327 KB30 KB200820252019Sumalroj, Supalakštevilka:1letnik:17str. 185-202ISSN:1855-3966COBISSID_HOST:18953817URN:URN:NBN:SI:doc-7RIUPI6FenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneadistance-regular graphpodkonstitutivna algebrarazdaljno-regularen grafsubconstituent algebraTerwilliger algebraTerwilligerjeva algebraA diagram associated with the subconstituent algebra of a distance-regular graph|In this paper we consider a distance-regular graph ?$\Gamma$?. Fix a vertex ?$x$? of ?$\Gamma$? and consider the corresponding subconstituent algebra ?$T = T(x)$?. The algebra ?$T$? is the ?$\mathbb{C}$?-algebra generated by the Bose-Mesner algebra ?$M$? of ?$\Gamma$? and the dual Bose-Mesner algebra ?$M^\ast$? of ?$\Gamma$? with respect to ?$x$?. We consider the subspaces ?$M, M^\ast, MM^\ast, M^\ast M, MM^\ast M, M^\ast MM^\ast, \dots$? along with their intersections and sums. In our notation, ?$MM^\ast$? means Span?$\{RS \mid R \in M, S \in M^\ast\}$?, and so on. We introduce a diagram that describes how these subspaces are related. We describe in detail that part of the diagram up to ?$MM^\ast + M^\ast M$?. For each subspace ?$U$? shown in this part of the diagram, we display an orthogonal basis for ?$U$? along with the dimension of ?$U$?. For an edge ?$U \subseteq W$? from this part of the diagram, we display an orthogonal basis for the orthogonal complement of ?$U$? in ?$W$? along with the dimension of this orthogonal complementV tem članku obravnavamo razdaljno-regularen graf ?$\Gamma$?. Fiksiramo vozlišce ?$x$? grafa ?$\Gamma$? in obravnavamo prirejeno Terwilligerjevo algebro ?$T = T(x)$?. Algebra ?$T$? je ?$\mathbb{C}$?-algebra, generirana z Bose-Mesnerjevo algebro ?$M$? grafa ?$\Gamma$? in dualno Bose-Mesnerjevo algebro ?$M^\ast$? grafa ?$\Gamma$? glede na ?$x$?. Obravnavamo podprostore ?$M, M^\ast, MM^\ast, M^\ast M, MM^\ast M, M^\ast MM^\ast, \dots$? ter njihove preseke in vsote. Pri tem ?$MM^\ast$? pomeni Span?$\{RS \mid R \in M, S \in M^\ast\}$? in tako dalje. Uvedemo diagram, ki opisuje, kako so ti podprostori povezani med seboj. Podrobno opišemo del tega diagrama vse do ?$MM^\ast + M^\ast M$?. Za vsak podprostor ?$U$?, prikazan v tem delu diagrama, podamo njegovo ortogonalno bazo ter njegovo dimenzijo. Za povezavo ?$U \subseteq W$? iz tega dela diagrama podamo ortogonalno bazo ortogonalnega komplementa podprostora ?$U$? v prostoru ?$W$? ter dimenzijo tega ortogonalnega komplementaTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije