Homogeneous coherent configurations from spherical buildings and other edge-coloured graphs

TL;DR

This paper generalizes the concept of distance-regular graphs to a broader class of edge-coloured graphs, including buildings and affine planes, establishing new links between their algebraic structures and symmetry properties.

Contribution

It introduces a framework for constructing coherent configurations from edge-coloured graphs with a distance function, extending classical association scheme theory to new geometric contexts.

Findings

Bose-Mesner algebra is isomorphic to the adjacency algebra of the graph.

Strong transitivity implies the existence of a canonical distance function.

Affine planes with strongly transitive groups are necessarily Desarguesian.

Abstract

We study a class of edge-coloured graphs, including the chamber systems of buildings and other geometries such as affine planes, from which we build coherent configurations (also known as non-commutative association schemes). The condition we require is that the graph be endowed with a certain distance function, taking its values in the adjacency algebra (itself generated by the adjacency operators). When all the edges are of the same colour, the condition is equivalent to the graph being distance-regular, so our result is a generalization of the classical fact that distance-regular graphs give rise to association schemes. The Bose-Mesner algebra of the coherent configuration is then isomorphic to the adjacency algebra of the graph. The latter is more easily computed, and comes with a "small" set of generators, so we are able to produce examples of Bose-Mesner algebras with particularly simple presentations. When a group acts "strongly transitively", in a certain sense, on a graph, we show that a distance function as above exists canonically; moreover, when the graph is (the chamber system of) a building, we show that strong transitivity is equivalent to the usual condition involving pairs of incident chambers and apartments. We study affine planes in detail. These are not buildings, yet the machinery developed allows us to state and prove some results which are directly analogous to classical facts in the theory of projective planes (which {\em are} buildings). In particular, we prove that an affine plane with a group acting strongly transitively on it must be Desarguesian.