The displacement and split decompositions for a $Q$-polynomial distance-regular graph

Abstract

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter at least three and standard module $V$. We introduce two direct sum decompositions of $V$. We call these the displacement decomposition for $\Gamma$ and the split decomposition for $\Gamma$. We describe how these decompositions are related.