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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
##### Keywords:
set-valued tableaux
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##### References:
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