2020/02/25 by Ichikawa, Takumi, Tachikawa, Yuji
#FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Representation Theory (math.RT) #Strongly Correlated Electrons (cond-mat.str-el)
paper · doi:10.48550/arxiv.2002.10642
It is well-known that the value of the Frobenius-Schur indicator |G|-1 ∑g∈ G χ(g2)=±1 of a real irreducible representation of a finite group G determines which of the two types of real representations it belongs to, i.e. whether it is strictly real or quaternionic. We study the extension to the case when a homomorphism φ:G→ ℤ/2ℤ gives the group algebra ℂ[G] the structure of a superalgebra. Namely, we construct of a super version of the Frobenius-Schur indicator whose value for a real irreducible super representation is an eighth root of unity, distinguishing which of the eight types of irreducible real super representations described in [Wall1964] it belongs to. We also discuss its significance in the context of two-dimensional finite-group gauge theories on pin- surfaces.