Journal article, 2014

Let F denote an algebraically closed field, and fix a nonzero q∈F that is not a root of unity. We consider the q-tetrahedron algebra ⊠q over F. It is known that each finite-dimensional irreducible ⊠q-module of type 1 is a tensor product of evaluation modules. This paper contains a comprehensive description of the evaluation modules for ⊠q. This description includes the following topics. Given an evaluation module V for ⊠q, we display 24 bases for V that we find attractive. For each basis we give the matrices that represent the ⊠q-generators. We give the transition matrices between certain pairs of bases among the 24. It is known that the cyclic group $\Z_4$ acts on ⊠q as a group of automorphisms. We describe what happens when V is twisted via an element of $\Z_4$. We discuss how evaluation modules for ⊠q are related to Leonard pairs of q-Racah type.

Leonard pair

Tetrahedron algebra

Equitable presentation

Kanazawa University

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

University of Wisconsin Madison

0024-3795 (ISSN)

Vol. 451 107-168Mathematics

Basic sciences

10.1016/j.laa.2014.03.019