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

Abstract

In this paper, we describe the nucleus of the Johnson graph $\Gamma =J(N,D)$ with $N>2D$. Let $X$ denote the vertex set of $\Gamma$. Let $A\in {\text{Mat}}_X(\mathbb{C})$ denote the adjacency matrix of $\Gamma$. Let $\{E_i{\}}_{i=0}^D$ denote the $Q$-polynomial ordering of the primitive idempotents of $A$. Fix $x\in X$, and consider the corresponding dual adjacency matrix $A^{\ast }$ and dual primitive idempotents $\{E_i^{\ast }{\}}_{i=0}^D$. The subalgebra $T$ of ${\text{Mat}}_X(\mathbb{C})$ generated by $A$, $A^{\ast }$ is called the subconstituent algebra of $\Gamma$ with respect to $x$. Let $V={\mathbb{C}}^X$ denote the standard module of $\Gamma$. For $0\le i\le D$ define \[ {\mathcal N}_i = (E^*_0 V + E^*_1 V + \cdots + E^*_i V) \cap (E_0 V + E_1 V + \cdots + E_{D-i} V). \] It is known that the sum $\mathcal{N}={\sum }_{i=0}^D{\mathcal{N}}_i$ is direct, and $\mathcal{N}$ is a $T$-module. The $T$-module $\mathcal{N}$ is called the nucleus of $\Gamma$ with respect to $x$. For $0\le i\le D$ we construct a basis for ${\mathcal{N}}_i$ and a basis for $E_i^{\ast }\mathcal{N}$. From this we obtain two bases of $\mathcal{N}$. We give a combinatorial interpretation of these two bases. We give the transition matrices between these two bases. We also give the action of $A$, $A^{\ast }$ on these bases.