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Representation Theory of General Linear Supergroups in Characteristic 2

Serina Hu

TL;DR

The paper develops a representation theory for general linear groups in the tensor category Ver_4^+ at characteristic 2, where objects are of the form $m\mathbf{1}+nP$ and $\mathrm{GL}(m\mathbf{1}+nP)$ arises as the characteristic-2 reduction of $\mathrm{GL}(m+n|n)$. It first classifies irreducibles for $\mathrm{GL}(P)$, then reduces the irreducible spectrum of $\mathrm{GL}(m+nP)$ to pairs of irreducibles for $\mathrm{GL}(m)$ and $\mathrm{GL}(nP)$, with the latter governed by $n$-tuples of $\mathrm{GL}(P)$-irreducibles arranged by nonincreasing degree. Subgroups such as $\mathbb{G}'_a$, $\mathbb{G}'_m$, and parabolic-like groups are analyzed, and explicit structures and tensor products for $\mathrm{GL}(P)$ and $\mathrm{GL}(1+P)$ are obtained. A Steinberg-style tensor product conjecture for $\mathrm{GL}(m+nP)$ is proposed, utilizing both even and odd Frobenius twists, signaling a path toward a full high-ridelity super-Lie theory in characteristic 2. These results provide a concrete, non-Frobenius-exact framework to study representations in a Verlinde-type setting and highlight a rich interaction between highest-weight theory, PBW decompositions, and Frobenius-twisted constructions in Ver_4^+.$

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 $m\mathbf{1}+nP$ 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.

Representation Theory of General Linear Supergroups in Characteristic 2

TL;DR

The paper develops a representation theory for general linear groups in the tensor category Ver_4^+ at characteristic 2, where objects are of the form and arises as the characteristic-2 reduction of . It first classifies irreducibles for , then reduces the irreducible spectrum of to pairs of irreducibles for and , with the latter governed by -tuples of -irreducibles arranged by nonincreasing degree. Subgroups such as , , and parabolic-like groups are analyzed, and explicit structures and tensor products for and are obtained. A Steinberg-style tensor product conjecture for is proposed, utilizing both even and odd Frobenius twists, signaling a path toward a full high-ridelity super-Lie theory in characteristic 2. These results provide a concrete, non-Frobenius-exact framework to study representations in a Verlinde-type setting and highlight a rich interaction between highest-weight theory, PBW decompositions, and Frobenius-twisted constructions in Ver_4^+.$

Abstract

We develop representation theory of general linear groups in the category , the simplest tensor category which is not Frobenius exact. Since is a reduction of the category of supervector spaces to characteristic (by a result of Venkatesh, arXiv:1507.05142), these groups may be viewed as general linear supergroups in characteristic . More precisely, every object in has the form where is the indecomposable projective, and is the reduction to characteristic of . We explicitly describe the irreducible representations of and then use this description to classify the irreducible representations of for general . We also define some subgroups of and classify their irreducible representations. Finally, we conjecture a Steinberg tensor product theorem for involving the square of the Frobenius map.
Paper Structure (23 sections, 37 theorems, 98 equations, 1 table)

This paper contains 23 sections, 37 theorems, 98 equations, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Symmetric Tensor Categories
  5. Commutative algebras and (affine) group schemes in tensor categories
  6. The fundamental group of a symmetric tensor category
  7. Distribution Algebras
  8. The category $\mathop{\mathrm{Ver}}\nolimits_4^+$
  9. $\mathop{\mathrm{Ver}}\nolimits_4^+$ as reduction of $\mathop{\mathrm{sVec}}\nolimits$ to characteristic 2
  10. The fundamental group of $\mathop{\mathrm{Ver}}\nolimits_4^+$
  11. Highest Weights
  12. Examples
  13. Symmetric algebras
  14. $\mathbb G_a, \mathbb G_m$, and friends
  15. General linear groups and subgroups
  16. ...and 8 more sections

Key Result

Proposition 2.9

$\mathop{\mathrm{Dist}}\nolimits(G)$ is a cocommutative Hopf algebra. Moreover, $(I/I^2)^* \subset \mathop{\mathrm{Dist}}\nolimits(G)$ is a Lie subalgebra.

Theorems & Definitions (89)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Proposition 2.9
  • proof
  • ...and 79 more