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Physica Scripta

U q ( sl 3 ) 3j-symbols for tensor product of symmetric representations

Ayaz Ahmed, P. Ramadevi, Shoaib Akhtar

Published Nov 10, 2023• 0 citations
Publisher

Abstract

Abstract We propose an algebraic procedure to obtain U q ( sl 3 ) quantum 3j-symbols (quantum Clebsch–Gordan coefficients) appearing in the decomposition of tensor product of symmetric representations. In fact, these symbols will be useful to write the spectral parameter dependent R-matrix elements for any bi-partite vertex model whose edges carry states of the symmetric representations.

DOI
10.1088/1402-4896/ad0bb5

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Abstract We propose an algebraic procedure to obtain U q ( sl 3 ) quantum 3j-symbols (quantum Clebsch–Gordan coefficients) appearing in the decomposition of tensor product of symmetric representations. In fact, these symbols will be useful to write the spectral parameter dependent R-matrix elements for any bi-partite vertex model whose edges carry states of the symmetric representations.
In this paper we continue the study of Q-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). Taking a special limit in this R-matrix we obtained new formulas for the Q-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct Q-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-)integer spins.
Relations are exploited among Clebsch–Gordan (CG) and Racah coefficients in the algebra suq(2), known as a deformation of su(2). These are used to show that the Yang–Baxter (YB) relation for the IRF (interaction round a face) model results from one of the symmetry relations for the 9-j symbol specific to suq(2), and that in an asymptotic limit this YB relation becomes the YB relation for the two-dimensional vertex model. The Racah coefficient, which has a particularly simple dependence on q, is efficiently used such that an asymptotic limit of the Racah coefficient is the CG coefficient and another limit gives the factorized S matrix of the vertex model.