On a Class of Non-commutative Imprimitive Association Schemes of Rank 6

Abstract

Many facts about group theory can be generalized to the context of the theory of association schemes. In particular, association schemes with fewer than 6 elements are all commutative. While there is a nonabelian group with 6 elements which is unique up to isomorphism, there are infinitely many isomorphism classes of noncommutative association schemes with 6 elements. All examples previously known to us are imprimitive, and fall into three classes which are reasonably well understood. In this paper, we construct a fourth class of noncommutative, imprimitive association schemes of rank 6.

Noncommutative reality-based algebras of rank 6

“…Example 5. The imprimitive rank 6 association schemes introduced by Drabkin and French [14] that arise from complete p-arrays for Mersenne primes p have linear character values…”

“…So by(14), 1/4 + s 2 4 = δ 4 − (δ 4 − 1)/2. Thus, s 4 = ± By renumbering b 4 and b 5 if necessary, we may assume that s 4= δ 4 −1 2 b 4 + δ 4 +1 2 b * 4 .…”