Computing Spectral Elimination Ideals

We present here an overview of the hypermatrix spectral decomposition deduced from the Bhattacharya-Mesner hypermatrix algebra. We describe necessary and sufficient conditions for the existence of a spectral decomposition. We further extend to hypermatrices the notion of resolution of identity and use them to derive hypermatrix analog of matrix spectral bounds. Finally we describe an algorithm for computing generators of the spectral elimination ideals which considerably improves on Groebner basis computation suggested in

Families of Association Schemes on Triples from Two-Transitive Groups

Association schemes on triples (ASTs) are ternary analogues of classical association schemes. Analogous to Schurian association schemes, ASTs arise from the actions of two-transitive groups. In this paper, we obtain the sizes and third valencies of the ASTs obtained from the two-transitive permutation groups by determining the orbits of the groups' two-point stabilizers. Specifically, we obtain these parameters for the ASTs obtained from the actions of $S_n$ and $A_n$, $PGU(3,q)$, $PSU(3,q)$, and $Sp(2k,2)$, $Sz(2^{2k+1})$ and $Ree(3^{2k+1})$, some subgroups of $A\Gamma L(k,n)$, some subgroups of $P\Gamma L(k,n)$, and the sporadic two-transitive groups. Further, we obtain the intersection numbers for the ASTs obtained from these subgroups of $P\Gamma L(k,n)$ and $A \Gamma L(k,n)$, and the sporadic two-transitive groups. In particular, the ASTs from these projective and sporadic groups are commutative.Comment: 20 pages, 5 table

Association schemes on triples from affine special semilinear groups

Association schemes on triples (ASTs) are 3-dimensional analogues of classical association schemes. If a group acts two-transitively on a set, the orbits of the action induced on the triple Cartesian product of that set yields an AST. By considering the actions of semidirect products of the affine special linear group ASL(k,n) with subgroups of the Galois group Gal(GF(n)), we obtain the sizes, third valencies, and intersection numbers of the ASTs obtained from subgroups of the affine special semilinear group.Comment: 6 page

Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems

Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations, including cometric Q-antipodal association schemes. After a review of imprimitivity, we show that an imprimitive association scheme is uniform if and only if it is dismantlable, and we cast these schemes in the broader context of certain --- uniform --- coherent configurations. We also give a third characterization of uniform schemes in terms of the Krein parameters, and derive information on the primitive idempotents of such a scheme. In the second half of the paper, we apply these results to cometric association schemes. We show that each such scheme is uniform if and only if it is Q-antipodal, and derive results on the parameters of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We revisit the correspondence between uniform indecomposable three-class schemes and linked systems of symmetric designs, and show that these are cometric Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class schemes in terms of only a few parameters, and show that any strongly regular graph with a ("non-exceptional") strongly regular decomposition gives rise to such a scheme. Hemisystems in generalized quadrangles provide interesting examples of such decompositions. We finish with a short discussion of five-class schemes as well as a list of all feasible parameter sets for cometric Q-antipodal four-class schemes with at most six fibres and fibre size at most 2000, and describe the known examples. Most of these examples are related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions, April 201

Uniformity in Association schemes and Coherent Configurations: Cometric Q-Antipodal Schemes and Linked Systems

2010 Mathematics Subject Classification. Primary 05E30, Secondary 05B25, 05C50, 51E12