Yang–Baxter Equations

Abstract

Let $F$ be a linear space, $R:F\otimes F\rightarrow F\otimes F$ an invertible linear map. It is well known that if $R=S_{(12)}:f_1\otimes f_2 \mapsto f_2 \otimes f_1$, then one can define a representation of the symmetric group ${{\mathrm{S}}}_n$ on $F^{\otimes n}$ by the following prescription: represent each element $\sigma \in {{\mathrm{S}}}_n$ as a product of transpositions of neighbors and apply $R_{i, i+1}=S_{(i, i+1)}$ instead of each $(i, i+1)$. Of course, such a decomposition is nonunique but the resulting linear operator does not depend on it.