Galois correspondence between permutation groups and cellular rings (association schemes)

Faradžev, I. A.; Иванов, А. А.; Klin, Mikhail

doi:10.1007/bf01787700

Cited by 33 publications

(17 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The textbook [1] as well as the lecture notes [45] are standard references for the foundations of the theory of association schemes. A more detailed exposition of the notions considered in this section can be found in [8,9].…”

Section: Basic Concepts and Duval's Main Resultsmentioning

confidence: 99%

Directed strongly regular graphs obtained from coherent algebras

Klin

Munemasa

Muzychuk

et al. 2004

Linear Algebra and its Applications

View full text Add to dashboard Cite

The notion of a directed strongly regular graph was introduced by A. Duval in 1988 as one of the possible generalizations of classical strongly regular graphs to the directed case. We investigate this generalization with the aid of coherent algebras in the sense of D.G. Higman. We show that the coherent algebra of a mixed directed strongly regular graph is a non-commutative algebra of rank at least 6. With this in mind, we examine the group algebras of dihedral groups, the flag algebras of a Steiner 2-designs, in search of directed strongly regular graphs. As a result, a few new infinite series of directed strongly regular graphs are constructed. In particular, this provides a positive answer to a question of Duval on the existence of a graph with certain parameter set having 20 vertices. One more open case ୋ This paper is a revised and shortened version of the preprint [29]. This preprint stimulated a new wave of interest in the investigations of d.s.r.g.'s, in particular results by L. Jorgensen, S. A. Hobart and T.J. Shaw, A. Duval and D. Iourinski. We refer to [11,21,25,30] where a part of recent investigations is discussed with more details. We also mention that the most of the current status of the theory of d.s.r.g.'s can be found in Andries Brouwer's home page

show abstract

Section: Basic Concepts and Duval's Main Resultsmentioning

confidence: 99%

Directed strongly regular graphs obtained from coherent algebras

Klin

Munemasa

Muzychuk

et al. 2004

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…This notion was introduced by Gol'fand and Klin [7]. For further results concerning this object see Ivanov [12] and a survey by Farad~ev et al [5]. The most remarkable theorems due to Ivanov [12] say that [V[ --m 2 and all irrefiexive Ri are graphs of Latin square type Lk,(m) or of negative Latin square type NLk,(m), (series K.1 or K.5 from [11]).…”

Section: Terminology and Statement Of Resultsmentioning

confidence: 99%

Skew-symmetric association schemes with two classes and strongly regular graphs of type L 2n?1(4n?1)

Ṗasechnik

1992

Acta Appl Math

View full text Add to dashboard Cite

A construction of a pair of strongly regular graphs Fn and Fin of type L2~-1 (4n -1) from a pair of skew-symmetric association schemes W, W t of order 4n -1 is presented. Examples of graphs with the same parameters as Fn and F~, i.e., of type L2n-1 (4n -1), were known only if 4n -1 = pS, where p is a prime. The first new graph appearing in the series has parameters (v, k, A) = (225, 98, 45). A 4-vertex condition for relations of a skew-symmetric association scheme (very similar to one for the strongly regular graphs) is introduced and is proved to hold in any case. This has allowed us to check the 4-vertex condition for Fn and Fin, thus to prove that Fn and F~ are not rank three graphs if n > 2. IntroductionThe main aim of this paper is to describe a construction of strongly regular graphs (SRG, for short) of type L:n-l(4n -1) from a pair of skew-symmetric Hadamard matrices of order 4n (Theorem 1 and Corollary 1). Examples of such graphs were known only if 4n -1 = pS, p is a prime. But it is well-known that skew-symmetric Hadamard matrices exist for a much wider range of n, cf., e.g., Hall [9]. In fact, the first new SRG, which appears in our series, has parameters (v, k, A) = (225, 98, 45). In the catalog of SRGs with small v due to Brouwer [2] there are no examples of graphs with the latter set of parameters. Some statements equivalent to Theorem 1 and Corollary 1 were announced by the author in [6].Questions concerning some further symmetry of our SRG are considered in the paper as well. There is the classification of rank three graphs due to Kantor and Liebler [13] and Liebeck [14], which uses the classification of finite simple groups. But we are able to prove (Theorem 2) using the so-called 4-vertex condition for an SRG, ( see, e.g., Hestenes and Higman [10]) that our graphs are not rank three graphs if n ) 2.Among the many different combinatorial objects associated with a skew-symmetric Hadamard matrix H of order 4n (see Reid and Brown [17], Delsarte [4], Szekeres [18], Hall [9]) there is a 2-class association scheme W = W(H) which we *Present address: Mathematics Subject Classification (1991). 05E30.

show abstract

“…We remark that the above proposition describes a Galois correspondence between S-rings over H and overgroups of H R in Sym(H ) (which is a particular case of a Galois correspondence between coherent configurations and permutation groups, see Faradzhev et al (1990);Weisfeiler (1976)). Note that, equality holds in (iii) if and only if S is Schurian, and equality holds in (iv) if and only if G is a 2-closed group.…”

Section: Proposition 21 Let S and B Arbitrary S-rings Over H And Lmentioning

confidence: 97%

A construction of the automorphism groups of indecomposable S-rings over $${Z_{2^n}}$$

Kovács

2011

Beitr Algebra Geom

View full text Add to dashboard Cite

show abstract

Galois correspondence between permutation groups and cellular rings (association schemes)

Cited by 33 publications

References 43 publications

Directed strongly regular graphs obtained from coherent algebras

Directed strongly regular graphs obtained from coherent algebras

Skew-symmetric association schemes with two classes and strongly regular graphs of type L 2n?1(4n?1)

A construction of the automorphism groups of indecomposable S-rings over $${Z_{2^n}}$$

Contact Info

Product

Resources

About