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.