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For a map $\mathcal M$ cellularly embedded on a connected and closedorientable surface, the bases of its Lagrangian (also known as delta-) matroid$\Delta(\mathcal M)$ correspond to the bases of a Lagrangian subspace $L$ ofthe 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 Lagrangiansubspace $L$ is said to be a representation of both $\mathcal M$ and$\Delta(\mathcal M)$. Furthermore, the bases of $\Delta(\mathcal M)$, whenunderstood as vertices of the hypercube $[-1,1]^n$, induce a polytope $\mathbfP(\Delta(\mathcal M))$ 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 commenton the action of $BC_n$ on $\mathcal M$ when $\mathcal M$ is understood adessin d'enfant.

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