Travis Scrimshaw, University of Queensland
"Stable Grothendieck polynomials and crystals"
Stable Grothendieck polynomials with n variables correspond to Schubert classes in the K-theory of the Grassmannian. They can be described as a sum over semistandard set-valued tableaux with max entry n and are known to decompose positively into Schur functions. A crystal is an edge-colored directed graph that encodes a representation of a (Drinfeld-Jimbo) quantum group. In this talk, we construct a type A crystal structure on semistandard set-valued tableaux, which yields the Schur positivity as a consequence, with the multiplicity (or coefficient) given by counting semistandard set-valued tableaux with a Yamanouchi condition. We use the uncrowding bijection of Buch to recover a multiplicity formula given by Lenart in terms of flagged increasing tableaux. As time permits, we will discuss some open problems in trying to construct a K-theory analog of crystals.
MC 5403