Abstract. Refined canonical stable Grothendieck polynomials were introduced by Hwang et al. There exist two combinatorial models for these polynomials: one using hook-valued tableaux and the other using pairs of a semistandard Young tableau and (what we call) an exquisite tableau. An uncrowding algorithm on hook-valued tableaux was introduced by Pan et al. In this paper, we discover a novel connection between the two models via the uncrowding and Goulden and Greene’s jeu de taquin algorithms, using a classical result of Benkart, Sottile, and Stroomer on tableau switching. This connection reveals a symmetry of the uncrowding algorithm defined on hook-valued tableaux. As a corollary, we obtain another combinatorial model for the refined canonical stable Grothendieck polynomials in terms of biflagged tableaux, which naturally appear in the characterization of the image of the uncrowding map.

We provide a general framework for computing mixing times of finite Markov chains whose semigroup's minimal ideal is left zero. Our analysis is based on combining results by Brown and Diaconis with our previous work on stationary distributions of finite Markov chains. Stationary distributions can be computed from the Karnofsky--Rhodes and McCammond expansion of the right Cayley graph of the finite semigroup underlying the Markov chain. Using loop graphs, which are planar graphs consisting of a straight line with attached loops, there are rational expressions for the stationary distribution in the probabilities. From these we obtain bounds on the mixing time. In addition, we provide a new Markov chain on linear extension of a poset with $n$ vertices, inspired by but different from the promotion Markov chain of Ayyer, Klee, and the last author. The mixing time of this Markov chain is $O(n \log n)$.