NoteInfinite family of nonschurian separable association schemes
Introduction
A coherent configuration on a finite set can be thought as a special partition of for which the diagonal of is a union of classes. The number is called the degree of . If the diagonal is exactly a class of the partitions then is an association scheme. The notion of a coherent configuration goes back to Higman who studied permutation groups via analyzing invariant binary relations [2] and Weisfeiler and Leman in connection with the graph isomorphism problem [5]. For more details on coherent configurations we refer the readers to [1].
There are two crucial properties of coherent configurations, namely, schurity and separability. We explain them below. The problem of determining whether a given coherent configuration is schurian (separable) is among the most fundamental problems in the theory of coherent configurations.
Let be a permutation group on the set . Then the partition of into the -orbits of defines a coherent configuration on . A coherent configuration on is called schurian if for some , or equivalently, if . The mapping is a Galois correspondence between schurian coherent configurations and the -closed permutation groups [1, Theorem 2.2.8].
Schurian coherent configuration is usually studied in the frames of the permutation group theory. However, most of coherent configurations are nonschurian. By this reason, the theory of coherent configurations is sometimes called “group theory without groups”. One of the motivations to study Schurian coherent configurations comes from the graph isomorphism problem which is equivalent to finding the orbits of the automorphism group of a given coherent configuration. For a schurian coherent configuration, the latter can be done efficiently (see [1, Section 2.2.3]).
Let and be coherent configurations. An algebraic isomorphism between and is a bijection from to that preserves intersection numbers. A combinatorial isomorphism between and is a bijection from to that maps the basis relations of to the basis relations of . Every combinatorial isomorphism induces the algebraic one. However, the converse statement does not hold in general. A coherent configuration is called separable if every algebraic isomorphism from to another coherent configuration is induced by a combinatorial isomorphism. In this case is determined up to isomorphism by the tensor of its intersection numbers. It follows that if is constructed from a graph by the Weisfeiler–Leman algorithm then the isomorphism between this graph and any other graph can be verified efficiently in a pure combinatorial way (see [1, Section 4.6.1]).
It seems natural to ask how the schurity and separability are related. Actually, there are infinite families of nonschurian nonseparable coherent configurations as well as schurian separable coherent configurations [1]. Infinite families of schurian nonseparable coherent configurations arise from finite affine and projective planes [1, Sections 2.5.1–2.5.2]. However, to the best of the author’s knowledge, no infinite family of nonschurian separable coherent configurations is known. The following question was asked in [1, p. 65].
Question Whether there exists an infinite family of nonschurian separable coherent configurations?
We give an affirmative answer to this question. More precisely, we prove the following theorem.
Theorem 1 For every prime , there exists a nonschurian association scheme of degree which is separable.
The author would like to thank Prof. I. Ponomarenko for drawing his attention to the problem and the anonymous referee for valuable comments which helped the author to improve the text significantly.
Section snippets
Construction
In this section we describe an infinite family of association schemes. In the next section it will be proved that they are nonschurian and separable. The schemes from this family are Cayley schemes. To make the description more clear, we describe firstly -rings (Schur rings) corresponding to them. In what follows, we freely use basic facts concerned with coherent configurations and -rings. For a background of coherent configurations and -rings, we refer the readers to the monograph [1] and
Proof of Theorem 1
We start with auxiliary lemma.
Lemma 3.1 Let be an association scheme, a thin parabolic of , and . If for every there exists such that then .
Proof Let us prove that for every . Suppose that is the class of , containing . By the condition of the lemma, there exists such that . Let such that . Note that because . So . Since and belong to the same class of , the pair belongs to . Therefore and hence
CRediT authorship contribution statement
Grigory Ryabov: Investigation, Writing - original draft.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The work is supported by Russian Scientific Fund (project No. 19-11-00039).
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