Fusion and Supercharacters

In this chapter the idea of fusion of the character table of a group is pursued in more detail. First the question of which groups have the property that their character table is a fusion of that of an abelian group is addressed. It proved difficult to answer this question but many results can be obtained. There is given an explicit description of the finite groups whose character tables fuse from a cyclic group. Then there is given an account of how the idea of fusion was independently discovered and used in the context of upper triangular groups UT_n(q) by Diaconis and Isaacs. Their motive was that whereas the character tables of UT_n(q) are “wild” certain fusions are not and random walks on the groups can be discussed. The interesting result that a fusion of the character table gives rise to a Hopf algebra is presented. There is also given a construction of a fusion of the character table of UT₄(q) by taking the class algebra of a loop constructed by an alternative multiplication on the elements on the elements of the group.