Symmetrizers for Schur superalgebras

For the Schur superalgebra S=S(m|n,r) over a ground field K of characteristic zero, we define the symmetrizer T^λ[i:j] of the ordered pairs of tableaux (T_i,T_j) of the shape λ. We 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 and T_j semistandard. In particular, A_λ,K≠0 if and only if λ is an (m|n)-hook partition. In this case, the S-superbimodule A_λ,K is identified as D_λ⊗_KD_λ^o, where D_λ and D_λ^o are left and right irreducible S-supermodules of the highest weight λ. We define modified symmetrizers T^λ{i:j} and show that their Z-span forms a Z-form A_λ,Z of A_λ,Q. We show that every modified symmetrizer T^λ{i:j} is a Z-linear combination of modified 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 and T_j semistandard.