Algebraic Graph Theory - Akihiro Munemasa

Monday, April 22, 2024 8:00 pm - 9:00 pm EDT (GMT -04:00)

Title: Abelian covers of association schemes with applications to SIC-POVM

Speaker: Akihiro Munemasa
Affiliation: Tohoku University
Location: Please contact Sabrina Lato for Zoom link.

Abstract: Godsil and Hensel (1992) developed a theory of abelian covers of complete graphs to construct antipodal distance-regular graphs of diameter 3. More recently, Coutinho, Godsil, Shirazi and Zhan (2016) showed that equiangular tight frames can be constructed from covers of complete graphs in terms of cyclic groups of prime order. In this talk, we introduce covers of (not necessarily symmetric) association scheme of d classes in terms of an (not necessarily cyclic) abelian group G of order d. As applications, we show that amorphic pseudocyclic association schemes of class 2^k on n vertices can be used to construct distance-regular antipodal 2^k-covers of the complete graph with n+1 vertices, by taking G to be the elementary abelian group of order 2^k. We also show that a fissioned generalized hexagon of order (2,2) can be used to construct a 7-class association scheme on 256 points, by taking G to be the cyclic group of order 4. This scheme represents 64 lines in the complex 8-dimensional space, forming a SIC-POVM. This talk is based on joint work with Jesse Lansdown.