Generalize the skew Hall–Littlewood–Schubert series to arbitrary multiplicity vectors A

Develop a generalized skew Hall–Littlewood–Schubert series for an arbitrary tuple A = (m_0, m_1, ..., m_n) of nonnegative integers by defining appropriate polynomials W_C(Y) attached to chains C in the poset P_A (whose elements are tuples (a_0, ..., a_n) with 0 ≤ a_i ≤ m_i ordered by the tableau order), and prove that the resulting series satisfies a self-reciprocity property analogous to that established for the case A = (r, 1, ..., 1).

Background

The paper introduces the skew Hall–Littlewood–Schubert series as a simultaneous generalization and refinement of generalized Igusa functions and Hall–Littlewood–Schubert series, and proves a self-reciprocity property for the case corresponding to A = (r, 1, ..., 1).

In the Further discussion, the authors point out a natural broader poset P_A whose elements are all tuples (a_0, ..., a_n) with 0 ≤ a_i ≤ m_i, ordered by the same tableau order. Interpreted as sub-multisets of {0^{m_0}, 1^{m_1},..., n^{m_n}}, this setting generalizes the one treated in the paper.

They then pose the problem of extending their construction—namely, defining the chain weight polynomials W_C(Y) and the associated series—to this general P_A and establishing a corresponding self-reciprocity property.