$FSZ$-groups and Frobenius-Schur indicators of quantum doubles

Abstract

We study the higher Frobenius-Schur indicators of the representations of the Drinfel'd double of a finite group G, in particular the question as to when all the indicators are integers. This turns out to be an interesting group-theoretic question. We show that many groups have this property, such as alternating and symmetric groups, P SL 2 (q), M 11 , M 12 and regular nilpotent groups. However we show there is an irregular nilpotent group of order 5 6 with non-integer indicators.

Examples of non-$FSZ$ $p$-groups for primes greater than three

“…The parameter d controlled the existence of negative indicators in the double of the groups under consideration in [10], in much the same way that Y p,j will dictate the existence of non-integer indicators here. More generally, such objects naturally arise when considering the F SZ property for groups of the form A ⋊ C where A is abelian and C is cyclic, such as in [8,Example 4.4]. Now that we understand how to take p j -th powers in S(p, j), we can begin investigating what Iovanov et al [8] called the F SZ p j property of S(p, j).…”

“…For our purposes, Theorem 1.8 below will be taken as our definition of the F SZ m properties, and therefore the F SZ property. Iovanov et al [8] also established that several large families of groups were F SZ, including but not limited to the symmetric groups S n ; P SL 2 (q) for a prime power q; and all regular p-groups.…”

“…Frobenius-Schur indicators are guaranteed to be algebraic integers in a certain cyclotomic field, and the Galois action on the field also acts on the indicators [9,Proposition 3.3]. In full generality these indicators need not even be real numbers [9,Example 7.5] [7], but in the case of D(G) they are guaranteed to be so [8,Remark 2.8]. All of the first examples computed in the case of group doubles [4,10,11] yielded indicator values in Z.…”

“…Iovanov et al [8] investigated this question, ultimately finding that there were exactly 32 non-isomorphic groups of order 5 6 with non-integer indicators. They dubbed the property of having all integer FS-indicators the F SZ property.…”

“…On the other hand, the regular wreath product Z p ≀ Z p is an irregular p-group for all primes p, and this was shown to be F SZ [8,Example 4.4], thereby establishing that the class of F SZ p-groups properly contains the class of regular p-groups. It is interesting to ask what can be said about the properties of irregular non-F SZ p-groups, or alternatively of irregular F SZ p-groups.…”