A family of isomorphic fusion algebras of twisted quantum doubles of finite groups

Abstract

Let D ω (G) be the twisted quantum double of a finite group, G, where ω ∈ Z 3 (G, C *). For each n ∈ N, there exists an ω such that D(G) and D ω (E) have isomorphic fusion algebras, where G is an extraspecial 2-group with 2 2n+1 elements, and E is an elementary abelian group with |E| = |G|. Res G C K α=χ⊗ψ M (1, α) ⊕ M (−1, α) 6. M (K, χ) ⊗ M (L, ψ) = Res Q ξ=Res Q χ⊗Res Q ψ M (KL, ξ) 7. M (K, χ) ⊗ M (, Λ) = M (K, λ 1) ⊕ M (K, λ −1) 8. M (K, χ) ⊗ M (K, λ) = M (1, Λ) ⊕ M (−1, Λ) 9. M (K, χ) ⊗ M (L, λ) = M (KL, λ 1) ⊕ M (KL, λ −1) 10. M (1 , Λ) ⊗ M (2 , Λ) = α∈ G M (1 2 , α) 11. M (1 , Λ) ⊗ M (K, λ 2) = χ∈ C K M (K, χ) 12. M (K, λ 1) ⊗ M (K, λ 2) = α(g K)= 1 2 M (1, α) β(g K)=− 1 2 M (−1, β) 13. M (K, λ 1) ⊗ M (L, λ 2) = χ∈ C KL M (KL, χ) Proof: We will prove only a few cases to give the reader an idea of the straightforward proof. Full details require properties outlined in Section 5.1 and can also be found in [7]. Case 1: Consider the trace of the element (x ⊗ e g) on the left hand side of the equation. Via the coproduct, ∆, this element acts with trace zero unless g = 1 2. In this case, the trace becomes α(x)β(x) = (α ⊗ β)(x). Cases 2 and 10 are similar. Case 4: Consider the trace of the element (g K ⊗ e g K) on the left hand side. The only term in ∆ (g K ⊗ e g K) that has nonzero action is (g K ⊗ e 1) ⊗ (g K ⊗ e 1 g K). Thus the trace is α(g K)λ 2 (g K) if 1 = 1 and α(g K)λ − 2 (g K) if 1 is −1. Case 5: Consider the trace of (x ⊗ e 1 G) on the left hand side. There are two terms that have nonzero action. If g K has order 2, then the terms are (x ⊗ e g K) ⊗ (x ⊗ e g K) and (x ⊗ e g K z) ⊗ (x ⊗ e g K z). Hence (x ⊗ e 1 G) acts as 2χ(x)ψ(x)δ x∈C K. This character thus decomposes into two representations, each of

An explicit fusion algebra isomorphism for twisted quantum doubles of finite groups

“…This was proved in [3] for p = 2, but much of that proof holds here. In particular, the two sets in question still have the same cardinality.…”

“…This article extends [3] to include odd primes. By demonstrating that D(G) and D ω (E) have isomorphic fusion algebras, we provide another family of examples involving (untwisted) quantum doubles of nonabelian groups and twisted quantum doubles of abelian groups with nonabelian cocycles (in the sense of [8]).…”

“…Representations of D(G) are induced from representations of centralizers of G. See [1,3,5,7] for details. Let K be a conjugacy class of G, g K ∈ K, and let C K = C G (g K ).…”

“…Let K be a noncentral conjugacy class. Then C K affords exactly p(p − 1) irreducible representations of dimension p n−1 (on which Z acts nontrivially) and p 2n−1 one-dimensional irreducibles (on which Z acts trivially) [3,Lemma A.2].…”