Quasisymmetric analogue of the Hall inner product for immanants

Determine how an analogue of the Hall bilinear function on symmetric functions can be applied to generalize immanants using quasisymmetric functions, despite the fact that the algebra of quasisymmetric functions is not self-dual.

Background

The paper discusses the use of the Hall inner product on the algebra of symmetric functions to define immanant-like matrix functions by pairing a fixed symmetric function with power-sum functions. Because the algebra of quasisymmetric functions is not self-dual, the corresponding pairing cannot be transferred directly. The unresolved issue is to identify an appropriate bilinear construction that would support an analogous theory of quasisymmetric immanants.

References

Since $\textsf{QSym}$ is not self-dual, as reviewed below, it is not clear how an analogue of the bilinear function in defineHall could be applied to generalize immanants with the use of quasisymmetric functions.

defineHall:

$\langle h_{\lambda}, m_{\mu} \rangle = \delta_{\lambda, \mu} = \begin{cases} 1 & \text{if $\lambda = \mu$,} \\ 0 & \text{if $\lambda \neq \mu$.} \end{cases} $

— Quasi-immanants  (2501.15667 - Campbell, 26 Jan 2025) in Section 2, Background