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 sln( 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 Ylambda(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 Ylambda(w). The set of all such tableaux is denoted Dl (w). Exploiting the method mentioned above for obtaining right keys, this thesis describes the entry at each location in any T ∈ Dl (w). Lastly, we will consider Dl (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.