Critical groups for Hopf algebra modules
Abstract
This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalizes the critical groups of complex finite group representations studied by Benkart, Klivans, Reiner and Gaetz. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations.
Cite
@article{arxiv.1704.03778,
title = {Critical groups for Hopf algebra modules},
author = {Darij Grinberg and Jia Huang and Victor Reiner},
journal= {arXiv preprint arXiv:1704.03778},
year = {2020}
}
Comments
24 pages. Comments are welcome! The ancillary file is a longer version with some more details and alternative proofs (compiled from the same source). The version on http://www.cip.ifi.lmu.de/~grinberg/algebra/ will probably get quicker updates. v3: Minor corrections