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

TL;DR

This paper studies the action of a stabilizer subgroup on the local graph of a Grassmann graph, revealing five orbits and their structure through Euclidean representations and eigenvalue analysis.

Contribution

It identifies five orbits of the stabilizer subgroup acting on the local graph of a Grassmann graph and characterizes their structure using Euclidean representations and eigenvalues.

Findings

Five orbits of the stabilizer subgroup are identified.

An equitable partition of the local graph is constructed.

Structure constants for the partition are computed.

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 $\Gamma(x)$. We show that there are five orbits. By construction, these five orbits give an equitable partition of $\Gamma(x)$; 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.