Representation Theory of General Linear Supergroups in Characteristic 2

Abstract

We develop representation theory of general linear groups in the category ${\text{Ver}}_4^{+}$, the simplest tensor category which is not Frobenius exact. Since ${\text{Ver}}_4^{+}$ is a reduction of the category of supervector spaces to characteristic $2$ (by a result of Venkatesh, arXiv:1507.05142), these groups may be viewed as general linear supergroups in characteristic $2$. More precisely, every object in ${\text{Ver}}_4^{+}$ has the form $m1+nP$ where $P$ is the indecomposable projective, and $\text{GL}(m1+nP)$ is the reduction to characteristic $2$ of $\text{GL}(m+n|n)$. We explicitly describe the irreducible representations of $\text{GL}(P)$ and then use this description to classify the irreducible representations of $\text{GL}(m1+nP)$ for general $m,n$. We also define some subgroups of $\text{GL}(m1+nP)$ and classify their irreducible representations. Finally, we conjecture a Steinberg tensor product theorem for ${\text{Ver}}_4^{+}$ involving the square of the Frobenius map.

32 pages