Bose-Mesner Algebras Research Articles

Homogeneous Coherent Configurations from Spherical Buildings and Other Edge-Coloured Graphs

We study a class of edge-coloured graphs, including the chamber systems of buildings and other geometries such as affine planes, from which we build homogeneous coherent configurations (also known as association schemes). The condition we require is that the graph be endowed with a certain distance function, taking its values in the adjacency algebra (itself generated by the adjacency operators). When all the edges are of the same colour, the condition is equivalent to the graph being distance-regular, so our result is a generalization of the classical fact that distance-regular graphs give rise to association schemes. The Bose-Mesner algebra of the coherent configuration is then isomorphic to the adjacency algebra of the graph. The latter is more easily computed, and comes with a “small” set of generators, so we are able to produce examples of Bose-Mesner algebras with particularly simple presentations. When a group G acts “strongly transitively”, in a certain sense, on a graph, we show that a distance function as above exists canonically; moreover, when the graph is a building, we show that strong transitivity is equivalent to the usual condition of transitivity on pairs of incident chambers and apartments. As an example, we show that the action of the Mathieu group $$M_{24}$$ on its usual geometry is not strongly transitive. We study affine planes in detail. These are not buildings, yet the machinery developed allows us to state and prove some results which are directly analogous to classical facts in the theory of projective planes (which are buildings). In particular, we prove a variant of the Ostrom-Wagner Theorem on Desarguesian planes which holds in both the projective and the affine case.

A duality of scaffolds for translation association schemes

Scaffolds are certain tensors arising in the study of association schemes, and have been (implicitly) understood diagrammatically as digraphs with distinguished “root” nodes and with matrix edge weights, often taken from Bose–Mesner algebras. In this paper, we first present a slight modification of Martin's conjecture (2021) concerning a duality of scaffolds whose digraphs are embedded in a closed disk in the plane with root nodes all lying on the boundary circle, and then show that this modified conjecture holds true if we restrict ourselves to the class of translation association schemes, i.e., those association schemes that admit abelian regular automorphism groups.

Scaffolds: A graph-theoretic tool for tensor computations related to Bose-Mesner algebras

We introduce a pictorial notation for certain tensors arising in the study of association schemes, based on earlier ideas of Terwilliger, Neumaier and Jaeger. These tensors, which we call “scaffolds”, obey a simple set of rules which generalize common linear-algebraic operations such as trace, matrix product and entrywise product. We first study an elementary set of “moves” on scaffolds and illustrate their use in combinatorics. Next we re-visit results of Dickie, Suzuki and Terwilliger. The main new results deal with the relationships among vector spaces of scaffolds with edge weights chosen from a fixed coherent algebra and various underlying diagrams. As one consequence, we provide simple descriptions of the Terwilliger algebras of triply regular and dually triply regular association schemes. We finish with a conjecture connecting the duality of Bose-Mesner algebras to the graph-theoretic duality of circular planar graphs.

Wedge-direct sums of table algebras and applications to association schemes, I

In the remarkable paper (Blau, J. Algebr. 396:220–271, 2013) on fusion rings, Blau introduced a very important operation on two table algebras (the Blau-construction). It is very difficult to describe the structure of table algebras obtained by recursively applying the Blau-construction. In this paper, we define the wedge-direct sum for a sequence of table algebras. Using the wedge-direct sum of a sequence of table algebras, we are able to not only give a clear description of the structure of table algebras obtained by recursively applying the Blau-construction but also obtain the characterization and classification of a class of p-table algebras. The structure of the Bose-Mesner algebras of the wreath products and wedge products of association schemes can also be clearly described in terms of the wedge-direct sum. As an application, we get some new and known results about the wreath products and wedge products of association schemes.

On pseudocyclic table algebras and applications to pseudocyclic association schemes

Pseudocyclic table algebras are generalizations of Bose-Mesner algebras of pseudocyclic association schemes. Constructions of pseudocyclic association schemes have been studied among others by Hollmann [Hh], Hollmann and Xiang [HX] and Mathon [M]. In this paper we prove some general properties of pseudocyclic association schemes via the study of pseudocyclic table algebras. In particular, we will show that subschemes and quotient schemes of pseudocyclic association schemes are also pseudocyclic association schemes, and combinatorial morphisms between any two t-pseudocyclic association schemes induce algebra homomorphisms between their Bose-Mesner algebras. Furthermore, we will prove that an algebra isomorphism between two t-pseudocyclic table algebras under some weak condition is an exact isomorphism. Properties of pseudocyclic P-polynomial table algebras will also be studied.

On structures of modular adjacency algebras of Johnson schemes

In this paper, we consider algebras over a field of characteristic p , which are generated by adjacency algebras of Johnson schemes. If the algebra is semisimple, the structure is the same as that of the well-known Bose-Mesner algebras. We determine the structure of the algebra when it is not semisimple.

