Bose-Mesner Algebra Research Articles

P-polynomial and bipartite coherent configurations

We introduce the notion of P-polynomial coherent configurations and show that they can have at most two fibers. We then introduce a class of two-fiber coherent configurations which have two distinguished bases for the coherent algebra, similar to the Bose-Mesner algebra of an association scheme. Examples of these bipartite coherent configurations include the P-polynomial class of distance-biregular graphs, as well as quasi-symmetric designs and strongly regular designs.

On the Terwilliger algebra of the group association scheme of Cn⋊C2

In 1992, Terwilliger introduced the notion of the Terwilliger algebra in order to study association schemes. The Terwilliger algebra of an association scheme A is the subalgebra of the complex matrix algebra, generated by the Bose-Mesner algebra of A and its dual idempotents with respect to a point x.In Bannai and Munemasa (1995) [3] determined the dimension of the Terwilliger algebra of abelian groups and dihedral groups, by showing that they are triply transitive (i.e., triply regular and dually triply regular). In this paper, we give a generalization of their results to the group association scheme of semidirect products of the form Cn⋊C2, where Cm is a cyclic group of order m≥2. Moreover, we will give the complete characterization of the Wedderburn components of the Terwilliger algebra of these groups.

Quantum walks on regular graphs with realizations in a system of anyons

We build interacting Fock spaces from association schemes and set up quantum walks on the resulting regular graphs (distance-regular and distance-transitive). The construction is valid for growing graphs and the interacting Fock space is well defined asymptotically for the growing graph. To realize the quantum walks defined on the spaces in terms of anyons we switch to the dual view of the association schemes and identify the corresponding modular tensor categories from the Bose–Mesner algebra. Informally, the fusion ring induced by the association scheme and a topological twist can be the basis for developing a modular tensor category and thus a system of anyons. Finally, we demonstrate the framework in the case of Grover quantum walk on distance-regular graph in terms of anyon systems for the graphs considered. In the dual perspective interacting Fock spaces gather a new meaning in terms of anyon collisions for the case of distance-regular graphs.

Factoring discrete-time quantum walks on distance regular graphs into continuous-time quantum walks

We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph X. Given that X has diameter d and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on X is a product of at most d commuting transition matrices of continuous-time quantum walks, each on some distance digraph of the line digraph of X. We also obtain a similar factorization for any graph X in a Bose Mesner algebra.

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.

Circulant association schemes on triples

Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an {\em association scheme on triples} (AST) and constructed examples of several families of ASTs. Many of their examples used 2-transitive permutation groups: the non-trivial ternary relations of the ASTs were sets of ordered triples of pairwise distinct points of the underlying set left invariant by the group; and the given permutation group was a subgroup of automorphisms of the AST. In this paper, we consider ASTs that do not necessarily admit 2-transitive groups as automorphism groups but instead a transitive cyclic subgroup of the symmetric group acts as automorphisms. Such ASTs are called {\em circulant} ASTs and the corresponding ternary relations are called {\em circulant relations}. We give a complete characterisation of circulant ASTs in terms of AST-regular partitions of the underlying set. We also show that a special type of circulant, that we call a {\em thin circulant}, plays a key role in describing the structure of circulant ASTs. We outline several open questions. 

IDEMPOTEN PRIMITIF SKEMA ASOSIASI GRUP UNTUK MATRIKS GRUP ATAS Z_3

Some combinatorial problems in Mathematics can be studied via association scheme. By this scheme, algebra structure called Bose-Mesner algebra can be obtained. In this article, we show the explicit forms of the idempotent primitive of an association scheme for the group of order 48. Keywords: Association scheme, idempotent primitif

Topological quantum structures from association schemes

Starting from an association scheme induced by a finite group and the corresponding Bose–Mesner algebra, we construct quantum Markov chains, their entangled versions, using the quantum probabilistic approach. Our constructions are based on the intersection numbers and their duals Krein parameters of the schemes. We make the connection for the first time between the fusion rules of anyonic particles evolving on a 2D surface to the Krein parameters of an association scheme. We consider braid group $$B_3$$ that describes the unitary dynamics of the anyons as the automorphism subgroup of the graphs. The dynamics induced by the fusions (and the adjoint splitting operations) may be viewed as the chain evolving on a growing graph and the braiding as automorphisms on a fixed graph. In our quantum probability framework, infinite iterations of the unitaries, which can encode algorithmic content for quantum simulations, can describe asymptotics elegantly if the particles are allowed to evolve coherently for a longer period. We define quantum states on the Bose–Mesner algebra which is also a von Neumann algebra as well as a Frobenius algebra to build the quantum Markov chains providing yet another perspective to topological computation, whereas frameworks such as Unitary Modular Categories can identify and characterize new anyonic systems our framework can build upon them within quantum probabilistic framework that are suitable for asymptotic analysis.

Bordered Complex Hadamard Matrices and Strongly Regular Graphs

We consider bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph. Examples include a complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph due to J. Wallis, F. Szollősi, and a family of Hadamard matrices given by Singh and Dubey. In this paper, we prove that there are no other bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph.

