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

We consider a type of distance-regular graph Γ =(X, R) Γ = ( X , R ) called a bilinear forms graph. We assume that the diameter D of Γ 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 Γ and the dual Bose-Mesner algebra of Γ 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.