2009/08/18 by Kazumasa Nomura, Paul Terwilliger, Nomura, Kazumasa +1
Computer Science · Engineering · Mathematics · #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:15A21
paper · pdf · doi:10.48550/arxiv.0908.2604
35 pages
arxiv created 2009/08/18 · openalex publication_date 2009/08/18 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · 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 \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 \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 F 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. We present a result which supports the conjecture. Given scalars \þi\i=0d, \þ^*i\i=0d in F that satisfy the known constraints on the eigenvalues of a tridiagonal pair, we define an F-algebra T by generators and relations. We consider the algebra e^*0Te^*0 for a certain idempotent e^*0 ∈ T. Let R denote the polynomial algebra over F involving d variables.We display a surjective algebra homomorphism μ: R → e^*0Te^*0. We conjecture that μ is an isomorphism. We show that this μ-conjecture implies the classification conjecture, and that the μ-conjecture holds for d≤ 5.