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

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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 ( R ) . For $$x\in X$$ x ∈ X , let $$\hat{x} \in V$$ x ^ ∈ V have x -coordinate 1 and all other coordinates 0. Let $$E \in \textrm{Mat}_X(\mathbb {R})$$ E ∈ Mat X ( R ) denote the primitive idempotent that corresponds to the second largest eigenvalue of the adjacency matrix of $$\Gamma$$ Γ . For a subset $$\Omega \subseteq X$$ Ω ⊆ X , define $$\widehat{\Omega }= \sum _{x \in \Omega } \hat{x}$$ Ω ^ = ∑ x ∈ Ω x ^ . We fix two vertices $$x,y \in X$$ x , y ∈ X and write $$k=\partial (x,y)$$ k = ∂ ( x , y ) . To avoid degenerate situations, we assume $$2 \le k \le D-1$$ 2 ≤ k ≤ D - 1 . Using y , we obtain an equitable partition $$\lbrace O_i \rbrace _{i=1}^6$$ { O i } i = 1 6 of the local graph $$\Gamma (x)$$ Γ ( x ) . By construction, $$O_1 = \Gamma (x) \cap \Gamma _{k-1}(y)$$ O 1 = Γ ( x ) ∩ Γ k - 1 ( y ) and $$O_6 = \Gamma (x) \cap \Gamma _{k+1}(y)$$ O 6 = Γ ( x ) ∩ Γ k + 1 ( y ) . We call $$\lbrace O_i \rbrace _{i=1}^6$$ { O i } i = 1 6 the y -partition of $$\Gamma (x)$$ Γ ( x ) . Let $$\lbrace O'_i \rbrace _{i=1}^6$$ { O i ′ } i = 1 6 denote the x -partition of $$\Gamma (y)$$ Γ ( y ) . According to the original balanced set condition, for $$i \in \lbrace 1,6\rbrace$$ i ∈ { 1 , 6 } , the vector $$E \widehat{O}_i - E \widehat{O}'_i$$ E O ^ i - E O ^ i ′ is a scalar multiple of $$E{\hat{x}}-E{\hat{y}}$$ E x ^ - E y ^ . We show that for $$1 \le i \le 6$$ 1 ≤ i ≤ 6 , the vector $$E \widehat{O}_i - E \widehat{O}'_i$$ E O ^ i - E O ^ i ′ is a scalar multiple of $$E{\hat{x}}-E{\hat{y}}$$ E x ^ - E y ^ . We investigate the consequences of this result.