The weak order on the hyperoctahedral group and the monomial basis for the Hopf algebra of signed permutations

TL;DR

This paper provides a combinatorial description of the weak order on the hyperoctahedral group and explores its implications for the structure of the Hopf algebra of signed permutations, including the monomial basis and its relation to quasi-symmetric functions.

Contribution

It introduces a new combinatorial characterization of the weak order on hyperoctahedral groups and analyzes the structure of shifted products and the monomial basis in the associated Hopf algebra.

Findings

Shifted products are disjoint unions of convex intervals.

The monomial basis maps to either zero or a monomial quasi-symmetric function of type B.

The characterization facilitates understanding of the algebraic and order-theoretic properties.

Abstract

We give a combinatorial description for the weak order on the hyperoctahedral group. This characterization is then used to analyze the order-theoretic properties of the shifted products of hyperoctahedral groups. It is shown that each shifted product is a disjoint union of some intervals, which can be convex embedded into a hyperoctahedral group. As an application, we investigate the monomial basis for the Hopf algebra $\mathfrak{H}Sym$ of signed permutations, related to the fundamental basis via M\"obius inversion on the weak order on hyperoctahedral groups. It turns out that the image of a monomial basis element under the descent map from $\mathfrak{H}Sym$ to the algebra of type $B$ quasi-symmetric functions is either zero or a monomial quasi-symmetric function of type $B$.