Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772997 | Linear Algebra and its Applications | 2017 | 69 Pages |
Abstract
A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let F denote an algebraically closed field, and fix a nonzero qâF that is not a root of unity. The universal double affine Hecke algebra (DAHA) HËq of type (C1â¨,C1) is the associative F-algebra defined by generators {ti±1}i=03 and relations (i) titiâ1=tiâ1ti=1; (ii) ti+tiâ1 is central; (iii) t0t1t2t3=qâ1. We consider the elements X=t3t0 and Y=t0t1 of HËq. Let V denote a finite-dimensional irreducible HËq-module on which each of X, Y is diagonalizable and t0 has two distinct eigenvalues. Then V is a direct sum of the two eigenspaces of t0. We show that the pair X+Xâ1, Y+Yâ1 acts on each eigenspace as a Leonard pair, and each of these Leonard pairs falls into a class said to have q-Racah type. Thus from V we obtain a pair of Leonard pairs of q-Racah type. It is known that a Leonard pair of q-Racah type is determined up to isomorphism by a parameter sequence (a,b,c,d) called its Huang data. Given a pair of Leonard pairs of q-Racah type, we find necessary and sufficient conditions on their Huang data for that pair to come from the above construction.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Kazumasa Nomura, Paul Terwilliger,