Strengthening the Balanced Set Condition for the Distance-Regular Graph of the Bilinear Forms

Abstract We consider a distance-regular graph $$\Gamma =(X, \mathcal {R})$$ Γ = ( X , R ) called the bilinear forms graph $$H_q(D,N-D)$$ H q ( D , N - D ) ; we assume $$N>2D\ge 6$$ N > 2 D ≥ 6 and $$q \not =2$$ q ≠ 2 . We show that $$\Gamma $$ Γ satisfies the following strengthened version of the balanced set condition. For a vertex $$x \in X$$ x ∈ X and $$0 \le i \le D$$ 0 ≤ i ≤ D , define $$\Gamma _i(x)=\lbrace y \in X\mid \partial (x,y)=i\rbrace $$ Γ i ( x ) = { y ∈ X ∣ ∂ ( x , y ) = i } , where $$\partial $$ ∂ denotes the path-length distance function. Abbreviate $$\Gamma (x)=\Gamma _1(x)$$ Γ ( x ) = Γ 1 ( x ) . Let $$V={\mathbb {R}}^X$$ V = R X denote the standard module for $$\textrm{Mat}_X(\mathbb {R})$$ Mat X