Lattices Associated with Subspaces in $d$-Bounded Distance-Regular Graphs

Abstract

Let $\mathrm{\Gamma }=(X,R)$ denote a $d$-bounded distance-regular graph with diameter $d\ge 3$. A regular strongly closed subgraph of $\mathrm{\Gamma }$ is said to be a subspace of $\mathrm{\Gamma }$. For $0\le i\le i+s\le d-1$, suppose ${\mathrm{\Delta }}_i$ and ${\mathrm{\Delta }}_0$ are subspaces with diameter $i$ and $i+s$, respectively, and with ${\mathrm{\Delta }}_i\subseteq {\mathrm{\Delta }}_0$. Let $\mathcal{L}(i,i+s;d)$ denote the set of all subspaces ${\mathrm{\Delta }}^{\prime }$ with diameters $\ge i$ such that $d({\mathrm{\Delta }}_0\cap {\mathrm{\Delta }}^{\prime })={\mathrm{\Delta }}_1$ and $d({\mathrm{\Delta }}_0+{\mathrm{\Delta }}^{\prime })=d({\mathrm{\Delta }}^{\prime })+s$ in $\mathrm{\Gamma }\$including\text{(}{\mathrm{\Delta }}_0$. If we partial order $\mathcal{L}(i,i+s;d)$ by ordinary inclusion (resp. reverse inclusion), then $\mathcal{L}(i,i+s;d)$ is a poset, denoted by ${\mathcal{L}}_0(i,i+s;d)$ (resp. ${\mathcal{L}}_R(i,i+s;d)$). In the present paper, we show that both ${\mathcal{L}}_0(i,i+s;d)$ and ${\mathcal{L}}_R(i,i+s;d)$ are atomic lattices, and classify their geometricity.