Codes and Expansions (CodEx) Seminar

Bill Martin (Worcester Polytechnic Institute)
Schoenberg's Theorem and Association Schemes

Consider a set of lines through the origin in \(\mathbb{R}^m\) admitting just two distinct angles between pairs. Examples include the diagonals of the dodecahedron and any set of real mutually unbiased bases. We are interested in the case where the unit vectors along these lines generate an association scheme. A set of \(v\) distinct unit vectors in \(\mathbb{R}^m\) with \(v \times v\) Gram matrix \(G\) generates an association scheme if the vector space spanned by its entrywise powers \(\{ J, G, G \circ G, G\circ G \circ G, \ldots \}\) is closed under matrix multiplication. Of particular interest are the cometric (or \(Q\)-polynomial) association schemes that arise in this way. Akin to DRACKNs, these give double covers of strongly regular graphs and there are many open questions about the existence of such objects. In this talk, we apply Schoenberg's Theorem to rule out feasible parameter sets for such schemes. The talk is mainly based on the 2019 thesis work of Brian Kodalen.