Hopf algebra structure of symmetric and quasisymmetric functions in superspace

Abstract

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

Quasisymmetric harmonics of the exterior algebra

“…. , 0) a composition of length r. The fermionic degree of F 1 r is exactly r. In [11], they show that a r,s exists and express it as a sum of ±1, but they do not give an explicit formula. Furthermore, they indicate that a r,s = (−1) rs a s,r .…”

“…Quasisymmetric functions generate a commutative subalgebra. In [11], the authors showed that the quasisymmetric functions in one set of commuting variables and one set of anticommuting variables forms a graded Hopf algebra. This implies that the quasisymmetric functions in one set of anticommuting variables are closed under multiplication and the space is spanned by one element at each non-negative degree.…”

Quasisymmetric harmonics of the exterior algebra

“…In [11], the authors showed that the quasisymmetric functions in one set of commuting variables and one set of anticommuting variables forms a graded Hopf algebra. This implies that the quasisymmetric functions in one set of anticommuting variables are closed under multiplication and the space is spanned by one element at each nonnegative degree.…”

“…In the notation of [11], where is a composition of length r . The “ ” over a part in [11] is to indicate a fermionic component. Therefore, the fermionic degree of is exactly r .…”

“…Motivated by physics, Desrosiers, Lapointe, and Mathieu [9, 10] introduced symmetric functions with one set of commuting and one set of anticommuting variables known as symmetric function in superspace. The commuting variables encode bosons, whereas the anticommuting ones encode fermions; hence, the anticommuting variables are sometimes referred to as “fermionic variables.” The Hopf algebra structure of the ring of symmetric functions in superspace was extended to quasisymmetric functions in superspace [11] and so a natural question is to extend the study of coinvariants of polynomial rings with commuting and anticommuting variables to the quotients of these polynomial rings by the ideal generated by “super” quasisymmetric polynomials.…”