The weak order on Weyl posets

TL;DR

This paper introduces a lattice structure on subsets of finite root systems that generalizes the weak order on Coxeter groups, connecting it to well-known polytopes like permutahedra and associahedra.

Contribution

It extends the weak order to all subsets of root systems and explores subposets related to key combinatorial and geometric objects.

Findings

The lattice structure extends the weak order to all root subsets.

Subposets correspond to elements, intervals, and faces of key polytopes.

Results generalize recent findings to all finite crystallographic root systems.

Abstract

We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice induced by antisymmetric closed subsets of roots is again a lattice. We then study further subposets of this lattice which naturally correspond to the elements, the intervals and the faces of the permutahedron and the generalized associahedra of the corresponding Weyl group. These results extend to arbitrary finite crystallographic root systems the recent results of G. Chatel, V. Pilaud and V. Pons on the weak order on posets and its induced subposets.