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

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.