Fusion Formulas and Fusion Procedure for the Yang-Baxter Equation

We use the fusion formulas of the symmetric group and of the Hecke algebra to construct solutions of the Yang–Baxter equation on irreducible representations of \(\mathfrak {gl}_{N}\), \(\mathfrak {gl}_{N|M}\), \(U_{q}(\mathfrak {gl}_{N})\) and \(U_{q}(\mathfrak {gl}_{N|M})\). The solutions are obtained via the fusion procedure for the Yang–Baxter equation, which is reviewed in a general setting. Distinguished invariant subspaces on which the fused solutions act are also studied in the general setting, and expressed, in general, with the help of a fusion function. Only then, the general construction is specialised to the four situations mentioned above. In each of these four cases, we show how the distinguished invariant subspaces are identified as irreducible representations, using the relevant fusion formula combined with the relevant Schur–Weyl duality.