Strengthening the balanced set condition for the distance-regular graph of the bilinear forms
Paul Terwilliger, Jason Williford
TL;DR
This work analyzes the bilinear forms graph $\Gamma = H_q(D,N-D)$ with $N>2D\ge 6$ and $q\neq 2$, establishing a strengthened version of the balanced set condition (bbalanced) for the $\theta_1$-eigenspace $EV$. By constructing the $y$-partition of the local graph $\Gamma(x)$ into six parts and defining the subspace $S=\mathrm{Span}\{E\widehat{O}_i\}_{i=1}^6$, the authors show that all six vectors satisfy $E\widehat{O}_i - E\widehat{O}'_i \in \mathrm{Span}\{E\hat{x}-E\hat{y}\}$, extending previous BSC results. They decompose $S$ into symmetric and antisymmetric parts, identify a distinguished vector $\omega$ in $\mathrm{Sym}(S)$, and analyze the Norton algebra structure to relate the strengthened condition to inner products and product closures within $EV$. The work provides a deeper algebraic understanding of the eigenstructure and local-partitions of $\Gamma$, with implications for related algebras and potential classifications. This Norton-algebra perspective offers a new framework for exploring balanced set-type conditions in distance-regular graphs beyond the original BSC.
Abstract
We consider a distance-regular graph $Γ=(X, \mathcal R)$ called the bilinear forms graph $H_q(D,N-D)$; we assume $N>2D\geq 6$ and $q \not=2$. We show that $Γ$ satisfies the following strengthened version of the balanced set condition. For a vertex $x \in X$ and $0 \leq i \leq D$ define $Γ_i(x)=\lbrace y \in X\vert \partial(x,y)=i\rbrace$, where $\partial$ denotes the path-length distance function. Abbreviate $Γ(x)=Γ_1(x)$. 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 $Γ$. For a subset $Ω\subseteq X$ define $\widehat Ω= \sum_{x \in Ω} \hat x$. 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 $Γ(x)$. By construction $O_1 = Γ(x) \cap Γ_{k-1}(y)$ and $O_6 = Γ(x) \cap Γ_{k+1}(y)$. We call $\lbrace O_i \rbrace_{i=1}^6$ the $y$-partition of $Γ(x)$. Let $\lbrace O'_i \rbrace_{i=1}^6$ denote the $x$-partition of $Γ(y)$. 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.