Year: 2025
Type: article
Abstract: In this article we determine all fusions of the association scheme $\mathcal{A} \otimes \mathcal{A}$, where $\mathcal{A}$ is the symmetric rank $3$ association scheme corresponding to a strongly regular graph. This includes both guaranteed fusions, which are fusions for all symmetric rank $3$ association schemes $\mathcal{A}$, and specific case fusions, which exist only under restrictions on the parameters of the association scheme. Along the sections we will determine the fusions of wreath products of strongly regular graphs and the fusions of the tensor square of a symmetric rank $3$ table algebra. This is an extension to the recent work of the authors and Meagher, where we solved the same problem for the generalized Hamming scheme $H(2,\mathcal{A})$ of the association scheme obtained from a strongly regular graph. The main results of this article prove that the families of strongly regular graphs for which $\mathcal{A} \otimes \mathcal{A}$ has a special case fusion are the same families for which $H(2,\mathcal{A})$ has a special case fusion; and that the imprimitive strongly regular graphs are the only family of strongly regular graphs for which the wreath product $\mathcal{A} \wr \mathcal{A}$ has a special case fusion.
DOI: 10.37236/13380
Authors: Allen Herman, Neha Joshi
Institutions: University of Regina, Arizona State University
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4409348349