2008/05/10 by Tatsuro Ito, Paul Terwilliger, Ito, Tatsuro +1 · 3 citations
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.0805.1465
34 pages
arxiv created 2008/05/10 · openalex publication_date 2008/05/10 · 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. 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. The pair A,A^* is called \it sharp whenever dim V0=1. It is known that if K is algebraically closed then A,A^* is sharp. Assuming A,A^* is sharp, we use the data Φ=(A; \Vi\i=0d; A^*; \V^*i\i=0d) to define a polynomial P in one variable and degree at most d. We show that P remains invariant if Φ is replaced by (A;\Vd-i\i=0d; A^*; \V^*i\i=0d) or (A;\Vi\i=0d; A^*; \V^*d-i\i=0d) or (A^*; \V^*i\i=0d; A; \Vi\i=0d). We call P the \it Drinfel'd polynomial of A,A^*. We explain how P is related to the classical Drinfel'd polynomial from the theory of Lie algebras and quantum groups. We expect that the roots of P will be useful in a future classification of the sharp tridiagonal pairs. We compute the roots of P for the case in which Vi and V^*i have dimension 1 for 0 ≤ i ≤ d.