Keys and alternating sign matrices

TL;DR

This paper proves that the key of an alternating sign matrix coincides with the key of its associated monotone triangle, providing a direct method to compute keys of Young tableaux and exploring related implications.

Contribution

It establishes the equivalence of two notions of keys for alternating sign matrices and introduces a straightforward computation method for Young tableau keys.

Findings

Keys of alternating sign matrices and their monotone triangles are identical.

A direct computation method for Young tableau keys is developed.

The results have implications for the structure and analysis of Young tableaux and related combinatorial objects.

Abstract

Lascoux and Sch\"utzenberger introduced a notion of key associated to any Young tableau. More recently Lascoux defined the key of an alternating sign matrix by recursively removing all -1's in such matrices. But alternating sign matrices are in bijection with monotone triangles, which form a subclass of Young tableaux. We show that in this case these two notions of keys coincide. Moreover we obtain an elegant and direct way to compute the key of any Young tableau, and discuss consequences of our result.