General SXP rule for Hamel–King symmetric functions

Determine whether a general SXP rule exists for the Hamel–King symmetric function X_λ(z;q), analogous to the SXP rules for symplectic and orthogonal universal characters.

Background

Section 5.2 discusses SXP-type rules for universal symplectic and orthogonal characters and explains their connection with branching coefficients from classical representation theory. The paper observes that these rules do not immediately extend to the more general Hamel–King symmetric function X_λ(z;q), leaving unresolved whether an analogous expansion exists in that general setting.

References

As this section shows, SXP rules for symplectic and orthogonal characters are intimately connected with the representation theory of their associated groups. Thus, it is not clear if there exists a general SXP rule for the symmetric function X'1(z; q) in the same manner.

— Character factorisations, $z$-asymmetric partitions and plethysm  (2501.18520 - Albion, 30 Jan 2025) in Section 5.2, final paragraph