Permutation-type solutions to the Yang - Baxter and other n-simplex equations

We study permutation-type solutions to n-simplex equations, that is, solutions whose matrix form can be written as with some matrix A and vector B, both over . With this ansatz the equations of the n-simplex equation reduce to a matrix equation over . We have completely analysed the 2-, 3- and 4-simplex equations in the generic D case. The solutions show interesting patterns that seem to continue to still higher simplex equations.