Grassmannian-parameterized solutions to direct-sum polygon and simplex equations

We consider polygon and simplex equations, of which the simplest nontrivial examples are pentagon (5-gon) and Yang-Baxter (2-simplex), respectively. We examine the general structure of (2n + 1)-gon and 2n-simplex equations in direct sums of vector spaces. Then, we provide a construction for their solutions, parameterized by elements of the Grassmannian Gr(n + 1, 2n + 1).