The nucleus of the Grassmann graph Jq(N,D)

Let Fq denote a finite field with q elements. Let N and D denote integers with N>D ≥ 1. Let V denote an N-dimensional vector space over Fq. The Grassmann graph Jq(N,D) is the graph with vertex set X that consists of the D-dimensional subspaces of 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 Jq(N,D) with respect to x is the subalgebra of MatX(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 N=N(x) denote the span of all irreducible T-modules with displacement 0. We call N the nucleus of Jq(N,D) with respect to x. In this paper, we study the structure of N. Specifically, we present a formula for the dimension of N, construct two explicit bases for N, and describe the action of A and A^* on these bases. To obtain these results, we use the projective geometry Pq(N), consisting of all subspaces of V, as a key tool.