0/1-Polytopes related to Latin squares autotopisms

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.