The algebra of extended peaks

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.