Strengthening the balanced set condition for the distance-regular graph of the bilinear forms
We consider a distance-regular graph $\Gamma=(X, \mathcal R)$ called the bilinear forms graph $H_q(D,N-D)$; we assume $N>2D\geq 6$ and $q \not=2$. We show that $\Gamma$ satisfies the following strengthened version of the balanced set condition. For a vertex $x \in X$ and $0 \leq i \leq D$ define $\Gamma_i(x)=\lbrace y \in X\vert \partial(x,y)=i\rbrace$, where $\partial$ denotes the path-length distance function. Abbreviate $\Gamma(x)=\Gamma_1(x)$. Let $V={\mathbb R}^X$ denote the standard module for ${\rm Mat}_X(\mathbb R)$. For $x\in X$ let $\hat x \in V$ have $x$-coordinate 1 and all other coordinates 0. Let $E \in {\rm Mat}_X(\mathbb R)$ denote the primitive idempotent that corresponds to the second largest eigenvalue of the adjacency matrix of $\Gamma$. For a subset $\Omega \subseteq X$ define $\widehat \Omega = \sum_{x \in \Omega} \hat x$. We fix two vertices $x,y \in X$ and write $k=\partial(x,y)$. To avoid degenerate situations, we assume $2 \leq k \leq D-1$. Using $y$ we obtain an equitable partition $\lbrace O_i \rbrace_{i=1}^6$ of the local graph $\Gamma(x)$. By construction $O_1 = \Gamma (x) \cap \Gamma_{k-1}(y)$ and $O_6 = \Gamma(x) \cap \Gamma_{k+1}(y)$. We call $\lbrace O_i \rbrace_{i=1}^6$ the $y$-partition of $\Gamma(x)$. Let $\lbrace O'_i \rbrace_{i=1}^6$ denote the $x$-partition of $\Gamma(y)$. According to the original balanced set condition, for $i \in \lbrace 1,6\rbrace$ the vector $ E \widehat O_i - E \widehat O'_i$ is a scalar multiple of $E{\hat x}-E{\hat y}$. We show that for $1 \leq i \leq 6$ the vector $ E \widehat O_i - E \widehat O'_i$ is a scalar multiple of $E{\hat x}-E{\hat y}$. We investigate the consequences of this result.