Séminaire Lotharingien de Combinatoire, 84B.33 (2020), 12 pp.

Jennifer Morse, Jianping Pan, Wencin Poh and Anne Schilling

Crystal for Stable Grothendieck Polynomials

Abstract. We introduce a type A crystal structure on decreasing factorizations on 321-avoiding elements in the 0-Hecke monoid which we call *-crystal. This crystal is a K-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the *-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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