An Odd Presentation of $W(\mathrm{E}_6)$

TL;DR

This paper provides a geometric interpretation of the odd presentation of the Weyl group $W(E_6)$ by relating it to the action on the moduli space of real cubic surfaces and their tessellations.

Contribution

It offers a geometric understanding of $W(E_6)$'s odd presentation through its action on moduli spaces and tessellations, connecting algebraic and geometric perspectives.

Findings

Geometric meaning of $W(E_6)$ presentation derived from moduli space action.

Connection established between group presentation and tessellation of moduli space.

Insight into the period map's role in understanding $W(E_6)$.

Abstract

The Weyl group $W(\mathrm{E}_6)$ has an odd presentation due to Christopher Simons as factor group of the Coxeter group on the Petersen graph by deflation of the free hexagons. The goal of this paper is to give a geometric meaning for this presentation, coming from the action of $W(\mathrm{E}_6)$ on the moduli space of marked maximally real cubic surfaces and its natural tessellation as seen through the period map of Allcock, Carlson and Toledo.