We construct a pair of non-commutative rank 8 association schemes from a rank 3 non-symmetric association scheme. For the pair, two association schemes have the same character table but different Frobenius-Schur indicators. This situation is similar to the pair of the dihedral group and the quaternion group of order 8. We also determine the structures of adjacency algebras of them over the rational number field.
We give a sufficient condition for a non-commutative association scheme to have a fusion association scheme, and construct non-commutative association schemes from symmetric balanced generalized weighing matrices and generalized Hadamard matrices. We then apply the criterion to these non-commutative association schemes to obtain symmetric fusion association schemes.
Leading towards the classification of primitive commutative association schemes as the ultimate goal, Bannai and some of his school have been trying to * identify the major sources of (primitive) commutative association schemes, * collect known group-case primitive commutative association schemes, and * compute their character tables over the last twenty years. The construction of their character tables are important first step for a systematic study of such association schemes and towards the classification of those schemes. In this talk, we briefly survey the progress made in this direction …
For prime powers $q$ and $q+\varepsilon$ where $\varepsilon\in\{1,2\}$, an affine resolvable design from $\mathbb{F}_q$ and Latin squares from $\mathbb{F}_{q+\varepsilon}$ yield a set of symmetric designs if $\varepsilon=2$ and a set of symmetric group divisible designs if $\varepsilon=1$. We show that these designs derive commutative association schemes, and determine their eigenmatrices.
We prove an Assmus-Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with $s$ classes). This in particular generalizes the Assmus-Mattson-type theorems for $\mathbb{Z}_4$-linear codes due to Tanabe (2003) and Shin, Kumar, and Helleseth (2004), as well as the original theorem by Assmus and Mattson (1969). The weights of a code are $s$-tuples of non-negative integers in this case, and the conditions in our theorem for obtaining $t$-designs from the code involve concepts from polynomial interpolation in $s$ variables. The Terwillig…
Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.
We show that taking the set of primitive idempotents of commutative association schemes is a functor from the category of commutative association schemes with surjective morphisms to the category of finite sets with surjective partial functions. We then consider projective systems of commutative association schemes consisting of surjections (which we call profinite association schemes), for which Bose-Mesner algebra is defined, and describe a Delsarte theory on such schemes. This is another method for generalizing association schemes to those on infinite sets, related with the approach by Barg…
Association schemes are combinatorial objects that allow us solve problems in several branches of mathematics. They have been used in the study of permutation groups and graphs and also in the design of experiments, coding theory, partition designs etc. In this paper we show some techniques for computing properties of association schemes. The main framework arises from the fact that we can characterize completely the Bose-Mesner algebra in terms of a zero-dimensional ideal. A Gröbner basis of this ideal can be easily derived without the use of Buchberger algorithm in an efficient way. From thi…
Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand …
We classify the rank two commutative semifields which are 8-dimensional over their center $\mathbb{F}_{q}$. This is done using computational methods utilizing the connection to linear sets in $\mathrm{PG}(2,q^{4})$. We then apply our methods to complete the classification of rank two commutative semifields which are 10-dimensional over $\mathbb{F}_{3}$. The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.
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…
An association scheme is amorphous if it has as many fusion schemes as possible. Symmetric amorphous schemes were classified by A. V. Ivanov [A. V. Ivanov, Amorphous cellular rings II, in Investigations in algebraic theory of combinatorial objects, pages 39--49. VNIISI, Moscow, Institute for System Studies, 1985] and commutative amorphous schemes were classified by T. Ito, A. Munemasa and M. Yamada [T. Ito, A. Munemasa and M. Yamada, Amorphous association schemes over the Galois rings of characteristic 4, European J. Combin., 12(1991), 513--526]. A scheme is called skew-symmetric if the diagon…
We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci--Del Lungo--Pergola--Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations.
In this paper, we study commutative weakly distance-regular digraphs whose attached association schemes are regular, and give a characterization of mixed arcs. As an application, we classify such digraphs of diameter 2.
In this article we determine feasible parameter sets for (what could potentially be) commutative association schemes with noncyclotomic eigenvalues that are of smallest possible rank and order. A feasible parameter set for a commutative association scheme corresponds to a standard integral table algebra with integral multiplicities that satisfies all of the parameter restrictions known to hold for association schemes. For each rank and involution type, we generate an algebraic variety for which any suitable integral solution corresponds to a standard integral table algebra with integral multip…
An irreducible character $χ$ of an association scheme is called nonlinear if the multiplicity of $χ$ is greater than $1$. The main result of this paper gives a characterization of commutative association schemes with at most two nonlinear irreducible characters. This yields a characterization of finite groups with at most two nonlinear irreducible characters. A class of noncommutative association schemes with at most two nonlinear irreducible character is also given.
For any positive integer $m$, the complete graph on $2^{2m}(2^m+2)$ vertices is decomposed into $2^m+1$ commuting strongly regular graphs, which give rise to a symmetric association scheme of class $2^{m+2}-2$. Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived.
