0/1-Polytopes related to Latin squares autotopisms

TL;DR

This paper investigates the geometric structure of subsets of Latin squares with specific autotopisms, representing them as 0/1-polytopes, and analyzes their dimensions for squares up to order 9.

Contribution

It characterizes the polyhedral structure of Latin squares with given autotopisms and determines their dimensions for small orders, providing new insights into their geometric properties.

Findings

Polytope structure is generated by a specific lower-dimensional polytope.

Dimension of these polytopes is explicitly calculated for Latin squares up to order 9.

Provides a formula relating cycle structures of autotopisms to polytope dimensions.

Abstract

The set LS(n) of Latin squares of order $n$ can be represented in $\mathbb{R}^{n^3}$ as a $(n-1)^3$-dimensional 0/1-polytope. Given an autotopism $\Theta=(\alpha,\beta,\gamma)\in\mathfrak{A}_n$, we study in this paper the 0/1-polytope related to the subset of LS(n) having $\Theta$ in their autotopism group. Specifically, we prove that this polyhedral structure is generated by a polytope in $\mathbb{R}^{((\mathbf{n}_{\alpha}-\mathbf{l}_{\alpha}^1)\cdot n^2 + \mathbf{l}_{\alpha}^1\cdot \mathbf{n}_{\beta}\cdot n)-(\mathbf{l}_{\alpha}^1\cdot \mathbf{l}_{\beta}^1\cdot (n -\mathbf{l}_{\gamma}^1) + \mathbf{l}_{\alpha}^1\cdot \mathbf{l}_{\gamma}^1\cdot (\mathbf{n}_{\beta} -\mathbf{l}_{\beta}^1) + \mathbf{l}_{\beta}^1\cdot \mathbf{l}_{\gamma}^1\cdot (\mathbf{n}_{\alpha} -\mathbf{l}_{\alpha}^1))}$, where $\mathbf{n}_{\alpha}$ and $\mathbf{n}_{\beta}$ are the number of cycles of $\alpha$ and $\beta$, respectively, and $\mathbf{l}_{\delta}^1$ is the number of fixed points of $\delta$, for all $\delta\in \{\alpha,\beta,\gamma\}$. Moreover, we study the dimension of these two polytopes for Latin squares of order up to 9.