Some orbits of a two-vertex stabilizer in a Grassmann graph

TL;DR

This work analyzes the action of ${\rm Stab}(x,y)$ on the local graph $\Gamma(x)$ of the Grassmann graph $J_q(n,k)$ for $n>2k$, establishing a precisely five-orbit partition known as the $y$-partition. Using a Euclidean representation linked to the second largest eigenvalue $\theta_1$, the authors construct and study vectors $A_{xy}^{+},A_{xy}^{0},A_{xy}^{-}$ that lie in ${\rm Fix}(x,y)$ and express them in both geometric and combinatorial bases, deriving intricate inner products and structure constants. They show the five orbits $\mathcal{B}_{xy},\mathcal{C}_{xy},\mathcal{A}_{xy}^{+},\mathcal{A}_{xy}^{0},\mathcal{A}_{xy}^{-}$ form an equitable partition of $\Gamma(x)$ and introduce a 5×5 orbit-adjacency matrix $\mathcal{M}_i$ whose eigenvalues coincide with the local graph spectrum of $\Gamma(x)$. The results deepen the understanding of stabilizer actions on Grassmann graphs and illustrate how orbit structure aligns with the global and local spectral data of distance-regular graphs.

Abstract

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$ that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$ consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. Let $\partial$ denote the path-length distance function of $J_q(n,k)$. Pick vertices $x,y$ of $J_q(n,k)$ such that $1<\partial(x,y)<k$. Let $\text{Stab}(x,y)$ denote the subgroup of $GL(V)$ that stabilizes both $x$ and $y$. In this paper, we investigate the orbits of $\text{Stab}(x,y)$ acting on the local graph $Γ(x)$. We show that there are five orbits. By construction, these five orbits give an equitable partition of $Γ(x)$; we find the corresponding structure constants. In order to describe the five orbits more deeply, we bring in a Euclidean representation of $J_q(n,k)$ associated with the second largest eigenvalue of $J_q(n,k)$. By construction, for each orbit its characteristic vector is represented by a vector in the associated Euclidean space. We compute many inner products and linear dependencies involving the five representing vectors.

Theorems & Definitions (69)

• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Definition 3.5
• Lemma 4.1
• Lemma 5.1
• Lemma 5.2
• Definition 5.3
• Definition 6.1