On the quasisymmetric functions in superspace

TL;DR

This paper extends classical quasisymmetric functions to superspace, introducing new algebraic structures, bases, and Hopf superalgebras, with explicit formulas and applications to fundamental functions.

Contribution

It characterizes the algebra of quasisymmetric functions in superspace, constructs related Hopf superalgebras, and develops explicit product and coproduct formulas.

Findings

Characterization of the algebra as invariants under a symmetric group action

Introduction of the Hopf superalgebra of superpermutations

Explicit formulas for products and coproducts in the bases

Abstract

Quasisymmetric functions in superspace were introduced as a natural extension of classical quasisymmetric functions involving both commuting and anticommuting variables. In this paper, we first provide a characterization of the algebra of quasisymmetric functions in superspace as an algebra of invariants under a quasisymmetrizing action of the symmetric group. Furthermore, we complete the superspace analogue of the classical hierarchy of combinatorial Hopf algebras by introducing the algebra of quasisymmetric functions in noncommuting variables in superspace. We endow this algebra with a Hopf superalgebra structure and thoroughly investigate its $Q$-basis and monomial basis, which are indexed by set supercompositions. By restricting to the minimal elements of the underlying poset, we construct the Hopf superalgebra of superpermutations, serving as the superspace analogue of the Malvenuto--Reutenauer algebra. We provide explicit product and coproduct formulas for these bases in terms of super-shuffles and global descents. Finally, via an abelianization morphism, we apply these noncommutative structures to derive a product formula for fundamental quasisymmetric functions in superspace.