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Critical groups for Hopf algebra modules

Published online by Cambridge University Press:  07 November 2018

DARIJ GRINBERG ,
JIA HUANG  and
VICTOR REINER
DARIJ GRINBERG
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A. e-mail: darijgrinberg@gmail.com
JIA HUANG
Affiliation:
Department of Mathematics and Statistics, University of Nebraska at Kearney, Kearney, NE 68849, U.S.A. e-mail: huangj2@unk.edu
VICTOR REINER
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A. e-mail: reiner@math.umn.edu
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Abstract

This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalises the critical groups of complex finite group representations studied in [1, 11]. 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 3 , May 2020 , pp. 473 - 503
Copyright
Copyright © Cambridge Philosophical Society 2018

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Footnotes

†

Supported by NSF grants DMS-1148634, 1601961.

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