Quantum $3j$-symbols for $U_q(\mathfrak{sl}_3)$

Abstract

We propose an algebraic expression for $U_q(\mathfrak{sl}_3)$ quantum $3j$ symbols (quantum Clebsch-Gordan coefficients) appearing in the decomposition of tensor product of symmetric representations. Our compact form 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.