Search for commutative fusion schemes in noncommutative association schemes

An association scheme X is a finite set X together with a set of binary relations R0, R1,…, Rd that satisfy certain regularity conditions. The adjacency algebra (Bose-Mesner algebra) associated with the scheme X consists of integral matrices closed under matrix addition, multiplication and the Hadamard product. Call a free Z-module supporting two multiplications in this way a “double product” algebra. An association scheme Y on X is a fusion scheme of X if each relation of Y is the union of relations of X . The study of fusion schemes in commutative association schemes is well established method to build new association schemes with specified properties. This work develops methods to construct commutative fusion schemes when X is not commutative. A principal method is to exploit “double product” homomorphic images. This leads to “lifting” issues that are addressed combinatorially. These methods are applied to the centralizer algebra of SlblSlb , where Sn denotes the symmetric group of degree n and Slb is the direct product of l copies of S b. The case l = b = 3 is studied in detail and several new commutative fusion schemes are found in this 55-dimensional algebra.