Combinatorics of K-Theoretic Jeu de Taquin

Abstract

Thomas and Yong [5] introduced a theory of jeu de taquin which extended Schutzenberger's [4] for Young tableaux. The extended theory computes structure constants for the K-theory of (type A) Grassmannians using combinatorial machinery similar to that for cohomology. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety G/P.

In this dissertation, we see that the root-system uniform rule is well-defined for certain G/P other than the Grassmannian. This gives rise to combinatorially defined rings which are conjecturally isomorphic to K(G/P). Although we do not prove that these rings are isomorphic to K(G/P), we do produce a ``Pieri rule" for computing the product of a general class with a generating class in the type B combinatorial case. We also investigate some symmetries which support the conjectural isomorphism. Moreover, our results combined with recent work of Buch and Ravikumar [1] imply that this conjecture is in fact true.

Lenart [2] gave a Pieri rule for the type A K-theory, demonstrating that the Pieri structure constants are binomial coefficients. In contrast, using techniques of [3], we show that type B Pieri structure constants have no such simple closed forms.

References:

[1] A. Buch and V. Ravikumar: Pieri rules for the K-theory of cominuscule Grassmannians, arXiv:1005.2605, 2010.

[2] C. Lenart: Combinatorial aspects of K-theory of Grassmannians. Ann. Combin. 4 (2000), 67--82.

[3] M. Petkovsek and H. Wilf and D. Zeilberger: A=B. A K Peters, Ltd. (1996).

[4] M.-P. Schutzenberger: Combinatoire et representation du groupe symetrique. Springer-Verlag Berlin, Lec. Notes in Math. 579 (1977), 59--113.

[5] H. Thomas and A. Yong: A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus. Algebra Number Theory 3 (2009), no. 2, 121--148.