Representation Theory of General Linear Supergroups in Characteristic 2

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 $m\mathbf{1}+nP$ where $P$ is the indecomposable projective, and $\text{GL}(m\mathbf{1}+nP)$ 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 $\text{GL}(m\mathbf{1}+nP)$ for general $m,n$. We also define some subgroups of $\text{GL}(m\mathbf{1}+nP)$ and classify their irreducible representations. Finally, we conjecture a Steinberg tensor product theorem for $\text{Ver}_4^+$ involving the square of the Frobenius map.