An action of the Coxeter group $B{C}_n$ 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 ${\mathbb{Q}}^E\oplus {\mathbb{Q}}^{E^{\ast }}$, where $E$ and $E^{\ast }$ 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 $B{C}_n$. In this paper we study the action of the Coxeter group $B{C}_n$ on $\mathcal{M}$, $L$, $\Delta (\mathcal{M})$ and $\mathbf{P}(\Delta (\mathcal{M}))$. We also comment on the action of $B{C}_n$ on $\mathcal{M}$ when $\mathcal{M}$ is understood a dessin d'enfant.

arXiv commentThis is the new version of the paper titled "Representations of Lagrangian matroids associated to partial duals of maps on surfaces". Submitted