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

2022/05/03 by Yu, Houyi · 1 citation
#05E05 #20F55 #Combinatorics (math.CO) #FOS: Mathematics #G.2.1 #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2205.01266

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 \mathfrakHSym of signed permutations, related to the fundamental basis via Möbius inversion on the weak order on hyperoctahedral groups. It turns out that the image of a monomial basis element under the descent map from \mathfrakHSym to the algebra of type B quasi-symmetric functions is either zero or a monomial quasi-symmetric function of type B.

Cited by

• Combinatorial equivalence of separable elements in types A and B

Related

• Wachs permutations, Bruhat order and weak order
• Structure of the Malvenuto-Reutenauer Hopf algebra of permutations
• On Hyperoctahedral Enumeration System, Application to Signed Permutations
• Avoiding 2-letter signed patterns
• The absolute order on the hyperoctahedral group