On Montgomery's Question on Frobenius–Schur Indicators and Gauge Equivalence: A Negative Answer for Group Algebras

In the open questions of the Oberwolfach mini-workshop on infinite dimensional Hopf algebras (Report 20/2014), Montgomery asks whether two semisimple Hopf algebras over \(\mathbb{C}\) with the same set of Frobenius–Schur indicator values have gauge equivalent, that is tensor equivalent, representation categories. She notes that the question still seems to be open for group algebras, and records R. Ng's variant that also requires equal fusion rules. Read literally as a set of numbers, the datum cannot be what is meant, and we read it as the family of indicator sequences of the simple modules. We show that the answer is negative for group algebras, also in this strongest reading and with Ng's condition: if two groups of odd order form a Brauer pair (they are not isomorphic but have identical character tables and power maps), then there is a bijection between the simple modules of their group algebras that preserves all higher indicators and the fusion rules, the exponents are equal, and the representation categories are not tensor equivalent. The last step is the theorem of Etingof and Gelaki that groups of odd order are determined by their tensor categories of representations, which we cite. Brauer pairs of odd order are classical (Dade; Eick and Müller; Lewis; Nenciu); for a self-contained instance we give two Camina groups of order \(3^6\), of a kind described by Lewis, and prove all their properties in the text. For general semisimple Hopf algebras, without Ng's condition, an elementary pair already answers the question in the negative (the functions on the Heisenberg group of order \(27\) and the group algebra of \((\mathbb{Z}/3)^3\), Remark 5.1); we did not find it in print, but it is in the spirit of an observation of Mignard and Schauenburg on pointed fusion categories, and we claim nothing new about it. The main content of the note is the group-algebra case, with or without equal fusion rules. The corpus record of the question also contains Lentner's Question 6(c) and Lu's Question 7, which are not settled here. No printed answer to the question was found in the sources we searched; this is not a proof of novelty, and experts may regard the argument as folklore. This is an unrefereed note.