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Evaluation modules for the q-tetrahedron algebra

Tatsuro Ito ; Hjalmar Rosengren (Institutionen för matematiska vetenskaper, matematik) ; Paul Terwilliger
Linear Algebra and its Applications (0024-3795). Vol. 451 (2014), p. 107-168.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: Equitable presentation; Leonard pair; Tetrahedron algebra

Preprint available at http://arxiv.org/abs/1308.3480

Denna post skapades 2013-12-12. Senast ändrad 2014-12-01.
CPL Pubid: 188995


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