Symmetrizers for Schur superalgebras

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

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