Tremain equiangular tight frames

Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.Comment: 11 page

Tremain Equiangular Tight Frames

Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph

Hypercubes, Leonard triples and the anticommutator spin algebra

This paper is about three classes of objects: Leonard triples, distance-regular graphs and the modules for the anticommutator spin algebra. Let \K denote an algebraically closed field of characteristic zero. Let $V$ denote a vector space over \K with finite positive dimension. A Leonard triple on $V$ is an ordered triple of linear transformations in $\mathrm{End}(V)$ such that for each of these transformations there exists a basis for $V$ with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. The Leonard triples of interest to us are said to be totally B/AB and of Bannai/Ito type. Totally B/AB Leonard triples of Bannai/Ito type arise in conjunction with the anticommutator spin algebra $\mathcal{A}$, the unital associative \K-algebra defined by generators $x,y,z$ and relations$$xy+yx=2z,\qquad yz+zy=2x,\qquad zx+xz=2y.$$ Let $D\geq0$ denote an integer, let $Q_{D}$ denote the hypercube of diameter $D$ and let $\tilde{Q}_{D}$ denote the antipodal quotient. Let $T$ (resp. $\tilde{T}$) denote the Terwilliger algebra for $Q_{D}$ (resp. $\tilde{Q}_{D}$). We obtain the following. When $D$ is even (resp. odd), we show that there exists a unique $\mathcal{A}$-module structure on $Q_{D}$ (resp. $\tilde{Q}_{D}$) such that $x,y$ act as the adjacency and dual adjacency matrices respectively. We classify the resulting irreducible $\mathcal{A}$-modules up to isomorphism. We introduce weighted adjacency matrices for $Q_{D}$, $\tilde{Q}_{D}$. When $D$ is even (resp. odd) we show that actions of the adjacency, dual adjacency and weighted adjacency matrices for $Q_{D}$ (resp. $\tilde{Q}_{D}$) on any irreducible $T$-module (resp. $\tilde{T}$-module) form a totally bipartite (resp. almost bipartite) Leonard triple of Bannai/Ito type and classify the Leonard triple up to isomorphism.Comment: arXiv admin note: text overlap with arXiv:0705.0518 by other author

Three-point bounds for energy minimization

Three-point semidefinite programming bounds are one of the most powerful known tools for bounding the size of spherical codes. In this paper, we use them to prove lower bounds for the potential energy of particles interacting via a pair potential function. We show that our bounds are sharp for seven points in RP^2. Specifically, we prove that the seven lines connecting opposite vertices of a cube and of its dual octahedron are universally optimal. (In other words, among all configurations of seven lines through the origin, this one minimizes energy for all potential functions that are completely monotonic functions of squared chordal distance.) This configuration is the only known universal optimum that is not distance regular, and the last remaining universal optimum in RP^2. We also give a new derivation of semidefinite programming bounds and present several surprising conjectures about them.Comment: 30 page