2008/01/04 by Kazumasa Nomura, Paul Terwilliger, Nomura, Kazumasa +1 · 3 citations
Engineering · Mathematics · #05E30 #33C45 #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #graph theory and CDMA systems #math.CO #math.RA #msc:05E30 #msc:33C45
paper · pdf · doi:10.48550/arxiv.0801.0621
18 pages
arxiv created 2008/01/04 · openalex publication_date 2008/01/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K denote a field and let V denote a vector space over K with finite positive dimension. Let End(V) denote the K-algebra consisting of all K-linear transformations from V to V. We consider a pair A,A^* ∈ End(V) that satisfy (i)--(iv) below: (i) Each of A,A^* is diagonalizable. (ii) There exists an ordering \Vi\i=0d of the eigenspaces of A such that A^* Vi ⊆ Vi-1 + Vi + Vi+1 for 0 ≤ i ≤ d, where V-1=0 and Vd+1=0. (iii) There exists an ordering \V^*i\i=0δ of the eigenspaces of A^* such that A V^*i ⊆ V^*i-1 + V^*i + V^*i+1 for 0 ≤ i ≤ δ, where V^*-1=0 and V^*δ+1=0. (iv) There is no subspace W of V such that AW ⊆ W, A^* W ⊆ W, W ≠ 0, W ≠ V. We call such a pair a \em tridiagonal pair on V. Let E^*0 denote the element of End(V) such that (E^*0-I)V^*0=0 and E^*0V^*i=0 for 1 ≤ i ≤ d. Let D (resp. D^*) denote the K-subalgebra of End(V) generated by A (resp. A^*). In this paper we prove that the span of E^*0 D D^*DE^*0 equals the span of E^*0D E^*0DE^*0, and that the elements of E^*0 D E^*0 mutually commute. We relate these results to some conjectures of Tatsuro Ito and the second author that are expected to play a role in the classification of tridiagonal pairs.