The Uniqueness of the $\mathrm{L}_2$ Association Scheme

“…The fifth column lists the number of generously transitive groups of degree n which are 2-closed, or equivalently the number of Schurian association schemes. The smallest number of points in a nonSchurian association scheme is 16; an example of such a scheme is the Shrikhande strongly regular graph [31].…”

Section: History and Terminologymentioning

confidence: 99%

Coherent configurations, association schemes and permutation groups

Cameron¹

2003

Groups, Combinatorics and Geometry

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Coherent configurations are combinatorial objects invented for the purpose of studying finite permutation groups; every permutation group which is not doubly transitive preserves a non-trivial coherent configuration. However, symmetric coherent configurations have a much longer history, having been used in statistics under the name of association schemes.The relationship between permutation groups and association schemes is quite subtle; there are groups which preserve no non-trivial association scheme, and other groups for which there is not a unique minimal association scheme. This paper gives a brief outline of the theory of coherent configurations and association schemes, and reports on some recent work on the connection between association schemes and permutation groups.

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“…Thus we have proved that ~ defines an association scheme with parameters v = s 2 , n x -2(8 -1) and p\ x = s -2 on the set of cosets of Σ But a well-known result of Shrikhande [16] implies that this is an Z/ 2 -association scheme (note that s Φ 4 by assumption), i.e., the cosets can be labelled P ίό (i, j = 1, , s) such that P iά ~ P hk if and only if i = h or j = k and (i, i) ^ (fe, Λ). We want to show that T lf " -, T s with Γ^ = Ui -^y i s a system of s transversals of Σ; then this may be used to extend Σ to a net of deficiency 1 by Lemma 2.11, which may then be completed using We remark that the case of deficiency 2 for μ = 1 had already been settled by Shrikhande [17].…”

Section: Re(a(p)\a(q))ϋ(a(q)\ A(p))mentioning

confidence: 77%