A crystal for stable Grothendieck polynomials

Abstract: We construct a type A crystal, which we call the *-crystal, whose character is the stable Grothendieck polynomials for fully-commutative permutations. This crystal is a K-theoretic generalization of Morse-Schilling crystal on decreasing factorizations. Using the residue map, we showed that this crystal intertwines with the crystal on set-valued tableaux given by Monical, Pechenik and Scrimshaw. We also proved that this crystal is isomorphic to that of pairs of semistandard Young tableaux using a newly defined insertion called the *-insertion. The insertion offers a combinatorial interpretation to the Schur positivity of the stable Grothendieck polynomials for fully-commutative permutations. Furthermore, the *-insertion has interesting properties in relation to row Hecke insertion and the uncrowding algorithm. This is joint work with Jennifer Morse, Jianping Pan and Anne Schilling.

combinatorics

Audience: researchers in the topic

York University Applied Algebra Seminar