MSC:
05E15 Combinatorial aspects of groups and algebras (MSC2010)
14M15 Grassmannians, Schubert varieties, flag manifolds
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
05E99 Algebraic combinatorics
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