2009/08/21 by Kazumasa Nomura, Paul Terwilliger, Nomura, Kazumasa +1
Mathematics · #15A21 #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA) #math.CO #math.RA #msc:15A21
paper · pdf · doi:10.48550/arxiv.0908.3151
11 pages
arxiv created 2009/08/21 · arxiv updated 2009/12/01
Let \K denote a field and let V denote a vector space over \K 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. We say the pair A,A^* is \it sharp whenever dim V0=1. It is known that if \K is algebraically closed then A,A^* is sharp. A conjectured classification of the sharp tridiagonal pairs was recently introduced by T. Ito and the second author. Shortly afterwards we introduced a conjecture, called the \em μ-conjecture, which implies the classification conjecture. In this paper we show that the μ-conjecture holds in a special case called q-Racah.