Simple Procedures for Left and Right Keys of Semi-Standard Young Tableaux

TL;DR

This paper introduces new, simple procedures for computing the left and right keys of semi-standard Young tableaux, which are important for understanding Demazure modules, by combining known lifts in a novel way.

Contribution

The paper presents new, straightforward procedures for determining the left and right keys of semi-standard Young tableaux, simplifying existing methods.

Findings

Procedures involve repeated Deodhar lifts for keys.

Keys encode characters of Demazure modules.

Procedures are simpler than existing methods.

Abstract

We give simple procedures to obtain the left and right keys of a semi-standard Young tableau. Keys derive their interest from the fact that they encode the characters of Demazure and opposite Demazure modules for the general and special linear groups. Given the importance of keys, there are indeed several procedures available in the literature to determine them. In comparison, our procedures are new (to the best of our knowledge) and especially simple. Having said that, we hasten to add that there is nothing new in any individual ingredient that goes into our procedures. These ingredients are all routine, straightforward, and (in any case) occur in the literature. But they never quite seem to have been put together as done here. Our procedures end up repeatedly performing the Deodhar lifts, maximal lifts for the left key and minimal lifts for right key. Together with the well known fact that keys can be obtained by such repeated lifts, this justifies the procedures. The relevance of Deodhar lifts to combinatorial models for Demazure characters is well known in Standard Monomial Theory. Right and left keys appear respectively as initial and final directions of Lakshmibai-Seshadri paths in Littelmanns Path Model Theory.