Representation Theory of General Linear Supergroups in Characteristic 2

We develop representation theory of general linear groups in the category Ver₄⁺, the simplest tensor category which is not Frobenius exact. Since Ver₄⁺ 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 Ver₄⁺ has the form m1+nP where P is the indecomposable projective, and GL(m1+nP) is the reduction to characteristic $2$ of GL(m+n|n). We explicitly describe the irreducible representations of GL(P) and then use this description to classify the irreducible representations of GL(m1+nP) for general $m,n$. We also define some subgroups of GL(m1+nP) and classify their irreducible representations. Finally, we conjecture a Steinberg tensor product theorem for Ver₄⁺ involving the square of the Frobenius map.