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

TL;DR

This paper explores how the Coxeter group BC_n acts on maps on surfaces, Lagrangian matroids, and their representations, revealing symmetries and structural properties related to these mathematical objects.

Contribution

It introduces the action of the Coxeter group BC_n on maps, Lagrangian matroids, and their representations, connecting surface embeddings with root system symmetries.

Findings

BC_n acts on Lagrangian matroids and their representations

The induced polytopes have edges parallel to BC_n root system

The action extends to dessins d'enfant 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.