Set-theoretical solutions to the Yang–Baxter relation from factorization of matrix polynomials and $\theta$-functions

New set-theoretical solutions to the Yang–Baxter Relation are constructed. These solutions arise from the decompositions “in different order” of matrix polynomials and $\theta$-functions. We also construct a “local action of the symmetric group” in these cases, generalizations of the action of the symmetric group $S_N$ given by the set-theoretical solution.