Self-similarity in cubic blocks of $\mathcal R$-operators

TL;DR

This paper investigates the self-similarity properties of cubic blocks assembled from linear operators in tensor spaces, revealing decompositions and invariances that extend to non-invertible cases and include Boltzmann weights.

Contribution

It demonstrates that in three dimensions, cubic blocks of $\\mathcal R$-operators decompose into tensor products, extending previous permutation-type results without requiring invertibility or Yang-Baxter relations.

Findings

Decomposition of 3D blocks into tensor products of similar operators.

Extension to non-invertible operators and algebraic generalizations.

Existence of non-trivial self-similarity involving Boltzmann weights.

Abstract

Cubic blocks are studied assembled from linear operators $\mathcal R$ 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. One small difference is that we do not require $A$ and, consequently, $\mathcal R$ to be invertible; more importantly, no relations on $\mathcal R$ are required of the type of Yang--Baxter or its higher analogues. It is shown that, in $d=3$ dimensions, a $p^n\times p^n\times p^n$ block decomposes into the tensor product of operators similar to the initial $\mathcal R$. One generalization of this involves commutative algebras over $F$ and allows to obtain, in particular, results about spin configurations determined by a four-dimensional $\mathcal R$. 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.