Representation Theory of General Linear Supergroups in Characteristic 2

2024/06/14 by Serina Hu, Hu, Serina · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2406.10201

openalex publication_date 2024/06/14 · openalex created_date 2024/06/18 · openalex updated_date 2026/08/02

Abstract

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.

Cited by

• Group schemes and their Lie algebras over a symmetric tensor category
• Homogeneous spaces in tensor categories

Related

• Tensor Representations of the General Linear Supergroup
• Deligne's category Rep(GL_δ) and representations of general linear supergroups
• Representations of General Linear Groups in the Verlinde Category
• The highest weight theory for Representations of General Linear groups in the Verlinde categories in positive characteristic
• Model structures, categorial quotients and representations of super commutative Hopf algebras II, The case Gl(m,n)