The Combinatorics of Macdonald's Dn1 Operator

Jeffrey Remmel

Séminaire lotharingien de combinatoire, 54A (2005-2007)

Résumé

To prove the existence of the Macdonald polynomials {P_\la(x;q,t)}, \la a partition of n, Macdonald [Séminaire Lotharingien Combin. 20 (1988), Article B20a; "Symmetric functions and Hall polynomials", 2nd ed., Clarendon Press, New York, 1995] introduced an operator D_n¹ and proved that for any Schur function s_\la(x₁, ..., x_n), D_n¹ s_\la(x₁, ..., x_n) = \sum_\mu d_\la,\mu(q,t) s_\mu(x₁, ..., x_n) where the sum runs over all partitions \mu of n which are less than or equal to \la in the dominance order and the d_\la,\mu(q,t) are polynomials in q and t with integer coefficients. We give an explicit combinatorial formula for the d_\la,\mu(q,t)'s.