The nucleus of the Grassmann graph $J_q(N,D)$

Abstract

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $N$ and $D$ denote integers with $N>D \ge 1$. Let $\mathcal{V}$ denote an $N$-dimensional vector space over $\mathbb{F}_q$. The Grassmann graph $J_q(N,D)$ is the graph with vertex set $X$ that consists of the $D$-dimensional subspaces of $\mathcal{V}$. Two vertices are adjacent whenever their intersection has dimension $D-1$. Fix a vertex $x$ in $X$. The Terwilliger algebra $T=T(x)$ of $J_q(N,D)$ with respect to $x$ is the subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^* = A^*(x)$. It is known that an irreducible $T$-module $W$ has certain parameters called the endpoint $r$, the dual endpoint $t$, and the diameter $d$. The displacement of $W$ is defined to be the integer $r+t-D+d$. Let $\mathcal{N}=\mathcal{N}(x)$ denote the span of all irreducible $T$-modules with displacement 0. We call $\mathcal{N}$ the nucleus of $J_q(N,D)$ with respect to $x$. In this paper, we study the structure of $\mathcal{N}$. Specifically, we present a formula for the dimension of $\mathcal{N}$, construct two explicit bases for $\mathcal{N}$, and describe the action of $A$ and $A^*$ on these bases. To obtain these results, we use the projective geometry $P_q(N)$, consisting of all subspaces of $\mathcal{V}$, as a key tool.