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

Let Γ denote a Q-polynomial distance-regular graph with diameter D ≥ 3 and standard module V . Recently Ito and Terwilliger introduced four direct sum decompositions of V ; we call these the (μ, ν)–split decompositions of V , where μ, ν ∈ {↓, ↑}. In this paper we show that the (↓, ↓)–split decomposition and the (↑, ↑)–split decomposition are dual with respect to the standard Hermitian form on V . We also show that the (↓, ↑)–split decomposition and the (↑, ↓)–split decomposition are dual with respect to the standard Hermitian form on V .