Quasisymmetric harmonics of the exterior algebra

TL;DR

This paper explores the structure of quasisymmetric polynomials within an algebra of anticommuting variables, revealing their algebraic properties, bases indexed by ballot sequences, and the Hilbert series of related quotient spaces.

Contribution

It introduces the concept of quasisymmetric polynomials in fermionic variables, proves their subalgebra structure, and characterizes bases and Hilbert series related to quotient spaces.

Findings

Quasisymmetric polynomials form a commutative subalgebra in the fermionic setting.

A basis of the quotient space is indexed by ballot sequences, with a specific Hilbert series.

The ideal generated by these polynomials has a basis indexed by sequences breaking the ballot condition.

Abstract

We study the ring of quasisymmetric polynomials in $n$ anticommuting (fermionic) variables. Let $R_n$ denote the polynomials in $n$ anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in $R_n$ form a commutative sub-algebra of $R_n$. (2) There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$ \text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,$$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$. (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition