Applications of the Isolating Fusion Algorithm to Table Algebras and Association Schemes

Let $\mathbf{B}$ be a basis for an $r$-dimensional algebra $A$ over a field or commutative ring with unity. The semifusions of $\mathbf{B}$ are the partitions of $\mathbf{B}$ whose characteristic functions form the basis of a subalgebra of $A$, and fusions are semifusions that respect a given involution on $A$. In this paper, we give an algorithm for computing a minimal semifusion (or fusion) of $\mathbf{B}$ that isolates a prescribed list of disjoint sums of basis elements of $\mathbf{B}$, when such a semifusion (or fusion) exists. We apply this algorithm to three problems: (1) computing the fusion lattices for small association schemes of a given order; (2) producing explicit realizations of association schemes with transitive automorphism groups; and (3) producing examples of non-Schurian fusions of Schurian association schemes whose adjacency matrices have noncyclotomic eigenvalues. The latter is of interest to the open question asking whether association schemes with transitive automorphsm groups can have noncyclotomic character values.