Noncommutative standard table algebras with at most one nontrivial multiplicity

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

“…In searching the database of small association schemes, we find several examples noncommutative association schemes of rank 7 that have 5 * -invariant relations. It is possible, as Antonou and Blau do in [1], to construct noncommutative rank 7 RBAs with 3 * -invariant basis elements as partial wreath products of nonreal rank 3 RBAs with the noncommutative rank 5 RBAs discussed in [6]. As for noncommutative rank 7 RBAs with only one * -invariant element, the above analysis shows their character table will have the form…”

“…Non-integral RBAs with postive degree map will exist, as it is relatively straightforward to produce an admissible character table (with rational character values and multiplicities), and use it to construct a representation affording χ following the approach of Antonou and Blau in [1]. For example, we have constructed a noncommutative rank 7 RBA with postive degree map that has this (admissible) character table: (Here the real quaternion algebra H is considered in its usual basis {1, i, j, k} with the involution (t + xi + yj + zk) * = t − xi − yj − zk.…”