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 R1n ii...in = ∏n α=1 δ jα Aαiβ+Bα with some n × n matrix A and vector B, both over ZD. With this ansatz the D n(n+1) equations of the n-simplex equation reduce to an [12n(n+1)+1]× [ 1 2n(n+1)+1] matrix equation over ZD. We have completely analyzed the 2-, 3and 4-simplex equations in the generic D case. The solutions show interesting patterns that seem to continue to still higher simplex equations.