The nucleus of the Johnson graph J(N,D)

This paper is about the nucleus of the Johnson graph $\Gamma =J(N,D)$ with $N>2D$. The nucleus is described as follows. Let X denote the vertex set of Γ. Let $A\in {\mathrm{Mat}}_X\left(C\right)$ denote the adjacency matrix of Γ. Let ${\left\{E_i\right\}}_{i=0}^D$ denote the Q-polynomial ordering of the primitive idempotents of A. Fix $x\in X$. The corresponding dual adjacency matrix $A^{⁎}$ is the diagonal matrix in ${\mathrm{Mat}}_X\left(C\right)$ such that for $y\in X$ the $(y,y)$-entry of $A^{⁎}$ is equal to the $(x,y)$-entry of $\left|X\right|E_1$. For $0\le i\le D$ the diagonal matrix $E_i^{⁎}\in {\mathrm{Mat}}_X\left(C\right)$ is the projection onto the ith subconstituent of Γ with respect to x. The matrices ${\left\{E_i^{⁎}\right\}}_{i=0}^D$ are the primitive idempotents of $A^{⁎}$. The subalgebra T of ${\mathrm{Mat}}_X\left(C\right)$ generated by A, $A^{⁎}$ is called the subconstituent algebra of Γ with respect to x. Let $V=C^X$ denote the standard module of Γ. For $0\le i\le D$ define$N_i=(E_0^{⁎}V+E_1^{⁎}V+\cdots +E_i^{⁎}V)\cap (E_{0}V+E_{1}V+\cdots +E_{D-i}V).$ It is known that the sum $N={\sum }_{i=0}^D{N}_i$ is direct, and $N$ is a T-module. This T-module $N$ is called the nucleus of Γ with respect to x. The sum $N={\sum }_{i=0}^D{E}_i^{⁎}N$ is direct. In this paper we consider the following vectors in V. For a subset $\alpha \subseteq x$ let ${\alpha }^{\vee }\in V$ denote the characteristic vector of the set of vertices in X that contain α. Let ${\alpha }^N\in V$ denote the characteristic vector of the set of vertices in X whose intersection with x is equal to α. We show that (i) the vectors ${\left\{{\alpha }^{\vee }\right\}}_{\alpha \subseteq x}$ form a basis of $N$; (ii) the vectors ${\left\{{\alpha }^N\right\}}_{\alpha \subseteq x}$ form a basis of $N$; (iii) for $0\le i\le D$, the vectors $\left\{{\alpha }^N\right|\alpha \subseteq x,\left|\alpha \right|=D-i\}$ form a basis of $E_i^{⁎}N$; (iv) for $0\le i\le D$, the vectors $\left\{{\alpha }^{\vee }\right|\alpha \subseteq x,\left|\alpha \right|=D-i\}$ form a basis of $N_i$. We give the transition matrices between the bases ${\left\{{\alpha }^{\vee }\right\}}_{\alpha \subseteq x}$ and ${\left\{{\alpha }^N\right\}}_{\alpha \subseteq x}$. We give the actions of A, $A^{⁎}$ on the bases ${\left\{{\alpha }^{\vee }\right\}}_{\alpha \subseteq x}$ and ${\left\{{\alpha }^N\right\}}_{\alpha \subseteq x}$. For $0\le i\le D$ we characterize the basis vectors for $E_i^{⁎}N$ in terms of the connected components of a certain graph obtained by adjusting the edges of the subgraph of Γ induced on the ith subconstituent of Γ with respect to x.