Spin polynomial functors and representations of Schur superalgebras

TL;DR

This paper develops categories of polynomial functors on supervector spaces, establishing their equivalence with supermodule categories over Schur superalgebras, and explores Sergeev duality in this context.

Contribution

It introduces and relates categories of strict polynomial functors on supervector spaces to supermodules over Schur superalgebras, extending classical representation theory to superalgebra settings.

Findings

Categories of polynomial functors are equivalent to supermodule categories over Schur superalgebras.

Established conditions for the equivalence when dimensions are sufficiently large.

Discussed connections to Sergeev duality within the supercategory framework.

Abstract

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