Commutative Association Schemes Research Articles

Generalised Classes in Groups and Association Schemes: Duals of Results on Characters and Sharpness

A result of Strunkov on generalised conjugacy classes of groups is most conveniently expressed in terms of the P-matrix of an association scheme. This result is dual to a result of Blichfeldt on permutation characters, which has been shown by Cameron and Kiyota to hold for arbitrary characters and led to the definition of sharp characters. We show that the result carries over to arbitrary (commutative) association scheme classes, and introduce the idea of a sharp class in an association scheme.

The Irreducible Modules of the Terwilliger Algebras ofDoob Schemes

Let Y be any commutative association scheme and we fix any vertex x of Y. Terwilleger introduced a non-commutative, associative, and semi-simple C-algebraT=T(x) for Y and x in [4]. We call T the Terwilliger (or subconstituent) algebra ofY with respect to x.Let \(W( \subset C^{|X|} ) \) be an irreducible T(x)-module. W is said to be thin if W satisfies a certain simple condition.Y is said to be thin with respect to x if each irreducible T(x) -module is thin. Y is said to be thin if Y is thin with respect to each vertex in X.The Doob schemes are direct product of a number of Shrikhande graphs and some complete graphsK 4 . Terwilliger proved in [4] that Doob scheme is not thin if the diameter is greater than two. I give the irreducible T(x)-modules of Doob schemes.

Commutative Association Schemes Whose Symmetrizations Have Two Classes

If a symmetric association scheme of class two is realized as the symmetrization of a commutative association scheme, then it either admits a unique symmetrizable fission scheme of class three or four, or admits three fission schemes, two of which are class three and one is of class four. We investigate the classification problem for symmetrizable (commutative) association schemes of two-class symmetric association schemes. In particular, we give a classification of association schemes whose symmetrizations are obtained from completely multipartite strongly regular graphs in the notion of wreath product of two schemes. Also the cyclotomic schemes associated to Paley graphs and their symmetrizable fission schemes are discussed in terms of their character tables.

Commutative association schemes whose symmetrizations have two classes

Commutative association schemes whose symmetrizations have two classes

Class 3 association schemes whose symmetrizations have two classes

For a given commutative association scheme L , by fusing all the non-self-paired relations pairwise with their symmetric counterparts, we can obtain a new symmetric association scheme L̃. In this paper, we investigate when a symmetric association scheme L̃ of class 2 admits a symmetrizable commutative fission scheme L of class 3. By studying the feasible parameter sets obtained from the acceptible fission character tables of given symmetric schemes, we discover that there are infinitely many symmetric association schemes of class 2, each of which admits one or two symmetrizable commutative fission schemes of class 3. We then characterize and classify some such symmetric schemes and their symmetrizable commutative fission schemes. Also we prove that none of the Hamming schemes H(2, q) with q ≥ 3, and none of the Johnson schemes J(ν, 2) with ν ≥ 5 and ν ≢ 7(mod 8) have such commutative fission schemes of class 3. H(2, 2) has a symmetrizable commutative fission scheme which is isomorphic to the group scheme L ( Z 4), and J(4, 2) has one such fission scheme coming from the action of the alternating group A 4 on the set of two-element subsets from a four-element set.

THE TERWILLIGER ALGEBRAS OF THE GROUP ASSOCIATION SCHEMES OF S5 AND A5

Terwilliger proposed a method for studying commutative association schemes by introducing a non-commutative, semi-simple C-algebra, whose structure reflects the combinatorial nature of the corresponding scheme, and applied the method to the P and Q polynomial schemes. In this paper, we continue the initial investigation of Bannai and Munemasa of the Terwilliger algebras of group association schemes. In particular, we determine the structure of the Terwilliger algebras of the group schemes of S 5 and A 5.

Character Tables of Fission Schemes and Fusion Schemes

For given commutative association schemes H = (X, {Ri}0⩽i⩽d and Y = (X, {Sj}0⩽j⩽e, if {Ri}0⩽i⩽d is a refinement of {Sj}0⩽j⩽e, then we say that X is a fission scheme of Y. It is shown that if each class Sj of Y is either equal to Ri for some i or obtained by fusing two Ri's in a simple uniform pattern, then the character table of one is determined by the other. This fact is proved by orthogonality relation of the character tables. Then the character table of the group scheme coming from the action of Chevalley group G2 (q) on the set Ω± of hyperplanes of type O±6(q) in the seven-dimensional orthogonal geometry over GF(q) is calculated from the character table of its fusion scheme, which is obtained from the action of O7(q) on Ω±, respectively. Further application of the main result is discussed in various other examples.

The Subconstituent Algebra of an Association Scheme, (Part I)

We introduce a method for studying commutative association schemes with “many” vanishing intersection numbers and/or Krein parameters, and apply the method to the P- and Q-polynomial schemes. Let Y denote any commutative association scheme, and fix any vertex x of Y. We introduce a non-commutative, associative, semi-simple $$\mathbb{C}$$ -algebra T = T(x) whose structure reflects the combinatorial structure of Y. We call T the subconstituent algebra of Y with respect to x. Roughly speaking, T is a combinatorial analog of the centralizer algebra of the stabilizer of x in the automorphism group of Y. In general, the structure of T is not determined by the intersection numbers of Y, but these parameters do give some information. Indeed, we find a relation among the generators of T for each vanishing intersection number or Krein parameter. We identify a class of irreducible T-moduIes whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say Y is thin if every irreducible T(y)-module is thin for every vertex y of Y. We compute the possible thin, irreducible T-modules when Y is P- and Q-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If Y is assumed to be thin, then “sufficiently large dimension” means “dimension at least four”. We give a combinatorial characterization of the thin P- and Q-polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible T-modules actually occur. We close with some conjectures and open problems.

Subschemes of some association schemes

By using the character tables of known commutative association schemes, we can, in many instances, construct some new commutative association schemes as their subschemes. In particular, we obtain many new non-group-case association schemes from the group association schemes of finite simple groups. Although examples worked out in this paper are limited to some special cases, it might be possible that this phenomenon occurs universally for all the group association schemes of finite Chevalley groups. The subschemes obtained in this way could be considered as an association scheme version of Weyl groups for Chevalley groups.

Splitting fields of association schemes

The splitting field K of a commutative association scheme is the extension of the rationals by the adjunction of all eigenvalues of the association scheme. Let L be a subfield of K containing all the Krein parameters. It is shown that the Galois group of K/L is contained in the center of the Galois group of K/ℚ. In particular, if the Krein parameters are all rational, then the eigenvalues are contained in a cyclotomic number field.

The character table of the commutative association scheme coming from the action of $GL(n,q)$ on non-incident point-hyperlane pairs

The character table of the commutative association scheme coming from the action of $GL(n,q)$ on non-incident point-hyperlane pairs

Algebraic decompositions of commutative association schemes

Algebraic decompositions of commutative association schemes

Tannaka-Krein duality for association schemes

A duality theorem is formulated for noncommutative association schemes. This duality theorem contains as special cases (1) the Delsarte-Tamaschke duality theorem (which was essentially obtained by Kawala in 1942) for commutative association schemes, and (2) the Tannaka-Krein duality theorem for arbitrary finite groups.