Isocategorical groups

By Pavel Etingof, Shlomo Gelaki. 2000

51 citations recorded by the source corpus.

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?

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5e9e3b4ef9adcd5345e0c80e935e80ef6a3e8488

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On finite symmetries and their gauging in two dimensions (2017, background): “It is also known that there are cases where two distinct groups lead to the same symmetry category [16, 17].”
Modular invariants for group-theoretical modular data. I (2009, background): “ivalence Rep(G) → Rep(Q) gives rise to an equivalence of ribbon categories Z(G) → Z(Q). Monoidal equivalences between categories of representations of ﬁnite groups were described in [9, 10] (see also [15]). According to [10], monoidal equivalences Rep(G) → Rep(Q) correspond to the following data: a ”
Module categories, weak Hopf algebras and modular invariants (2001, background): “([15]) Two finite groups G1, G2 are called isocategorical if Rep(G1) is equivalent to Rep(G2) as a monoidal category.”