Leonard pairs, spin models, and distance-regular graphs

A Leonard pair is an ordered pair of diagonalizable linear maps on a finite-dimensional vector space, that each act on an eigenbasis for the other one in an irreducible tridiagonal fashion. In the present paper we consider a type of Leonard pair, said to have spin. The notion of a spin model was introduced by V.F.R. Jones to construct link invariants. A spin model is a symmetric matrix over C that satisfies two conditions, called the type II and type III conditions. It is known that a spin model W is contained in a certain finite-dimensional algebra N(W), called the Nomura algebra. It often happens that a spin model W satisfies W∈M⊆N(W), where M is the Bose-Mesner algebra of a distance-regular graph Γ; in this case we say that Γ affords W. If Γ affords a spin model, then each irreducible module for every Terwilliger algebra of Γ takes a certain form, recently described by Caughman, Curtin, Nomura, and Wolff. In the present paper we show that the converse is true; if each irreducible module for every Terwilliger algebra of Γ takes this form, then Γ affords a spin model. We explicitly construct this spin model when Γ has q-Racah type. The proof of our main result relies heavily on the theory of spin Leonard pairs.

Fractional revival and association schemes

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose–Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.

Study the metal-insulator transitions of bilayer graphene: Abelian group schemes approach

Abstract Bilayer graphene (BLG), as a two-dimensional crystalline form of carbon, with a controllable band gap, has been proposed as an alternative to graphene for nanoscale electronics. Through modeling single-electron transport in BLG, using Abelian group schemes Z m × Z m as a novel approach, the present paper has numerically studied metal-insulator transition using random matrix theory in doped AA-stacking BLG. In this mathematical technique, initially, Abelian group schemes and the underlying graph of honeycomb periodic lattice of the single-layer graphene were constructed. Then, the tight-binding Hamiltonian matrix of this layer was represented by the adjacency matrices of the graph. Through using these matrices, it is possible to generate m-dimensional Bose-Mesner algebra. Finally, by applying the wreath product of these matrices, the Hamiltonian was calculated. In fact, instead of the hopping integral between individual layers, the mathematical combination of their underlying graphs was considered. This would be generalized to more than two layers. Numerical results indicated that insulator to metal transition occurred at the threshold doping value C = 0.5 % , which has already been achieved. After this threshold value, by increasing doping, a fast quantum chaotic transition from Poisson to Wigner energy-level statistic distribution was observed. The paper has suggested that a sufficient difference in the concentration of dopant atoms at individual layers would be an efficient way of controlling metal-insulator transitions.

The perfect matching association scheme

We revisit the Bose-Mesner algebra of the perfect matching association scheme. Our main results are: 1. An inductive algorithm, based on solving linear equations, to compute the eigenvalues of the orbital basis elements given the central characters of the symmetric groups. 2. Universal formulas, as content evaluations of symmetric functions, for the eigenvalues of fixed orbitals. 3. An inductive construction of an eigenvector (the so called first Gelfand-Tsetlin vector) in each eigenspace leading to a different inductive algorithm (not using central characters) for the eigenvalues of the orbital basis elements.

A diagram associated with the subconstituent algebra of a distance-regular graph

In this paper we consider a distance-regular graph Γ . Fix a vertex x of Γ and consider the corresponding subconstituent algebra T = T ( x ) . The algebra T is the ℂ -algebra generated by the Bose-Mesner algebra M of Γ and the dual Bose-Mesner algebra M * of Γ with respect to x . We consider the subspaces M , M * , M M * , M * M , M M * M , M * M M * , … along with their intersections and sums. In our notation, M M * means Span{ R S ∣ R ∈ M , S ∈ M * } , and so on. We introduce a diagram that describes how these subspaces are related. We describe in detail that part of the diagram up to M M * + M * M . For each subspace U shown in this part of the diagram, we display an orthogonal basis for U along with the dimension of U . For an edge U ⊆ W from this part of the diagram, we display an orthogonal basis for the orthogonal complement of U in W along with the dimension of this orthogonal complement.

Complex Hadamard Matrices Attached to a 3-Class Nonsymmetric Association Scheme

In this paper we classify complex Hadamard matrices contained in the Bose-Mesner algebra of nonsymmetric 3-class association schemes. As a consequence of our classification, we have two infinite families and some small examples of complex Hadamard matrices contained in the Bose-Mesner algebra of a self-dual fission of a complete multipartite graph.

Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

Abstract We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

An A-invariant subspace for bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.

On a class of quaternary complex Hadamard matrices

We introduce a class of regular unit Hadamard matrices whose entries consist of two complex numbers and their conjugates for a total of four complex numbers. We then show that these matrices are contained in the Bose–Mesner algebra of an association scheme arising from skew Paley matrices.

On Mitchell's embedding theorem for a quasi-schemoid

A quasi-schemoid is a small category whose morphisms are colored with appropriate combinatorial data. In this paper, Mitchell's embedding theorem for a tame schemoid is established. The result allows us to give a cofibrantly generated model category structure to the category of chain complexes over a functor category with a schemoid as the domain. Moreover, a notion of Morita equivalence for schemoids is introduced and discussed. In particular, we show that every Hamming scheme of binary codes is Morita equivalent to the association scheme arising from the cyclic group of order two. In an appendix, we construct a new schemoid from an abstract simplicial complex, whose Bose–Mesner algebra is closely related to the Stanley–Reisner ring of the given complex.

Distance-regular graphs where the distance-d graph has fewer distinct eigenvalues

Let the Kneser graph K of a distance-regular graph Γ be the graph on the same vertex set as Γ, where two vertices are adjacent when they have maximal distance in Γ. We study the situation where the Bose–Mesner algebra of Γ is not generated by the adjacency matrix of K. In particular, we obtain strong results in the so-called ‘half antipodal’ case.