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

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

2007-05-01

arxiv.org/abs/0705.0167v1

Mathematics

Combinatorics

14 pages

Scientific paper

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D \geq 3$ 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 $\mu, \nu \in \lbrace \downarrow, \uparrow \rbrace$. 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 $V$.

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