Tridiagonal pairs and the [formula omitted]-tetrahedron algebra

Abstract

The q -tetrahedron algebra ⊠ q was recently introduced and has been studied in connection with tridiagonal pairs. In this paper we further develop this connection. Let K denote an algebraically closed field and let q denote a nonzero scalar in K that is not a root of unity. Let V denote a vector space over K with finite positive dimension and let A , A ∗ denote a tridiagonal pair on V . Let { θ i } i = 0 d , (resp. { θ i ∗ } i = 0 d ) denote a standard ordering of the eigenvalues of A (resp. A ∗ ). Ito and Terwilliger have shown that when θ i = q 2 i - d and θ i ∗ = q d - 2 i ( 0 ⩽ i ⩽ d ) there exists an irreducible ⊠ q -module structure on V such that the ⊠ 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 θ i = q 2 i - d and θ i ∗ = q 2 i - d + cq d - 2 i for 0 ⩽ i ⩽ d . In this case we associate to A , A ∗ a polynomial P in one variable and prove the following theorem as our main result. Theorem The following are equivalent: 1. [(i)] There exists a ⊠ 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 ⊠ q . 2. [(ii)] P ( q 2 d - 2 ( q - q - 1 ) - 2 ) ≠ 0 . Suppose (i) and (ii) hold. Then the ⊠ q -module structure on V is unique and irreducible.