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Equivariant \(K\)-theory of Grassmannians. (English) Zbl 1369.14060
Summary: We address a unification of the Schubert calculus problems solved by A. S. Buch [Acta Math. 189, No. 1, 37–78 (2002; Zbl 1090.14015)] and A. Knutson and T. Tao [Duke Math. J. 119, No. 2, 221–260 (2003; Zbl 1064.14063)]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant \(K\)-theory of Grassmannians with respect to the basis of Schubert structure sheaves. This rule is positive in the sense of D. Anderson et al. [J. Eur. Math. Soc. (JEMS) 13, No. 1, 57–84 (2011; Zbl 1213.19003)] and in a stronger form. Our work is based on the combinatorics of genomic tableaux and a generalization of M.-P. Schützenberger’s jeu de taquin [Lect. Notes Math. 579, 59–113 (1977; Zbl 0398.05011)]. Using our rule, we deduce the two other combinatorial rules for these coefficients. The first is a conjecture of H. Thomas and A. Yong [“Equivariant Schubert calculus and jeu de taquin”, Ann. Inst. Fourier (Grenoble) (to appear), arXiv:1207.3209]. The second (found in a sequel to this paper) is a puzzle rule, resolving a conjecture of Knutson and Vakil from 2005.

MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
05E05 Symmetric functions and generalizations
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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