Stability transfer for quasi-symmetric bases and 0-Hecke representation stability

Generalize the methods for transferring stability between symmetric-function bases to transfers between quasi-symmetric-function bases, and determine whether the generalized methods establish 0-Hecke representation stability for sequences in the sense of Laudone.

Background

The paper develops criteria for transferring coefficient stability between homogeneous bases of symmetric functions and applies them to representation stability for symmetric groups. It notes that an analogous Frobenius characteristic map exists from the Grothendieck ring of 0-Hecke algebras to quasi-symmetric functions, where representation stability is expressed through stabilization of multiplicities of certain $(P,\omega)$-partition generating functions. Extending the basis-transfer framework to quasi-symmetric functions could yield corresponding stability results for 0-Hecke modules.