Tensor product representations for orthosymplectic Lie superalgebras

We derive a general result about commuting actions on certain objects in braided rigid monoidal categories. This enables us to define an action of the Brauer algebra on the tensor space $V^{\otimes k}$ which commutes with the action of the orthosymplectic Lie superalgebra $\spo(V)$ and the orthosymplectic Lie color algebra $\spo(V,\beta)$. We use the Brauer algebra action to compute maximal vectors in $V^{\otimes k}$ and to decompose $V^{\otimes k}$ into a direct sum of submodules $T^\lambda$. We compute the characters of the modules $T^\lambda$, give a combinatorial description of these characters in terms of tableaux, and model the decomposition of $V^{\otimes k}$ into the submodules $T^\lambda$ with a Robinson-Schensted-Knuth type insertion scheme.