Tridiagonal pairs and the q-tetrahedron algebra

Details Tridiagonal pairs and the q-tetrahedron algebra Tridiagonal pairs and the q-tetrahedron algebra

2008-06-05

arxiv.org/abs/0806.0901v3

Linear Algebra and its Applications, vol. 431, #5-7 (2009), 903-925

Mathematics

Representation Theory

30 pages, bibliography added (references were missing in first version), published in Linear Algebra and its Applications

Scientific paper

In this paper we further develop the connection between tridiagonal pairs and the q-tetrahedron algebra $\boxtimes_q$. Let V denote a finite dimensional vector space over an algebraically closed field and let A, A^* denote a tridiagonal pair on V. For $0 \leq i \leq d$ let $\theta_i$ (resp. $\theta^*_i$) denote a standard ordering of the eigenvalues of A (resp. A^*). Fix a nonzero scalar q which is not a root of unity. T. Ito and P. Terwilliger have shown that when $\theta_i = q^{2i-d}$ and $\theta^*_i = q^{d-2i}$ there exists an irreducible $\boxtimes_q$-module structure on V such that the $\boxtimes_q$ generators x_{01}, x_{23} act as A, A^* respectively. In this paper we examine the case in which there exists a nonzero scalar c in K such that $\theta_i = q^{2i-d}$ and $\theta^*_i = q^{2i-d} + c q^{d-2i}$. In this case we associate to A,A^* a polynomial P and prove the following equivalence. The following are equivalent: (i) There exists a $\boxtimes_q$-module structure on V such that x_{01} acts as A and x_{30} + cx_{23} acts as A^*, where x_{01}, x_{30}, x_{23} are standard generators for $\boxtimes_q$. (ii) P(q^{2d-2} (q-q^{-1})^{-2}) \neq 0. Suppose (i),(ii) hold. Then the $\boxtimes_q$-module structure on V is unique and irreducible.

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