Hamming graphs in Nomura algebras

Let A be an association scheme on q ≥ 3 vertices. We show that the Bose–Mesner algebra of the generalized Hamming scheme H ( n , A ) , for n ⩾ 2 , is not the Nomura algebra of any type II matrix. This result gives examples of formally self-dual Bose–Mesner algebras that are not the Nomura algebras of type II matrices.

Group-Like Algebras and Their Representations

The concept of “group-like algebras” was defined by the author as a special class of bF algebras. They generalize scheme rings (Bose–Mesner algebras) of noncommutative association schemes. We develop the representation theory for group-like algebras and symmetric bF algebras. We also study group-like algebras of association-scheme type with dimension 2 and 3.

Calculating effective resistances on underlying networks of association schemes

Recently, in the work of Jafarizadeh et al. [J. Phys, A: Math. Theor. 40, 4949 (2007); e-print arXiv:0705.2480], calculation of effective resistances on distance-regular networks was investigated, where in the first paper, the calculation was based on stratification and Stieltjes functions associated with the network, whereas in the latter one a recursive formula for effective resistances was given based on the Christoffel–Darboux identity. In this paper, evaluation of effective resistances on more general networks that are underlying networks of association schemes is considered, where by using the algebraic combinatoric structures of association schemes such as stratification and Bose–Mesner algebras, an explicit formula for effective resistances on these networks is given in terms of the parameters of the corresponding association schemes. Moreover, we show that for particular underlying networks of association schemes with diameter d such that the adjacency matrix A possesses d+1 distinct eigenvalues, all of the other adjacency matrices Ai, i≠0, 1 can be written as polynomials of A, i.e., Ai=Pi(A), where Pi is not necessarily of degree i. Then, we use this property for these particular networks and assume that all of the conductances except for one of them, say, c≡c1=1, are zero to give a procedure for evfor a galuating effective resistances on these networks. The preference of this procedure is that one can evaluate effective resistances by using the structure of their Bose–Mesner algebra without any need to know the spectrum of the adjacency matrices.

Inheritance of hyper-duality in imprimitive Bose–Mesner algebras

We prove the following result concerning the inheritance of hyper-duality by block and quotient Bose–Mesner algebras associated with a hyper-dual pair of imprimitive Bose–Mesner algebras. Let M and M ˜ denote Bose–Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of M with respect to p to the subconstituent algebra of M ˜ with respect to p ˜ . Also suppose that M is imprimitive with respect to a subset I of Hadamard idempotents, so M ˜ is dual imprimitive with respect to the subset Ψ ( I ) of primitive idempotents, where Ψ : M → M ˜ is the formal duality associated with ψ . Let B denote the block Bose–Mesner algebra of M on the block containing p, and let Q ˜ denote the quotient Bose–Mesner algebra of M ˜ with respect to Ψ ( I ) . Then there is a hyper-duality from the subconstituent algebra of B with respect to p to the subconstituent algebra of Q ˜ with respect to p ˜ .

Investigation of continuous-time quantum walk on root lattice [formula omitted] and honeycomb lattice

The continuous-time quantum walk (CTQW) on root lattice A n (known as hexagonal lattice for n = 2 ) and honeycomb one is investigated by using spectral distribution method. To this aim, some association schemes are constructed from abelian group Z m ⊗ n and two copies of finite hexagonal lattices, such that their underlying graphs tend to root lattice A n and honeycomb one, as the size of the underlying graphs grows to infinity. The CTQW on these underlying graphs is investigated by using the spectral distribution method and stratification of the graphs based on Terwilliger algebra, where we get the required results for root lattice A n and honeycomb one, from large enough underlying graphs. Moreover, by using the stationary phase method, the long time behavior of CTQW on infinite hexagonal lattice is approximated with finite ones for finite distances from the origin while for large distances, the scaling behavior of the probability distribution is deduced. Also, it is shown that, the probability amplitudes at finite times possess the point group symmetry in the sense that two lattice points have the same probability amplitudes if and only if they belong to the same orbit (stratum of the graph) of the point group of the lattice. Apart from physical results, it is shown that the Bose–Mesner algebras of our constructed association schemes (called n-variable P-polynomial) can be generated by n commuting generators, where raising, flat and lowering operators (as elements of Terwilliger algebra) are associated with each generator. A system of n-variable orthogonal polynomials which are special cases of generalized Gegenbauer polynomials is constructed, where the probability amplitudes are given by integrals over these polynomials or their linear combinations. Finally the supersymmetric structure of finite honeycomb lattices is revealed.

