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.

In the literature there are several determinant formulas for Schur functions: the Jacobi--Trudi formula, the dual Jacobi--Trudi formula, the Giambelli formula, the Lascoux--Pragacz formula, and the Hamel--Goulden formula, where the Hamel--Goulden formula implies the others. In this paper we use an identity proved by Bazin in 1851 to derive determinant identities involving Macdonald's 9th variation of Schur functions. As an application we prove a determinant identity for factorial Schur functions conjectured by Morales, Pak, and Panova. We also obtain a generalization of the Hamel--Goulden formula, which contains a result of Jin, and prove a converse of the Hamel--Goulden theorem and its generalization.

Let $\pi$ be a set partition of $[n]=\{1,2,\ldots,n\}$. The standard representation of $\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i