\(K\)-theoretic Schubert calculus for \(OG(n,2n+1)\) and jeu de taquin for shifted increasing tableaux. (English) Zbl 1348.14127

Clifford, Edward; Thomas, Hugh; Yong, Alexander

J. Reine Angew. Math. 690, 51-63 (2014).

The authors prove a Littlewood-Richardson rule for the \(K\)-theory of odd orthogonal Grassmannians. That is, consider the variety \(X\) of isotropic \(n\)-planes in \(C^{2n+1}\) (with respect to a non-degenerate symmetric bilinear form). In the \(K\)-theory ring of \(X\) a natural basis is given by the classes of the structure sheaves of geometrically defined subvarieties, the Schubert varieties. Schubert varieties are parametrized by combinatorial objects, called shifted Young diagrams. Multiplication in \(K\)-theory hence defines structure constants depending on three shifted Young diagrams. One would like to find a combinatorial rule (“Littlewood-Richardson rule”) calculating these structure constants from the three Yong diagrams. Based on earlier results (and conjectures) of Thomas-Yong, and a recent result by Buch-Ravikumar, the authors prove such a combinatorial rule.
Technically, Theorem 1.1 proves that a certain notion in the relevant “jeu de taquin” is well defined; then the strategy already outlined in Thomas-Yong proves the main result, the Littlewood-Richardson rule for the \(K\)-theory of odd orthogonal Grassmannians.

Reviewer: Richárd Rimányi (Chapel Hill)

MSC:

14N15	Classical problems, Schubert calculus
05E10	Combinatorial aspects of representation theory

Keywords:

Schubert calculus; Littlewood-Richardson rule; orthogonal Grassmannian; jeu de taquin