2010/01/12 by Tatsuro Ito, Kazumasa Nomura, Ito, Tatsuro +3 · 4 citations
Computer Science · Engineering · Mathematics · #05E30 #15A21 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Matrix Theory and Algorithms #Rings and Algebras (math.RA) #graph theory and CDMA systems #math.CO #math.RA #msc:05E30 #msc:15A21
paper · pdf · doi:10.48550/arxiv.1001.1812
36 pages
arxiv created 2010/01/12 · openalex publication_date 2010/01/12 · arxiv updated 2010/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V → V and A^*:V → V that satisfy the following conditions: (i) each of A,A^* is diagonalizable; (ii) there exists an ordering \lbrace Vi\rbracei=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 \lbrace V^*i\rbracei=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 \it tridiagonal pair on V. It is known that d=δ and for 0 ≤ i ≤ d the dimensions of Vi,Vd-i,V^*i, V^*d-i coincide. The pair A,A^* is called \it sharp whenever \rm dim V0=1. It is known that if F is algebraically closed then A,A^* is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture.