An equitable partition for the distance-regular graph of the bilinear forms

Paper

Abstract

We consider a type of distance-regular graph

called a bilinear forms graph. We assume that the diameter

is at least

. Fix adjacent vertices

. In our first main result, we introduce an equitable partition of

that has

subsets and the following feature: for every subset in the equitable partition, the vertices in the subset are equidistant to

and equidistant to

. This equitable partition is called the

-partition of

. By definition, the subconstituent algebra

is generated by the Bose-Mesner algebra of

and the dual Bose-Mesner algebra of

with respect to

. As we will see, for the

-partition of

the characteristic vectors of the subsets form a basis for a

-module

. In our second main result, we decompose

into an orthogonal direct sum of irreducible

-modules. This sum has five summands: the primary

-module and four irreducible

-modules that have endpoint one. We show that every irreducible

-module with endpoint one is isomorphic to exactly one of the nonprimary summands.