In:
Forum of Mathematics, Pi, Cambridge University Press (CUP), Vol. 5 ( 2017)
Kurzfassung:
We address a unification of the Schubert calculus problems solved by Buch [A Littlewood–Richardson rule for the $K$ -theory of Grassmannians, Acta Math . 189 (2002), 37–78] and Knutson and Tao [Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2) (2003), 221–260]. 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 Anderson et al. [Positivity and Kleiman transversality in equivariant $K$ -theory of homogeneous spaces, J. Eur. Math. Soc. 13 (2011), 57–84] and in a stronger form. Our work is based on the combinatorics of genomic tableaux and a generalization of Schützenberger’s [Combinatoire et représentation du groupe symétrique, in Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976 , Lecture Notes in Mathematics, 579 (Springer, Berlin, 1977), 59–113] jeu de taquin . Using our rule, we deduce the two other combinatorial rules for these coefficients. The first is a conjecture of Thomas and Yong [Equivariant Schubert calculus and jeu de taquin, Ann. Inst. Fourier (Grenoble) (2013), to appear]. The second (found in a sequel to this paper) is a puzzle rule, resolving a conjecture of Knutson and Vakil from 2005.
Materialart:
Online-Ressource
ISSN:
2050-5086
Sprache:
Englisch
Verlag:
Cambridge University Press (CUP)
Publikationsdatum:
2017
ZDB Id:
2723153-7
