Character theory for 2-representations
Abstract
Group representations very comfortably admit a higher categorical generalization. This includes induction and restriction for 2-representations that very directly parallel classical definitions. An appropriate character theory for 2-representations has been more opaque. In this paper we draw from a range of motivations to provide an intuitive character theory for 2-representations. It is built from the bicategorical trace developed to describe topological fixed point theory. We use the secondary traces of Ben-Zvi and Nadler which connect to Riemann-Roch and Lefschetz-type theorems in the context of differential graded categories and Fourier-Mukai transforms. Additional sources of motivation include Lipmans' work on Grothendieck duality and, crucially, the Bartlett and Ganter--Kapranov approaches to characters. (The second of these was in turn motivated by Hopkins, Kuhn and Ravenel's work on equivariant Morava E-theories.) Our characters are compatible with restriction and induction and, in cases that are sufficiently finite, admit a secondary character.
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BibTeX
@misc{ponto2026character,
title = {{Character theory for 2-representations}},
author = {Kate Ponto and Travis Wheeler},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21052},
eprint = {2609.21052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21052},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Character theory for 2-representations
AU - Kate Ponto
AU - Travis Wheeler
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21052
UR - https://arxiv.org/abs/2609.21052
AB - Group representations very comfortably admit a higher categorical generalization. This includes induction and restriction for 2-representations that very directly parallel classical definitions. An appropriate character theory for 2-representations has been more opaque. In this paper we draw from a range of motivations to provide an intuitive character theory for 2-representations. It is built from the bicategorical trace developed to describe topological fixed point theory. We use the secondary traces of Ben-Zvi and Nadler which connect to Riemann-Roch and Lefschetz-type theorems in the context of differential graded categories and Fourier-Mukai transforms. Additional sources of motivation include Lipmans' work on Grothendieck duality and, crucially, the Bartlett and Ganter--Kapranov approaches to characters. (The second of these was in turn motivated by Hopkins, Kuhn and Ravenel's work on equivariant Morava E-theories.) Our characters are compatible with restriction and induction and, in cases that are sufficiently finite, admit a secondary character.
ER -