The direct sum map on Grassmannians and jeu de taquin for increasing tableaux. (English) Zbl 1231.05280

Thomas, Hugh; Yong, Alexander

Int. Math. Res. Not. 2011, No. 12, 2766-2793 (2011).

Summary: The direct sum map Gr\((a, \mathbb C^n) \times \text{Gr}(b,\mathbb C^m) \to \text{Gr}(a+b,\mathbb C^{m+n})\) on Grassmannians induces a \(K\)-theory pullback that defines the splitting coefficients. We geometrically explain an identity from A. S. Buch [“Grothendieck classes of quiver varieties”, Duke Math. J. 115, No. 1, 75–103 (2002; Zbl 1052.14056)] between the splitting coefficients and the Schubert structure constants for products of Schubert structure sheaves. This is related to the topic of product and splitting coefficients for Schubert boundary ideal sheaves. Our main results extend jeu de taquin for increasing tableaux [H. Thomas and A. Yong, “A jeu de taquin theory for increasing tableaux, with applications to \(K\)-theoretic Schubert calculus”, Algebra Number Theory 3, No. 2, 121–148 (2009; Zbl 1229.05285)] by proving transparent analogues of M.-P. Schützenberger’s [“La Correspondance de Robinson”, Lect. Notes Math. 579, 59–113 (1977; Zbl 0398.05011)] fundamental theorems on well definedness of rectification. We then establish that jeu de taquin gives rules for each of these four kinds of coefficients.

MSC:

05E05	Symmetric functions and generalizations
05A17	Combinatorial aspects of partitions of integers
14N15	Classical problems, Schubert calculus