A duality between pairs of split decompositions for a $Q$-polynomial distance-regular graph

Abstract

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D\ge 3$ and standard module $V$. Recently Ito and Terwilliger introduced four direct sum decompositions of $V$; we call these the $(\mu ,\nu )$--{\it split decompositions} of $V$, where $\mu ,\nu \in \{\downarrow ,\uparrow \}$. In this paper we show that the ($\downarrow ,\downarrow$)--split decomposition and the ($\uparrow ,\uparrow$)--split decomposition are dual with respect to the standard Hermitian form on $V$. We also show that the ($\downarrow ,\uparrow$)--split decomposition and the ($\uparrow ,\downarrow$)--split decomposition are dual with respect to the standard Hermitian form on $V$.

14 pages