Counting edges of different types in a local graph of a Grassmann graph

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 $\Gamma(x)$ denote the local graph of $x$ in $J_q(n,k)$. In this paper we define three types of edges in $\Gamma(x)$, namely type $0$, type $+$, and type $-$; for adjacent $w,z\in \Gamma(x)$ such that $\partial(w,y)=\partial(z,y)$, the type of the edge $wz$ depends on the subspaces $w+z,w,z,w\cap z$ and their intersections with $y$. Our general goal is to count the number of edges in $\Gamma(x)$ for each type. Consider a two-vertex stabilizer $\text{Stab}(x,y)$ in $GL(V)$; it is known that the $\text{Stab}(x,y)$-action on $\Gamma(x)$ has five orbits. Pick two orbits $\mathcal{O},\mathcal{N}$ that are not necessarily distinct; for a given $w\in \mathcal{O}$, we find the number of vertices in $z\in \mathcal{N}$ such that the edge $wz$ has (i) type $0$, (ii) type $+$, (iii) type $-$. To find these numbers, we make heavy use of a subalgebra $\mathcal{H}$ of $\text{Mat}_{P}(\mathbb{C})$; the algebra $\mathcal{H}$ contains matrices that are closely related to the five orbits of the $\text{Stab}(x,y)$-action on $\Gamma(x)$.