A new family of posets generalizing the weak order on some Coxeter groups

TL;DR

This paper introduces a new family of posets constructed from acyclic digraphs and valuations, unifying several known order structures on Coxeter groups and wreath products, and associates quasi-symmetric functions to their elements.

Contribution

It generalizes the weak order on Coxeter groups and wreath products through a unified poset construction and computes their M"obius functions, connecting to classical symmetric functions.

Findings

The posets include weak orders on Coxeter groups of types A, B, and affine A.

The M"obius function values are explicitly computed for these posets.

Associated quasi-symmetric functions generalize Stanley and Lam's functions.

Abstract

We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its M\"obius function. We show that the weak order on Coxeter groups of type A, B, affine A, and the flag weak order on the wreath product $\mathbb{Z} \_r \wr S\_n$ introduced by Adin, Brenti and Roichman, are special instances of our construction. We conclude by associating a quasi-symmetric function to each element of these posets. In the $A$ and $\widetilde{A}$ cases, this function coincides respectively with the classical Stanley symmetric function, and with Lam's affine generalization.