20.500.12556/RUP-23328 Every Q-polynomial distance-regular graph is sharp over $\mathbb{R}$ Let $\Gamma$ 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. Naj bo $\Gamma$ $Q$-polinomski razdaljno regularen graf in naj bo $T=T(x)$ njegova Terwilligerjeva algebra glede na izbrano vozlišče $x$. Čeprav je že dolgo znano, da je vsak ireducibilen $T$-modul nad kompleksnimi števili oster (\emph{sharp}), ustrezen rezultat nad realnimi števili doslej ni bil dokazan. V tem delu pokažemo, da je vsak ireducibilen $T$-modul nad $\mathbb{R}$ prav tako oster. S tem dopolnimo izrek Nomure in Terwilligerja ter dokažemo, da je vsak $Q$-polinomski razdaljno regularen graf oster tako nad $\mathbb{R}$ kot tudi nad $\mathbb{C}$. Kot nadaljnje posledice dokažemo, da kompleksifikacija ireducibilnega realnega $T$-modula ostane reducibilna, opišemo izomorfnostne razrede modulov prek kompleksifikacije, določimo Wedderburnov razcep realne Terwilligerjeve algebre ter pokažemo, da je več naravno definiranih podalgeber komutativnih in sestavljenih izključno iz simetričnih matrik. Dobljeni rezultati osvetljujejo povezavo med realno in kompleksno teorijo reprezentacij Terwilligerjeve algebre ter prispevajo k boljšemu razumevanju strukture $Q$-polinomskih razdaljno regularnih grafov. distance-regular graphs Q-polynomial property Terwilliger algebra razdaljno-regularen graf Q-polinomial lastnost Terwilligerjeva algebra true false true Angleški jezik Slovenski jezik Neznano 2026-07-17 11:46:49 2026-07-17 11:46:52 2026-07-18 03:05:52 0000-00-00 00:00:00 2026 0 0 str. 1-36 article 106223 Vol. 224 Nov. 2026 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17 0097-3165 10.1016/j.jcta.2026.106223 285240579 https://www.sciencedirect.com/science/article/pii/S009731652600066X 1 3f705354-81c4-11f1-9e8d-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=34114 Fakulteta za matematiko, naravoslovje in informacijske tehnologije 0 0 0