An action of the Coxeter group $BC_n$ on maps on surfaces, Lagrangian matroids and their representations

Abstract

For a map $\mathcal M$ cellularly embedded on a connected and closed orientable surface, the bases of its Lagrangian (also known as delta-) matroid $\Delta(\mathcal M)$ correspond to the bases of a Lagrangian subspace $L$ of the standard orthogonal space $\mathbb{Q}^E\oplus\mathbb{Q}^{E^*}$, where $E$ and $E^*$ are the edge-sets of $\mathcal M$ and its dual map. The Lagrangian subspace $L$ is said to be a representation of both $\mathcal M$ and $\Delta(\mathcal M)$. Furthermore, the bases of $\Delta(\mathcal M)$, when understood as vertices of the hypercube $[-1,1]^n$, induce a polytope $\mathbf P(\Delta(\mathcal M))$ with edges parallel to the root system of type $BC_n$. In this paper we study the action of the Coxeter group $BC_n$ on $\mathcal M$, $L$, $\Delta(\mathcal M)$ and $\mathbf P(\Delta(\mathcal M))$. We also comment on the action of $BC_n$ on $\mathcal M$ when $\mathcal M$ is understood a dessin d'enfant.