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

TL;DR

This paper introduces an algorithm for computing minimal semifusions and fusions in algebraic structures, with applications to association schemes, automorphism groups, and eigenvalue properties, advancing understanding of algebraic combinatorics.

Contribution

The paper presents a novel isolating fusion algorithm and demonstrates its use in analyzing association schemes and their automorphisms, including non-Schurian fusions.

Findings

Computed fusion lattices for small association schemes.

Produced explicit realizations with transitive automorphism groups.

Generated examples of non-Schurian fusions with noncyclotomic eigenvalues.

Abstract

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.