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 $A^*$ and dual primitive idempotents $\{ E^*_i\}_{ i=0}^D$. The subalgebra $T$ of $\text{ Mat}_X({ \mathbb C})$ generated by $A$, $A^*$ is called the subconstituent algebra of $\Gamma$ with respect to $x$. Let $V={ \mathbb C}^X$ denote the standard module of $\Gamma$. For $0 \leq i \leq D$ 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 $0 \leq i \leq D$ we construct a basis for ${ \mathcal N}_i$ and a basis for $E^*_i { \mathcal N}$. 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 $A$, $A^*$ on these bases.
Categories: math.RA