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The stable Grothendieck polynomials arise from enumerative geometry, as they can be used to calculate the intersection numbers for the Flag varieties. We introduce a new crystal structure on decreasing factorizations of 321-avoiding elements of the 0-Hecke monoid, whose generating functions are the stable Grothendieck polynomials. This crystal is a K-theoretic generalization of the crystal on decreasing factorizations in the symmetric group and it intertwines with the crystal on set-valued tableaux (through the residue map). We also introduced a new insertion algorithm that is associated with our crystal, with surprising connections to the Hecke insertion algorithm and the uncrowding algorithm for set-valued tableaux.

In this talk, I will introduce basic definitions and our results with easy-to-understand examples.

This talk is based on joint work of the same title with Jennifer Morse (University of Virginia), Wencin Poh and Anne Schilling.