Imprimitive cometric association schemes: Constructions and analysis

Abstract

Dualizing the "extended bipartite double" construction for distance-regular graphs, we construct a new family of cometric (or Q-polynomial) association schemes with four associate classes based on linked systems of symmetric designs. The analysis of these new schemes naturally leads to structural questions concerning imprimitive cometric association schemes, some of which we answer with others being left as open problems. In particular, we prove that any Q-antipodal association scheme is dismantlable: the configuration induced on any subset of the equivalence classes in the Q-antipodal imprimitivity system is again a cometric association scheme. Further examples are explored.

New families of Q-polynomial association schemes

“…In [11], a method of constructing Q -bipartite association schemes is given, called the "extended Q -bipartite double" of a scheme. In this section, we attempt to reverse this process to obtain new schemes from the Q -bipartite schemes constructed earlier.…”

“…In particular, no quadrangle of odd order can give rise to such a scheme. From [11] we have that this scheme has an extended Q -bipartite double, which in this case will be the original scheme. 2…”

“…to [11] and [10]. In addition, there are some examples recently discovered by Martin, Muzychuk, and van Dam [12] which are constructed from hemisystems of generalized quadrangles found in [5].…”

Bounds on s-Distance Sets with Strength t

“…Suppose l = s, namely the scheme (X, R) is Q-bipartite. Then, by [15,Corolalry 4.2], the image of the embedding of the scheme into the unit sphere is an antipodal set. Therefore it follows from [6] …”

Coherent configurations and triply regular association schemes obtained from spherical designs

“…The linked system of symmetric (2 2m , 2 2m−1 − 2 m−1 , 2 2m−2 − 2 m−1 ) designs constructs real MUB, see [17]. Let X be the embedding of Ω into the first eigenspace.…”