Adjacency Algebras Research Articles

Modular Adjacency Algebras of Hamming Schemes

To each association scheme G and to each field R, there is associated naturally an associative algebra, the so-called adjacency algebra RG of G over R. It is well-known that RG is semisimple if R has characteristic 0. However, little is known if R has positive characteristic. In the present paper, we focus on this case. We describe the algebra RG if G is a Hamming scheme (and R a field of positive characteristic). In particular, we show that, in this case, RG is a factor algebra of a polynomial ring by a monomial ideal.

Representations of Association Schemes and Their Factor Schemes

In the present paper we investigate the relationship between the complex representations of an association scheme G and the complex representations of certain factor schemes of G. Our first result is that, similar to group representation theory, representations of factor schemes over normal closed subsets of G can be viewed as representations of G itself. We then give necessary and sufficient conditions for an irreducible character of G to be a character of a factor scheme of G. These characterizations involve the central primitive idempotents of the adjacency algebra of G and they are obtained with the help of the Frobenius reciprocity low which we prove for complex adjacency algebras.

Locality of a modular adjacency algebra of an association scheme of prime power order

We will prove that a modular adjacency algebra of an association scheme of prime power order is a local algebra. Moreover, if the order is a prime, then the algebra is local symmetric.

Theory of Hecke algebras to association schemes

In the book entitled “Methods of Representation Theory” by Curtis and Reiner they discuss character tables of Hecke algebras. This paper aims to generalize their argument on Hecke algebras to the adjacency algebra of association schemes.

MORPHISMS OF DOUBLE FROBENIUS ALGEBRAS. II

The paper continues the study of morphisms of double Frobenius algebras (which include finite-dimensional Hopf algebras, character algebras, and adjacency algebras of association schemes). Here we consider morphisms satisfying a certain regularity condition.

Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians

On the adjacency algebra of a distance-regular graph we introduce an analogue of the Gibbs state depending on a parameter related to temperature of the graph. We discuss a scaling limit of the spectral distribution of the Laplacian on the graph with respect to the Gibbs state in the manner of central limit theorem in algebraic probability, where the volume of the graph goes to ∞ while the temperature tends to 0. In the model we discuss here (the Laplacian on the Johnson graph), the resulting limit distributions form a one parameter family beginning with an exponential distribution (which corresponds to the case of the vacuum state) and consisting of its deformations by a Bessel function.

Semisimplicity of adjacency algebras of association schemes

In this paper, we investigate the semisimplicity of adjacency algebras of association schemes over positive characteristic fields. Our main result is that the Frame number characterizes semisimplicity of an adjacency algebra. In a sense, this is a generalization of Maschke's theorem.

Morphisms of double frobenius algebras. i

Double Frobenius algebras (or dF-algebras) were recently introduced by the author. The concept includes finite-dimensional Hopf algebras, adjacency algebras of (non-commutative) association schemes, and C-algebras (character algebras). In this paper we define certain kinds of morphisms of dF-algebras and develop their basic theory. The morphisms generalize homomorphisms of Hopf algebras and of C-algebras.

On Nakayama Automorphisms of Double Frobenius Algebras

Double Frobenius algebras (or dF-algebras) were recently introduced by the author. The concept generalizes finite-dimensional Hopf algebras, adjacency algebras of (non-commutative) association schemes, andC-algebras (or character algebras). This paper studies basic properties of various Nakayama automorphisms in a dF-algebra. As an application the theorem of D. E. Radford, stating that the antipodeSof a finite-dimensional Hopf algebra is of finite order, is generalized to dF-algebras, and so is his formula forS4nin terms of certain group-like elements.

Compact cellular algebras and permutation groups

Keeping in mind the generalization of Birkhoff''s theorem on doubly stochastic matrices we define compact cellular algebras and compact permutation groups. Arising in this connection weakly compact graphs extend compact graphs introduced by G.~Tinhofer. It is proved that compact algebras are exactly the centralizer algebras of compact groups. A developed technique enables to get non-trivial examples of compact algebras and groups as well as completely identify compact Frobenius groups and the adjacency algebras of Johnson''s and Hamming''s schemes. In particular, Petersen''s graph proves to be not compact, which answers the question by C.~Godsil. A simple polynomial-time isomorphism test for the class of compact cellular algebras (weakly compact graphs) is presented.

Compact cellular algebras and permutation groups

Having in mind the generalization of Birkhoff's theorem on doubly stochastic matrices we define compact cellular algebras and compact permutation groups. Arising in this connection weakly compact graphs extend compact graphs introduced by G. Tinhofer. It is proved that compact algebras are exactly the centralizer algebras of compact groups. The technique developed enables us to get nontrivial examples of compact algebras and groups as well as completely identify compact Frobenius groups and the adjacency algebras of Johnson's and Hamming's schemes. In particular, Petersen's graph proves to be not compact, which answers a question by C. Godsil. Simple polynomial-time isomorphism tests for the classes of compact cellular algebras and weakly compact graphs are presented.

