Equi-isoclinic subspaces, covers of the complete graph, and complex conference matrices

TL;DR

This paper explores the overlooked connection between equi-isoclinic subspaces and covers of the complete graph, using representation theory and complex conference matrices to construct new mathematical objects with potential applications.

Contribution

It clarifies and extends Godsil and Hensel's 1992 work by detailing the connection with equi-isoclinic subspaces and applying it to construct new configurations in real and complex spaces.

Findings

Construction of $q+1$ planes in $ ext{R}^{q+1}$ for even prime powers $q>2$

Identification of parameters previously unlisted in literature

New insights into the interplay between equi-isoclinic subspaces and complex conference matrices

Abstract

In 1992, Godsil and Hensel published a ground-breaking study of distance-regular antipodal covers of the complete graph that, among other things, introduced an important connection with equi-isoclinic subspaces. This connection seems to have been overlooked, as many of its immediate consequences have never been detailed in the literature. To correct this situation, we first describe how Godsil and Hensel's machine uses representation theory to construct equi-isoclinic tight fusion frames. Applying this machine to Mathon's construction produces $q+1$ planes in $\mathbb{R}^{q+1}$ for any even prime power $q>2$. Despite being an application of the 30-year-old Godsil-Hensel result, infinitely many of these parameters have never been enunciated in the literature. Following ideas from Et-Taoui, we then investigate a fruitful interplay with complex symmetric conference matrices.