Evaluation modules for the q-tetrahedron algebra
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


Tatsuro Ito

Kanazawa University

Hjalmar Rosengren

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Paul Terwilliger

University of Wisconsin Madison

Linear Algebra and Its Applications

0024-3795 (ISSN)

Vol. 451 107-168

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