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

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 $mathbb{Q}^Eoplusmathbb{Q}^{E^*}$, 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 denfant.