The notion of pseudocyclic association scheme is generalized to the non-commutative case. It is proved that any pseudocyclic scheme the rank of which is much more than the valency is the scheme of a Frobenius group and is uniquely determined up to isomorphism by its intersection number array. An immediate corollary of this result is that any scheme of prime degree, valency $k$ and rank at least $k^4$ is schurian.
We study cyclotomic association schemes over a finite commutative ring $R$ with identity. The main interest for us is to identify the normal cyclotomic schemes $C$, i.e. those for which $Aut(C)$ is a subgroup of the one-dimensional affine semilinear group over $R$. The problem is reduced to the case when the ring $R$ is local in which a necessary condition of normality in terms of the subgroup of $R^\times$ defining $C$, is given. This condition is proved to be sufficient for a class of local rings including the Galois rings of odd characteristic.
In this article, we investigate the existence and schurity problem of association schemes whose thin residues are isomorphic to an elementary abelian $p$-group of rank $2$.
We classify certain non-symmetric commutative association schemes. As an application, we determine all the primitive weakly distance-regular circulant digraphs.
An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete $4$-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known $2$-arc-transitive covers of the cube: the Möbius-Kantor graph, the Nauru graph, and the Foster g…
An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we study 4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of skew-symmetric amorphous association schemes, submitted for publication], we discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables and intersection numbers for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, …
In [3], Hanaki defined the Terwilliger algebras of association schemes over a commutative unital ring. In this paper, we call the Terwilliger algebras of association schemes over a field $\mathbb{F}$ the Terwilliger $\mathbb{F}$-algebras of association schemes and study the Terwilliger $\mathbb{F}$-algebras of quasi-thin association schemes. As main results, we determine the $\mathbb{F}$-dimensions, the semisimplicity, the Jacobson radicals, and the algebraic structures of the Terwilliger $\mathbb{F}$-algebras of quasi-thin association schemes. We also get some results with independent interes…
Let $(X,\mathcal{R})$ be a commutative association scheme and let $Γ=(X,R\cup R^\top)$ be a connected undirected graph where $R\in \mathcal{R}$. Godsil (resp., Brouwer) conjectured that the edge connectivity (resp., vertex connectivity) of $Γ$ is equal to its valency. In this paper, we prove that the deletion of the neighborhood of any vertex leaves behind at most one non-singleton component. Two vertices $a,b\in X$ are called "twins" in $Γ$ if they have identical neighborhoods: $Γ(a)=Γ(b)$. We characterize twins in polynomial association schemes and show that, in the absence of twins, the del…
We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the commutative case, reduce to rank two cluster algebras of affine type. The conjecture states that solutions are positive Laurent polynomials in the initial cluster variables. We prove this by use of a non-commutative version of the path models which we used for the commutative case.
We study polynomial generalizations of the Kontsevich automorphisms acting on the skew-field of formal rational expressions in two non-commuting variables. Our main result is the Laurentness and pseudo-positivity of iterations of these automorphisms. The resulting expressions are described combinatorially using a generalization of the combinatorics of compatible pairs in a maximal Dyck path developed by Lee, Li, and Zelevinsky. By specializing to quasi-commuting variables we obtain pseudo-positive expressions for rank 2 quantum generalized cluster variables. In the binomial case when all inter…
Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for t = Theta(sqrt(n)). Here we show that beating sqrt(n) would require substantially new algorithmic ideas, by proving a lower bound for the problem in the sum-of-squares (or Lasserre) hierarchy, the most powerful class of semi-definite programming algorithms we know of: r rounds of the sum-of-squares h…
We study the representations of non-commutative universal lattices and use them to compute lower bounds for the \TauC for the commutative universal lattices $G_{d,k}= \SL_d(\Z[x_1,...,x_k])$ with respect to several generating sets. As an application of the above result we show that the Cayley graphs of the finite groups $\SL_{3k}(\F_p)$ can be made expanders using suitable choice of the generators. This provides the first examples of expander families of groups of Lie type where the rank is not bounded and gives a natural (and explicit) counter examples to two conjectures of Alex Lubotzky and …
In this paper we will first present a generalization of the wedge product of association schemes to table algebras and give a necessary and sufficient condition for a table algebra to be the wedge product of two table algebras. Then we show that if the duals of two commutative table algebras are table algebras, then the dual of their wedge product is a table algebra, and is also isomorphic to the wedge product of the duals of those table algebras in the reverse order. Some applications to association schemes are also given.
It is well known that the adjacency algebra of an association scheme has the standard character. In this paper we first define the concept of standard character for C-algebras and we say that a C-algebra has the {\it standard character condition} if it has the standard character. Then we investigate some properties of C-algebras which have the standard character condition and prove that under some conditions a C-algebra has an adjacency algebra homomorphic image. In particular, we obtain a necessary and sufficient condition for which a commutative table algebra comes from an association scheme…
