Every Q-polynomial distance-regular graph is sharp over $\mathbb{R}$Fernández, Blas (Avtor) Lee, Jae-Ho (Avtor) Park, Jongyook (Avtor) distance-regular graphsQ-polynomial propertyTerwilliger algebraLet $\Gamma$ be a $Q$-polynomial distance-regular graph, and let $T=T(x)$ denote its Terwilliger algebra with respect to a fixed vertex $x$. While it has long been known that every irreducible $T$-module over the complex field is sharp, the corresponding result over the real field had remained unproved. In this work, we establish that every irreducible $T$-module over $\mathbb{R}$ is also sharp. This resolves the real analogue of a theorem of Nomura and Terwilliger and shows that every $Q$-polynomial distance-regular graph is sharp over both $\mathbb{R}$ and $\mathbb{C}$. As further consequences, we prove that the complexification of an irreducible real $T$-module remains irreducible, characterize isomorphism classes via complexification, determine the Wedderburn decomposition of the real Terwilliger algebra, and show that several naturally arising subalgebras are commutative and consist entirely of symmetric matrices. These results clarify the relationship between the real and complex representation theories of the Terwilliger algebra and provide new structural insight into $Q$-polynomial distance-regular graphs. 20262026-07-17 11:46:49Neznano23328UDK: 519.17ISSN pri članku: 0097-3165DOI: 10.1016/j.jcta.2026.106223COBISS.SI-ID: 285240579sl