The algebra of extended peaks

TL;DR

This paper introduces a new family of subalgebras within the ring of quasisymmetric functions, using a basis indexed by extended peak sets and involving a complex root of unity parameter.

Contribution

It develops a novel algebraic structure based on extended peak sets, extending previous work on $q$-deformed $P$-partitions and Gessel's fundamental functions.

Findings

Subalgebras admit bases indexed by extended peak sets.

Basis elements are $q$-analogues at roots of unity.

Connects extended peaks with intermediate permutation statistics.

Abstract

Building up on our previous works regarding $q$-deformed $P$-partitions, we introduce a new family of subalgebras for the ring of quasisymmetric functions. Each of these subalgebras admits as a basis a $q$-analogue to Gessel's fundamental quasisymmetric functions where $q$ is equal to a complex root of unity. Interestingly, the basis elements are indexed by sets corresponding to an intermediary statistic between peak and descent sets of permutations that we call extended peak.