An Open Problem in the Combinatorics of Macdonald Polynomials

Although a combinatorial interpretation of Macdonald polynomials involving llings of Ferrers diagrams of partitions has been known since 2004, a general combinatorial proof of a well-known symmetry property of these polynomials remains elusive. Our paper frames the problem and discusses an already-solved special case, that of one-dimensional (single-row or single-column) Ferrers diagrams, and then give our solution for a new special case, that of hookshaped Ferrers diagrams with standardized (non-multiset) llings. We then discuss remaining issues.