Explicit expansions for the dual shin basis

Derive explicit formulas expanding the dual shin basis of the algebra of quasisymmetric functions into its usual bases, thereby enabling the study of quasisymmetric dual shin functions in the quasi-immanant framework.

Background

The paper lists several Schur-like bases and their dual bases, including the dual shin basis. It states that the dual shin basis is insufficiently understood and that explicit expansions into standard quasisymmetric-function bases are unavailable. Consequently, the paper excludes dual shin functions from its analysis; obtaining such formulas would make it possible to investigate their associated quasi-immanants.

References

Little is known about the dual of the shin basis, and no explicit formulas are known for expanding this dual basis into the usual bases of $\textsf{QSym}$, so we omit consideration of quasisymmetric dual shin functions.

— Quasi-immanants  (2501.15667 - Campbell, 26 Jan 2025) in Section 3, Quasi-Schur second immanants