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.Comment: 30 pages, bibliography added (references were missing in first
version), published in Linear Algebra and its Application