Duchamp--Hivert--Thibon introduced the construction of a right
Hnβ(0)-module, denoted as MPβ, for any partial order P on the set [n].
This module is defined by specifying a suitable action of Hnβ(0) on the set
of linear extensions of P. In this paper, we refer to this module as the
poset module associated with P. Firstly, we show that β¨nβ₯0βG0β(P(n)) has a Hopf algebra structure that is isomorphic to the
Hopf algebra of quasisymmetric functions, where P(n) is the full
subcategory of mod-Hnβ(0) whose objects are direct sums of finitely
many isomorphic copies of poset modules and G0β(P(n)) is the
Grothendieck group of P(n). We also demonstrate how
(anti-)automorphism twists interact with these modules, the induction product
and restrictions. Secondly, we investigate the (type 1) quasisymmetric power
sum expansion of some quasi-analogues YΞ±β of Schur functions, where
Ξ± is a composition. We show that they can be expressed as the sum of the
P-partition generating functions of specific posets, which allows us to
utilize the result established by Liu--Weselcouch. Additionally, we provide a
new algorithm for obtaining these posets. Using these findings, for the dual
immaculate function and the extended Schur function, we express the
coefficients appearing in the quasisymmetric power sum expansions in terms of
border strip tableaux.Comment: 42 page