Every $Q$-polynomial distance-regular graph is sharp over $\mathbb{R}$

Abstract

Let $\Gamma$ denote a distance-regular graph with vertex set $X$ and diameter $D \geq 3$. Fix a vertex $x \in X$. Let the field $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $\operatorname{Mat}_X(\mathbb{F})$ denote the $\mathbb{F}$-algebra of matrices whose rows and columns are indexed by $X$ and all entries in $\mathbb{F}$. The Terwilliger algebra $T^\mathbb{F} = T^\mathbb{F}(x)$ is the subalgebra of $\operatorname{Mat}_X(\mathbb{F})$ generated by the adjacency matrix $A$ of $\Gamma$ and the dual primitive idempotents $\{E_i^*\}_{i=0}^D$ of $\Gamma$ with respect to $x$. Let $\{E_i\}_{i=0}^D$ denote the primitive idempotents of $A$. Assume that the ordering $\{E_i\}_{i=0}^D$ is $Q$-polynomial. Let $W$ denote an irreducible $T^\mathbb{F}$-module. We say that $W$ is sharp over $\mathbb{F}$ whenever $\dim (E_r^* W) = 1$, where $r$ is the endpoint of $W$. It is known, by Nomura and Terwilliger (2008), that every irreducible $T^\mathbb{C}$-module is sharp. In this paper, we prove that every irreducible $T^\mathbb{R}$-module is sharp. Once this is established, we obtain four additional results: (i) if $W$ is an irreducible $T^\mathbb{R}$-module, then its complexification $W^\mathbb{C}= W \otimes_{\mathbb{R}} \mathbb{C}$ is an irreducible $T^\mathbb{C}$-module; (ii) two irreducible $T^\mathbb{R}$-modules $W_1$ and $W_2$ are isomorphic if and only if their complexifications $W_1^\mathbb{C}$ and $W_2^\mathbb{C}$ are isomorphic as $T^\mathbb{C}$-modules; (iii) if $\bigoplus_{i=1}^h \operatorname{Mat}_{n_i}(\mathbb{C})$ is the Wedderburn decomposition of $T^\mathbb{C}$, then $\bigoplus_{i=1}^h \operatorname{Mat}_{n_i}(\mathbb{R})$ is the Wedderburn decomposition of $T^\mathbb{R}$; (iv) each of the subalgebras $E_1^* T E_1^*$, $E_1 T E_1$, $E_D^* T E_D^*$, and $E_D T E_D$ is commutative and every element of these algebras is a symmetric matrix.