Quasisymmetric harmonics of the exterior algebra

Abstract

AbstractWe study the ring of quasisymmetric polynomials innanticommuting (fermionic) variables. Let$R_n$denote the ring of polynomials innanticommuting 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 subalgebra 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$$ \begin{align*}\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,\end{align*} $$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.