Semisimplifications and representations of the General Linear Supergroup

Thorsten Heidersdorf; Rainer Weissauer

Semisimplifications and representations of the General Linear Supergroup

Representation Theory 2025-02-04 v4 Mathematical Physics math.MP

Authors: Thorsten Heidersdorf , Rainer Weissauer

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.

Keywords

representation theory of groups group representations automorphic representations

Cite

@article{arxiv.2501.05298,
  title  = {Semisimplifications and representations of the General Linear Supergroup},
  author = {Thorsten Heidersdorf and Rainer Weissauer},
  journal= {arXiv preprint arXiv:2501.05298},
  year   = {2025}
}

Comments

v4: Minor changes in Section 6 (reordering); v3: Added an appendix ("A variant of the $\eta$ functor"); v2: very minor changes (typos etc)

Semisimplifications and representations of the General Linear Supergroup

Abstract

Keywords

Cite

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