On the Terwilliger algebra of bipartite distance-regular graphs with Δb>2b> = 0 and cb>2b> = 1

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• 作者：Mark S. MacLean^a ; ^{macleanm@seattleu.edu" class="auth_mail" title="E-mail the corresponding author} ; &Scaron ; tefko Miklavič^b ; ¹ ; ^{stefko.miklavic@upr.si" class="auth_mail" title="E-mail the corresponding author} ; Safet Penjić^b ; ² ; ^{safet.penjic@iam.upr.si" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05E30 ; 05C50
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：496
• 期：Complete
• 页码：307-330
• 全文大小：503 K

文摘

Let Γ denote a bipartite distance-regular graph with diameter b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si1.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=326302fd0d04aadbf867cde4c711745c" title="Click to view the MathML source">D≥4$D\ge 4$ and valency b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si2.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=cce4dd3fbc22b8dd68ae71b0684aa8cb" title="Click to view the MathML source">k≥3$k\ge 3$. Let X denote the vertex set of Γ, and let A   denote the adjacency matrix of Γ. For b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si3.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=6023f8860fb6e4e41bbbbba335308947" title="Click to view the MathML source">x∈X$x\in X$ and for b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si108.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=c0747087566e7a4d17d438d49317230b" title="Click to view the MathML source">0≤i≤D$0\le i\le D$, let b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si5.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=b90fbd10c3f810a571dc5f9210ba3001" title="Click to view the MathML source">Γb>ib>(x) denote the set of vertices in X that are distance i from vertex x  . Define a parameter b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si6.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=bbc45b0934b1369d00b538b3060f4a0c" title="Click to view the MathML source">Δb>2b> in terms of the intersection numbers by b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si27.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=069b9934ac7232cd1951ff68ac36f8ca">. We first show that b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si169.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=a0d38a9088abb2c1feb85c4ecc52426b" title="Click to view the MathML source">Δb>2b>=0 implies that b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si170.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=df697011aff03bd63dfef12667b8e0f4" title="Click to view the MathML source">D≤5$D\le 5$ or b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si10.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=9b9ee5c6a501360b45e04a5a923673b2" title="Click to view the MathML source">cb>2b>∈{1,2}.

For b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si3.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=6023f8860fb6e4e41bbbbba335308947" title="Click to view the MathML source">x∈X$x\in X$ let b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si11.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=228f085f922143fab75cb07838d329a3" title="Click to view the MathML source">T=T(x)$T=T(x)$ denote the subalgebra of b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si12.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=8ec57517ccfa8052d1bda8fa23d99273" title="Click to view the MathML source">Matb>Xb>(C) generated by b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si13.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=85a92c473df159658dcb942949c2f8e7">, where for b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si108.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=c0747087566e7a4d17d438d49317230b" title="Click to view the MathML source">0≤i≤D$0\le i\le D$, b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si112.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=c14744e70850f155ceff3f1c298bd895"> represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T-module W   we mean b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si15.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=47aa7d642e700a9fe1d4625f5822ef19">.

In this paper we assume Γ has the property that for b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si16.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=1d7f649e29fc79978befa6e53c4ca78f" title="Click to view the MathML source">2≤i≤D−1$2\le i\le D-1$, there exist complex scalars b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si17.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=c8b9f883dd4661615cc2f9374de839c9" title="Click to view the MathML source">αb>ib>, b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si18.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=75ba86fb687b9af2ef592ec528ad9909" title="Click to view the MathML source">βb>ib> such that for all b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si19.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=5ff742b2c79d8c4dd1f9eeac8be77444" title="Click to view the MathML source">x,y,z∈X$x,y,z\in X$ with b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si20.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=6c8139ee4793bfdff8e8aeac895672c4" title="Click to view the MathML source">∂(x,y)=2$\partial (x,y)=2$, b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si21.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=4eef9c87443ca1fab5f52641c1d11d71" title="Click to view the MathML source">∂(x,z)=i$\partial (x,z)=i$, b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si22.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=762bc977d6ce703cf87d7adde5945b00" title="Click to view the MathML source">∂(y,z)=i$\partial (y,z)=i$, we have b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si23.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=4033859798e485c4253acb438d2501e5" title="Click to view the MathML source">αb>ib>+βb>ib>|Γb>1b>(x)∩Γb>1b>(y)∩Γb>i−1b>(z)|=|Γb>i−1b>(x)∩Γb>i−1b>(y)∩Γb>1b>(z)|. We additionally assume that b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si169.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=a0d38a9088abb2c1feb85c4ecc52426b" title="Click to view the MathML source">Δb>2b>=0 with b=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000550&_mathId=si24.gif&_user=111111111&_pii=S0024379516000550&_rdoc=1&_issn=00243795&md5=e88c74fa9cbef19ad8598002d607c069" title="Click to view the MathML source">cb>2b>=1.

Under the above assumptions we study the algebra T. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2. We give an orthogonal basis for this T-module, and we give the action of A on this basis.