New Descriptions of Demazure Tableaux and Right Keys, with Applications to Convexity

TL;DR

This paper introduces new methods for determining the right and left keys of semistandard Young tableaux, explores the structure of Demazure characters, and characterizes the convexity of associated tableau sets.

Contribution

It provides novel inspection-based techniques for computing tableau keys and establishes a necessary and sufficient condition for the convexity of Demazure tableau sets.

Findings

Methods to obtain right and left keys by inspection

Description of entries in Demazure tableau sets

Convexity condition for tableau sets

Abstract

The right key of a semistandard Young tableau is a tool used to find Demazure characters for $sl_n(\mathbb{C})$. This thesis gives methods to obtain the right and left keys by inspection of the semistandard Young tableau. Given a partition $\lambda$ and a Weyl group element $w$, there is a semistandard Young tableau $Y_\lambda(w)$ of shape $\lambda$ that corresponds to $w$. The Demazure character for $\lambda$ and $w$ is known to be the sum of the weights of all tableaux whose right key is dominated by $Y_\lambda(w)$. The set of all such tableaux is denoted $\mathcal{D}_\lambda(w)$. Exploiting the method mentioned above for obtaining right keys, this thesis describes the entry at each location in any $T \in \mathcal{D}_\lambda(w)$. Lastly, we will consider $\mathcal{D}_\lambda(w)$ as an integral subset of Euclidean space. The final results present a condition that is both necessary and sufficient for this subset to be convex.