Noncommutative reality-based algebras of rank 6

Abstract

We classify the RBA-bases of $6$-dimensional noncommutative semisimple algebras for which the algebra has a positive degree map. We show that these RBAs are parametrized by seven real numbers, the first four of which are positive and the remaining three arbitrary. Our classification gives formulas for their standard bases and structure constants. Using these we give a list of all noncommutative integral table algebras of rank 6 with order up to 150. Four in the list are primitive, but we show these cannot be realized as adjacency algebras of association schemes. In the last section of the paper we apply our methods to give a precise description of the noncommutative integral table algebras of rank 6 for which the multiplicity of both linear characters is 1.Comment: 26 pages, to appear in Comm. Algebr

A survey of semisimple algebras in algebraic combinatorics

“…• The structure constants must satisfy the "handshaking lemma": if b * j = b j , then for all b i ∈ B with δ(b i ) > 1, λ i jiδ(b i ) must be even (for a proof see [31,Lemma 7]).…”

The search for small association schemes with noncyclotomic eigenvalues

“…. , r − 1}, if i = j, then (b i ) i,j k j must be even (see [8,Lemma 7]); • realizability of all closed subsets and quotients;…”

Schur indices for noncommutative reality-based algebras with two nonreal basis elements

“…The authors discussed noncommutative rank 5 RBAs in [6] and noncommutative rank 6 RBAs in [5]. In both cases Frobenius-Schur indicator theory implies that B has only one pair of nonreal basis elements.…”

“…The only finite groups that satisfy the hypothesis of our main theorem are the dihedral groups of order 6 and 8, but there are infinite families of table algebras and association schemes that satisfy it. It applies, for example, to all noncommutative RBAs of rank 5 and 6 with positive degree map that were discussed in [6] and [5]. Noncommutative RBAs of rank 7 can have one, two, or three pairs of nonreal elements, our theorem only applies to those with one nonreal pair.…”