0/1-Polytopes related to Latin squares autotopisms

Abstract: The set LS(n) of Latin squares of order $n$ can be represented in $\mathbb{R}<sup>{n<sup>3}$ as a $(n-1)<sup>3$-dimensional 0/1-polytope. Given an autotopism $\Theta=(\alpha,\beta,\gamma)\in\mathfrak{A}<em>n$, 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 $$\mathbb{R}^{{((\mathbf{n}}</em>{\alpha}-\mathbf{l}<em>{\alpha}^1)\cdot n² + \mathbf{l}</em>{\alpha}^1\cdot \mathbf{n}<em>{\beta}\cdot n)-(\mathbf{l}</em>{\alpha}^1\cdot \mathbf{l}<em>{\beta}^1\cdot (n -\mathbf{l}</em>{\gamma}¹⁾ + \mathbf{l}<em>{\alpha}^1\cdot \mathbf{l}</em>{\gamma}^1\cdot (\mathbf{n}<em>{\beta} -\mathbf{l}</em>{\beta}¹⁾ + \mathbf{l}<em>{\beta}^1\cdot \mathbf{l}</em>{\gamma}^1\cdot (\mathbf{n}<em>{\alpha} -\mathbf{l}</em>{\alpha}<sup>1))}$$, where $\mathbf{n}<em>{\alpha}$ and $\mathbf{n}</em>{\beta}$ are the number of cycles of $\alpha$ and $\beta$, respectively, and $\mathbf{l}_{\delta}<sup>1$ 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.