On the number of irreducible modular representations of a P and Q polynomial scheme

Whereas results about the number of irreducible modular representations of finite groups are known, those of Bose–Mesner algebras of association schemes are not well understood. In this paper, we consider the number of irreducible modular representations of the adjacency algebra of some P and Q polynomial association schemes. These results should be helpful for investigating the structure of association schemes.

Bose–Mesner algebras attached to invertible Jones pairs

In 1989, Vaughan Jones introduced spin models and showed that they could be used to form link invariants in two different ways—by constructing representations of the braid group, or by constructing partition functions. These spin models were subsequently generalized to the so-called four-weight spin models by Bannai and Bannai; these could be used to construct partition functions, but did not lead to braid group representations in any obvious way. Jaeger showed that spin models were intimately related to certain association schemes. Yamada gave a construction of a symmetric spin model on 4 n vertices from each four-weight spin model on n vertices. In this paper, we build on recent work with Munemasa to give a different proof to Yamada's result, and we analyze the structure of the association scheme attached to this spin model.

On topological spin models and generalized Δ– Y transformations

We extend the exactly triply regular Bose–Mesner algebras and the associated Δ– Y transformations introduced by Jaeger [J. Algebraic Combin. 4 (1995) 103–144] to evaluate the partition function of a topological spin model. We also study the inverse problem for dual pairs of Bose–Mesner algebras in relation to topological spin models.

Type II Matrices and Their Bose-Mesner Algebras

Type II matrices were introduced in connection with spin models for link invariants. It is known that a pair of Bose-Mesner algebras (called a dual pair) of commutative association schemes are naturally associated with each type II matrix. In this paper, we show that type II matrices whose Bose-Mesner algebras are imprimitive are expressed as so-called generalized tensor products of some type II matrices of smaller sizes. As an application, we give a classification of type II matrices of size at most 10 except 9 by using the classification of commutative association schemes.

A Generalization of Strongly Regular Graphs

Motivated from an example of ridge graphs relating to metric polytopes, a class of connected regular graphs such that the squares of their adjacency matrices are in certain symmetric Bose-Mesner algebras of dimension 3 is considered in this paper as a generalization of strongly regular graphs. In addition to analysis of this prototype example defined over (MetP5)*, some general properties of these graphs are studied from the combinatorial view point.

On Even Generalized Table Algebras

Generalized table algebras were introduced in Arad, Fisman and Muzychuk (Israel J. Math. 114 (1999), 29–60) as an axiomatic closure of some algebraic properties of the Bose-Mesner algebras of association schemes. In this note we show that if all non-trivial degrees of a generalized integral table algebra are even, then the number of real basic elements of the algebra is bounded from below (Theorem 2.2). As a consequence we obtain some interesting facts about association schemes the non-trivial valencies of which are even. For example, we proved that if all non-identical relations of an association scheme have the same valency which is even, then the scheme is symmetric.

Association schemes generated by a non-symmetric relation of valency 2

We give some observations on association schemes with a relation of valency 2 from the representation theory of Bose–Mesner algebras and the basic structure theory. One of the applications of these observations is the classification of the association schemes with a non-symmetric relation of valency 2 if the cardinality of the point set is a product of two primes, and another is the proof of Li's conjecture that any finite simple group is a connected 2-DCI-group.

Bose-Mesner Algebras Associated with Four-Weight Spin Models

It is known that for each matrix W i and it's transpose t W i in any four-weight spin model (X, W 1, W 2, W 3, W 4; D), there is attached the Bose-Mesner algebra of an association scheme, which we call Nomura algebra. They are denoted by N(W i ) and N( t W i ) = N′(W i ) respectively. H. Guo and T. Huang showed that some of them coincide with a self-dual Bose-Mesner algebra, that is, N(W 1) = N′(W 1) = N(W 3) = N′(W 3) holds. In this paper we show that all of them coincide, that is, N(W i ), N′(W i ), i=1, 2, 3, 4, are the same self-dual Bose-Mesner algebra.

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let {\cal M} denote a Bose-Mesner algebra on a finite nonempty set i>X. Fix p \in X, and let {\cal M}^* and {\cal T} denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of {\cal M} with respect to i>p. By a hyper-duality of {\cal M}, we mean an automorphism ψ of {\cal T} such that \psi({\cal M}) = {\cal M}^*,\psi({\cal M}^*) = {\cal M}; \psi^{2}(A) ={}^{\rm t}\kern-1.3pt{A} for all A \in {\cal M}s and |X\,{|}\,\psi \rho is a duality of {\cal M}. {\cal M} is said to be hyper-self-dual whenever there exists a hyper-duality of {\cal M}. We say that {\cal M} is strongly hyper-self-dual whenever there exists a hyper-duality of {\cal M} which can be expressed as conjugation by an invertible element of {\cal T}. We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.