On Highly Closed Cellular Algebras and Highly Closed Isomorphisms

We define and study $m$-closed cellular algebras (coherent configurations) and $m$-isomorphisms of cellular algebras which can be regarded as $m$th approximations of Schurian algebras (i.e. the centralizer algebras of permutation groups) and of strong isomorphisms (i.e. bijections of the point sets taking one algebra to the other) respectively. If $m=1$ we come to arbitrary cellular algebras and their weak isomorphisms (i.e. matrix algebra isomorphisms preserving the Hadamard multiplication). On the other hand, the algebras which are $m$-closed for all $m\ge 1$ are exactly Schurian ones whereas the weak isomorphisms which are $m$-isomorphisms for all $m\ge 1$ are exactly ones induced by strong isomorphisms. We show that for any $m$ there exist $m$-closed algebras on $O(m)$ points which are not Schurian and $m$-isomorphisms of cellular algebras on $O(m)$ points which are not induced by strong isomorphisms. This enables us to find for any $m$ an edge colored graph with $O(m)$ vertices satisfying the $m$-vertex condition and having non-Schurian adjacency algebra. On the other hand, we rediscover and explain from the algebraic point of view the Cai-Fürer-Immerman phenomenon that the $m$-dimensional Weisfeiler-Lehman method fails to recognize the isomorphism of graphs in an efficient way.

On dualization of double frobenius algebras

Double Frobenius algebras (or dF-algebras) were recently introduced by the author. The concept generalizes finite-dimensional Hopf algebras, adjacency algebras of (non-commutative) association schemes, and C-algebras (or character algebras). In this paper we define a dualization construction of a dF-algebra, the so-called linear dual. We show that in the case of a Hopf algebra the linear dual gives the usual dual Hopf algebra and in the case of a C-algebra it essentially gives the usual Kawada’s dual.

On Pseudo-automorphisms and Fusions of an Association Scheme

wA pseudo-automorphism of an association scheme is an automorphism of its adjacency algebra with respect to both ordinary and Hadamard multiplications. An association scheme is said to be of G-type if it admits a group G of pseudo-automorphisms acting regularly on the set of adjacency matrices that are not the identity matrix. It is shown that an association scheme of G-type is amorphous if G is an elementary abelian 2-group.

On nonsymmetric P- and Q-polynomial association schemes

If a nonsymmetric P-polynomial association scheme, or equivalently, a distance-regular digraph, has diameter d and girth g, then d = g or d = g − 1, by Damerell's theorem. The dual of this theorem was proved by Leonard. In this paper, we prove that the diameter of a nonsymmetric P- and Q-polynomial association scheme is one less than its girth and its cogirth. We also give a structure theorem for a nonsymmetric Q-polynomial association scheme whose diameter is equal to its cogirth. We use self-duality and unimodality to show that the eigenvalues of a nontrivial nonsymmetric P- and Q-polynomial association scheme are quadratic over the rationals. The fact that the adjacency algebra becomes a C-algebra gives a necessary condition for the existence of a nonsymmetric P- and Q-polynomial association scheme. As an application, it is shown that the only nontrivial nonsymmetric P- and Q-polynomial association scheme with girth 5 is the directed 5 cycle.

Coherent algebras

Coherent algebras are defined to be the subalgebras of the matrix algebras M n ( C ) closed under Hadamard ( = coefficientwise) multiplication and containing the all 1 matrix, and are shown to be precisely the adjacency algebras of coherent configurations. Each such algebra has a type, which is a symmetric matrix with positive integer entries. The theory is illustrated by applications to quasisymmetric designs, which are essentially equivalent to coherent algebras of type 2 2 2 3 .

Intersection numbers for coherent configurations and the spectrum of a graph

It is shown that the intersection numbers of a coherent configuration are closely related to the determinant of a generic matrix for the corresponding adjacency algebra. This is important as both concepts provide isomorphism invariants for graphs.

Bilinear forms over a finite field, with applications to coding theory

Let Ω be the set of bilinear forms on a pair of finite-dimensional vector spaces over GF( q). If two bilinear forms are associated according to their q-distance (i.e., the rank of their difference), then Ω becomes an association scheme. The characters of the adjacency algebra of Ω, which yield the MacWilliams transform on q-distance enumerators, are expressed in terms of generalized Krawtchouk polynomials. The main emphasis is put on subsets of Ω and their q-distance structure. Certain q-ary codes are attached to a given X ⊂ Ω; the Hamming distance enumerators of these codes depend only on the q-distance enumerator of X. Interesting examples are provided by Singleton systems X ⊂ Ω, which are defined as t-designs of index 1 in a suitable semilattice (for a given integer t). The q-distance enumerator of a Singleton system is explicitly determined from the parameters. Finally, a construction of Singleton systems is given for all values of the parameters.