Cometric Association Schemes

Last modified: June 2, 2006

These tables are joint work with Jason Williford with significant contributions from Akihiro Munemasa and Mikhail Muzychuk.

Select items from the table at left.

Overview

The table at left lists cometric (or Q-polynomial) association schemes. The first goal is to include all association schemes known to me which are cometric but not metric or character-theoretic duals of metric translation schemes. As of 2011, I will aim to mix in to the list all small Q-polynomial distance-regular graphs, perhaps up to 1024 vertices. But no such vertex limit is to be assumed in the non-metric case. On the one hand, our exhaustive search seldom goes beyond 100 vertices; on the other hand, we may include interesting examples even if they have a large number of vertices.

Metric schemes correspond to distance-regular graphs. They are well-studied and carefully prepared tables exist giving the known examples and open parameter sets, including a notation as to which are cometric. (See the book by Brouwer, Cohen and Neumaier.) Using the character theory of finite abelian groups, each metric translation scheme (e.g., any coset graph of an additive completely regular code) has a dual scheme which is cometric. Moreover, most metric translation schemes arise in this way. Nevertheless, we will now include the smaller ones here.

I know of three infinite families of cometric schemes which are not metric or duals of metric schemes. These correspond to linked systems of symmetric designs, extended Q-bipartite doubles of these, and some Q-bipartite doubles of Hermitian forms dual polar spaces -- a special class of distance-regular graphs.

For all the (roughly two dozen) remaining examples known to me, I list at left the following

the number of classes, d
the number of vertices, |X|
the Krein array {b₀^*, b₁^*, . . . , b_d-1^*; c₁^*, c₂^*, . . . , c_d^* } where the tridiagonal matrix L₁^* with (k,j)-entry equal to the Krein parameter q_1,j^k has diagonal entries
a_j^*= m₁ - b_j^*- c_j^*
and b^* above the diagonal and c^* below the diagonal
the dimension, m₁, of the first eigenspace in the/a Q-polynomial ordering
the sequence of cosines in the first eigenspace (Q_i,1/ m₁ : i=0,1, . . . , d)
a designation as to whether the scheme is primitive, Q-bipartite, Q-antipodal, or both
a description of the scheme, comments or references