Browsing by Author "Joshi, Neha"

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    Fusions of association schemes
    (Faculty of Graduate Studies and Research, University of Regina, 2023-01) Joshi, Neha; Allen, Herman; Meagher, Karen; Fallat, Shaun; Floricel, Remus; Fan, Lisa; Sankey, Alyssa
    Since their introduction as symmetric coherent configurations by Bose and Mesner in 1959, association schemes have gained significant importance in algebraic combinatorics. An important breakthrough was achieved by Delsarte’s PhD thesis where he proved that many problems from coding theory, combinatorial design theory and statistics can be treated using the concept of association schemes [12]. Since its initial introduction, many algebraists and graph theorists have been studying the existence, construction and generalizations of various association schemes [1, 3, 7, 8, 17, 19, 28, 29, 30]. Because of their impressive construction, association schemes are useful to these subjects and there is always a search for new association schemes. One easy way to construct a new association scheme is by taking either direct or wreath products of two existing association schemes. One such example that we studied was given by Sankey in [32]. Another way to construct a new association scheme is by fusing specific relations of an existing association scheme. The resulting association scheme is known as a fusion (previously referred to as “subscheme” by mathematicians) [4]. The focus of this thesis is to examine a few important association schemes and classify them based on their fusions. It can be observed from literature that the nonexistence of nontrivial fusions is a rare phenomenon and this is the motivation behind this thesis. An association scheme, A, is said to be fusion-primitive if there does not exist any nontrivial fusions of A. In 1992, Muzychuk proved that there does not exist any nontrivial fusions of the Johnson scheme, J (n, k) for all n > 3k + 4 with k ≥ 4 [29]. In 1994, this result was further refined for all n > 3k + 1 by Uchida in [36]. In this thesis, we prove that the Johnson scheme does not have any nontrivial fusions for all n, n > 2k + 1 with k ≤ 20. In addition, we classify almost all of the multiplicity-free subgroups of the symmetric group based on their nontrivial fusions. The Hamming scheme, H(n, q) is an important example in coding theory [10]. The fusionprimitivity of the Hamming scheme has been discussed before (see [28]). In this thesis, we study the generalized Hamming scheme, H(n,A) and prove that it is fusion-imprimitive (that is, it always has a nontrivial fusion). We also classify all fusions of the generalized Hamming scheme, H(2,A) where A is the association scheme corresponding to a strongly-regular graph.