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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.

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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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