Abstract
Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated with stable canonical Grothendieck polynomials. In this paper, we define a novel uncrowding algorithm for hook-valued tableaux. The algorithm “uncrowds” the entries in the arm of the hooks, and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that “uncrowds” the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials.
Highlights
Set-valued tableaux play an important role in the K-theory of the Grassmannian
Stable symmetric Grothendieck polynomials Gλ can be viewed as a K-theory analog of the Schur functions sλ
We describe a novel uncrowding algorithm on hook-valued tableaux
Summary
Set-valued tableaux play an important role in the K-theory of the Grassmannian. They form a generalization of semistandard Young tableaux, where boxes may contain sets of integers rather than just integers [3]. The uncrowding algorithm on setvalued tableaux was originally developed by Buch [3, Theorem 6.11] to give a bijective proof of Lenart’s Schur expansion of symmetric stable Grothendieck polynomials [9] This uncrowding algorithm takes as input a set-valued tableau and produces a semistandard Young tableau (using the RSK bumping algorithm to uncrowd cells that contain more than one integer) and a flagged increasing tableau [9] ( known as an elegant filling [1,10,14]), which serves as a recording tableau. Chan and Pflueger [4] provide an expansion of stable Grothendieck polynomials indexed by skew partitions in terms of skew Schur functions Their proof uses a generalization of the uncrowding algorithm of Lenart [9], Buch [3], and Reiner et al [15] to skew shapes.
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