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 %$\Delta (\lambda {)}^{\ast }{\otimes }_K\nabla (\lambda )$, where $\Delta (\lambda {)}^{\ast }$ is the dual of the standard supermodule %and $\nabla (\lambda )$ is the costandard supermodule of the highest weight $\lambda$. $D_{\lambda }{\otimes }_K{D}_{\lambda }^o$, where $D_{\lambda }$ and $D_{\lambda }^o$ 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.