Affine matrix-ball construction and its relation to representation theory
In 1985, Shi found a generalization of the Robinson-Schensted algorithm to (extended) affine symmetric groups and described their Kazhdan-Lusztig cells in terms of combinatorics. Recently, Chmutov, Lewis, Pylyavskyy, and Yudovina developed its generalization, called the affine matrix-ball construction (abbreviated AMBC). It provides a bijection from an (extended) affine symmetric group to the set of triples (P,Q,?) where P and Q are row-standard Young tableaux of the same shape and ? is an integer vector satisfying certain inequalities. In this talk, I will briefly explain this algorithm, and discuss how this is related to representation of (extended) affine symmetric groups, especially the asymptotic Hecke algebras introduced by Lusztig. This work is joint with Pavlo Pylyavskyy.