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.