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Cubic blocks are studied assembled from linear operators R acting in the tensor product of d linear "spin" spaces. Such operator is associated with a linear transformation A in a vector space over a field F of a finite characteristic p, like "permutation-type" operators studied by Hietarinta [J. Phys. A: Math. Gen. 30, 4757–4771 (1997)]. One small difference is that we do not require A and, consequently, R to be invertible; more importantly, no relations on R are required of the type of Yang–Baxter or its higher analogues. It is shown that, in d = 3 dimensions, a pn × pn × pn block decomposes into the tensor product of operators similar to the initial R. One generalization of this involves commutative algebras over F and allows to obtain, in particular, results about spin configurations determined by a four-dimensional R. Another generalization deals with introducing Boltzmann weights for spin configurations; it turns out that there exists a non-trivial self-similarity involving Boltzmann weights as well. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]