Spin polynomial functors and representations of Schur superalgebras

Abstract:

We introduce categories of homogeneous strict polynomial functors, $\mathsf {Pol}^{\mathrm {I}}_{d,\Bbbk }$ and $\mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk }$, defined on vector superspaces over a field $\Bbbk$ of characteristic not equal 2. These categories are related to polynomial representations of the supergroups $GL(m|n)$ and $Q(n)$. In particular, we prove an equivalence between $\mathsf {Pol}^{\mathrm {I}}_{d,\Bbbk }$, $\mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk }$ and the category of finite dimensional supermodules over the Schur superalgebra $\mathcal {S}(m|n,d)$, $\mathcal {Q}(n,d)$ respectively provided $m,n \ge d$. We also discuss some aspects of Sergeev duality from the viewpoint of the category $\mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk }$.