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

A duality between pairs of split decompositions for a $Q$-polynomial distance-regular graph - Mathematics > Combinatorics - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D\geq 3$ and standard module $V$. Recently Ito and Terwilliger introduced fourdirect sum decompositions of $V$; we call these the $(\mu, u)$-{\it splitdecompositions} of $V$, where $\mu, u \in \lbrace \downarrow, \uparrow brace$. In this paper we show that the ($\downarrow,\downarrow$)-splitdecomposition and the ($\uparrow,\uparrow$)-split decomposition are dual withrespect 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 thestandard Hermitian form on $V$.

Autor: Joohyung Kim

Fuente: https://arxiv.org/