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Semisimplifications and representations of the General Linear Supergroup

Thorsten Heidersdorf, Rainer Weissauer

TL;DR

The paper addresses how to systematically semisimplify the finite-dimensional representation theory of GL$(m|n)$ by forming the semisimple quotient of the maximal nonzero-superdimension subcategory, showing it is equivalent to $Rep(H_{m|n})$, with a canonical decomposition $H_{m|n} \cong GL(m-n) \times H_{n|n}$. It develops the Duflo-Serganova functor as a key tool, constructs a tannakian tower via DS that relates $H_{m|n}$ to $H_{m-1|n-1}$, and proves a splicing theorem that decomposes negatively stable maximal atypical simples into classical and principal parts to reduce to the equal-rank case $m=n$. The work provides a detailed program of reconstruction, stabilization, and determinant analysis, yielding a concrete description of tensor products up to negligible summands and a product decomposition $H_{m|n} \cong GL(r) \times H_{n|n}$ that encodes both classical and principal data. The results have implications for fusion rules, block theory, and potential physical interpretations in supersymmetric contexts, since the semisimplified category encodes a hidden symmetry group governing representation-theoretic constraints. Overall, the article advances a coherent framework to understand and compute semisimple and near-semisimple structures in GL$(m|n)$-representation theory by bridging diagrammatic combinatorics, DS-cohomology, and tannakian duality.

Abstract

We study the semisimplification of the full karoubian subcategory generated by the irreducible finite dimensional representations of the algebraic supergroup $GL(m|n)$ over an algebraically closed field of characteristic zero. This semisimplification is equivalent to the representations of a pro-reductive group $H_{m|n}$. We show that there is a canonical decomposition $H_{m|n} \cong GL(m\!-\! n) \times H_{n|n}$, thereby reducing the determination of $H_{m|n}$ to the equal rank case $m\! =\! n$ which was treated in a previous paper.

Semisimplifications and representations of the General Linear Supergroup

TL;DR

The paper addresses how to systematically semisimplify the finite-dimensional representation theory of GL by forming the semisimple quotient of the maximal nonzero-superdimension subcategory, showing it is equivalent to , with a canonical decomposition . It develops the Duflo-Serganova functor as a key tool, constructs a tannakian tower via DS that relates to , and proves a splicing theorem that decomposes negatively stable maximal atypical simples into classical and principal parts to reduce to the equal-rank case . The work provides a detailed program of reconstruction, stabilization, and determinant analysis, yielding a concrete description of tensor products up to negligible summands and a product decomposition that encodes both classical and principal data. The results have implications for fusion rules, block theory, and potential physical interpretations in supersymmetric contexts, since the semisimplified category encodes a hidden symmetry group governing representation-theoretic constraints. Overall, the article advances a coherent framework to understand and compute semisimple and near-semisimple structures in GL-representation theory by bridging diagrammatic combinatorics, DS-cohomology, and tannakian duality.

Abstract

We study the semisimplification of the full karoubian subcategory generated by the irreducible finite dimensional representations of the algebraic supergroup over an algebraically closed field of characteristic zero. This semisimplification is equivalent to the representations of a pro-reductive group . We show that there is a canonical decomposition , thereby reducing the determination of to the equal rank case which was treated in a previous paper.
Paper Structure (49 sections, 41 theorems, 125 equations)

This paper contains 49 sections, 41 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Tensor product decomposition up to superdimension $0$
  3. The main theorems
  4. The universal semisimple quotient
  5. The tannakian tower
  6. Stabilization
  7. Splicing
  8. Proof of the main theorem
  9. Determinants
  10. Physical motivation
  11. The category $\mathcal{T}_{m|n}^+$
  12. The category $\mathcal{T} = \mathcal{T}_{m|n}$
  13. Irreducible representations
  14. Dualities
  15. The karoubian category $\mathcal{T}_{m|n}^+$
  16. ...and 34 more sections

Key Result

Theorem 1.1

The categories $\overline{\mathcal{T}}_{m|n}$ are semisimple tannakian categories, hence their Tannaka groups $H_{m|n}$ are projective limits of reductive algebraic groups over $k$. A fibre functor is provided by the composite of functors $d_{m|n}$. By Tannakian duality the functor $d_{m|n}$ induces a closed embedding of affine group schemes $H_{m-1|n-1} \hookrightarrow H_{m|n}$ over $k$ such tha

Theorems & Definitions (89)

  • Theorem 1.1: Theorem \ref{['thm:tannaka']} and Corollary \ref{['cor:fibre']}
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • ...and 79 more