Symmetrizers for Schur superalgebras

Abstract

For the Schur superalgebra $S=S(m|n,r)$ over a ground field $K$ of characteristic zero, we define symmetrizers $T^{\lambda}[i:j]$ of the ordered pairs of tableaux $T_i, T_j$ of the shape $\lambda$ and show that the $K$-span $A_{\lambda,K}$ of all symmetrizers $T^{\lambda}[i:j]$ has a basis consisting of $T^{\lambda}[i:j]$ for $T_i,T_j$ semistandard. The $S$-superbimodule $A_{\lambda,K}$ is identified as %$\Delta(\lambda)^*\otimes_K \nabla(\lambda)$, where $\Delta(\lambda)^*$ is the dual of the standard supermodule %and $\nabla(\lambda)$ is the costandard supermodule of the highest weight $\lambda$. $D_{\lambda}\otimes_K D^o_{\lambda}$, where $D_\lambda$ and $D^o_\lambda$ are left and right irreducible $S$-supermodules of the highest weight $\lambda$. We define modified symmetrizers $T^{\lambda}\{i:j\}$ and show that their $\mathbb{Z}$-span form a $\mathbb{Z}$-form $A_{\lambda,\mathbb{Z}}$ of $A_{\lambda, \mathbb{Q}}$. We show that every modified symmetrizer $T^\lambda\{i:j\}$ is a $\mathbb{Z}$-linear combination of symmetrizers $T^\lambda\{i:j\}$ for $T_i, T_j$ semistandard. Using modular reduction to a field $K$ of characteristic $p>2$, we obtain that $A_{\lambda,K}$ has a basis consisting of modified symmetrizers $T^\lambda\{i:j\}$ for $T_i, T_j$ semistandard.