Isocategorical groups

TL;DR

This paper investigates when two finite groups have equivalent tensor categories of representations over C without regard to the commutativity constraint, introducing the concept of isocategorical groups and classifying such groups.

Contribution

It introduces the notion of isocategorical groups, provides examples, and classifies groups that are isocategorical to a given group, using the theory of triangular Hopf algebras.

Findings

Example of isocategorical but non-isomorphic groups

Classification of groups isocategorical to a given group

Conditions under which isocategorical groups are actually isomorphic

Abstract

It is well known that if two finite groups have the same symmetric tensor categories of representations over C, then they are isomorphic. We study the following question: when do two finite groups G1,G2 have the same tensor categories of representations over C (without regard for the commutativity constraint). We call two groups with such property isocategorical. We give an example of two groups which are isocategorical but not isomorphic: the affine symplectic group of a vector space over the field of two elements, and an appropriate "affine pseudosymplectic group" introduced by R.Griess (containing the "pseudosymplectic group" of A.Weil). On the other hand, we give a classification of groups isocategorical to a given group. In particular, we show that if G has no nontrivial normal subgroups of order 2^{2m} then any group isocategorical to G must actually be isomorphic to G. The proofs use the theory of triangular Hopf algebras. We also apply the notion of isocategorical groups to studying the question: when are two triangular semisimple Hopf algebras isomorphic as Hopf algebras?