Uncrowding algorithm for hook-valued tableaux

TL;DR

This paper introduces a new uncrowding algorithm for hook-valued tableaux, linking them to set-valued and increasing tableaux, and applies it to expand canonical Grothendieck polynomials.

Contribution

It presents a novel uncrowding algorithm for hook-valued tableaux and demonstrates its compatibility with crystal operators, advancing combinatorial understanding of Grothendieck polynomials.

Findings

Uncrowding algorithm produces set-valued and increasing tableaux from hook-valued tableaux.

The algorithm intertwines with crystal operators, preserving combinatorial structures.

Applications include new expansions of canonical Grothendieck polynomials.

Abstract

Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to 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.