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

Abstract We consider a type of distance-regular graph = (X, R) Γ = (X, R) called a bilinear forms graph. We assume that the diameter D of Γ is at least 3. Fix adjacent vertices x, y X x, y ∈ X. In our first main result, we introduce an equitable partition of X that has 6D-2 6 D - 2 subsets and the following feature: for every subset in the equitable partition, the vertices in the subset are equidistant to x and equidistant to y. This equitable partition is called the (x, y) -partition of X. By definition, the subconstituent algebra T=T (x) T = T (x) is generated by the Bose-Mesner algebra of Γ and the dual Bose-Mesner algebra of Γ with respect to x. As we will see, for the (x, y) -partition of X the characteristic vectors of the subsets form a basis for a T -module U=U (x, y) U = U (x, y). In our second main result, we decompose U into an orthogonal direct sum of irreducible T -modules. This sum has five summands: the primary T -module and four irreducible T -modules that have endpoint one. We show that every irreducible T -module with endpoint one is isomorphic to exactly one of the nonprimary summands.