On Kernels of Descent Statistics

Abstract

The kernel $\mathcal{K}^{\operatorname{st}}$ of a descent statistic $\operatorname{st}$, introduced by Grinberg, is a subspace of the algebra $\operatorname{QSym}$ of quasisymmetric functions defined in terms of $\operatorname{st}$-equivalent compositions, and is an ideal of $\operatorname{QSym}$ if and only if $\operatorname{st}$ is shuffle-compatible. This paper continues the study of kernels of descent statistics, with emphasis on the peak set $\operatorname{Pk}$ and the peak number $\operatorname{pk}$. The kernel $\mathcal{K}^{\operatorname{Pk}}$ in particular is precisely the kernel of the canonical projection from $\operatorname{QSym}$ to Stembridge's algebra of peak quasisymmetric functions, and is the orthogonal complement of Nyman's peak algebra. We prove necessary and sufficient conditions for obtaining spanning sets and linear bases for the kernel $\mathcal{K}^{\operatorname{st}}$ of any descent statistic $\operatorname{st}$ in terms of fundamental quasisymmetric functions, and give characterizations of $\mathcal{K}^{\operatorname{Pk}}$ and $\mathcal{K}^{\operatorname{pk}}$ in terms of the fundamental basis and the monomial basis of $\operatorname{QSym}$. Our results imply that the peak set and peak number statistics are $M$-binomial, confirming a conjecture of Grinberg.