Séminaire Lotharingien de Combinatoire, B54As (2007), 55 pp.
Jeffrey Remmel
The Combinatorics of Macdonald's Dn1 Operator
Abstract.
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 Dn1 and proved that for any
Schur function s\la(x1, ..., xn),
Dn1 s\la(x1, ..., xn) =
\sum\mu d\la,\mu(q,t)
s\mu(x1, ..., xn)
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.
Received: December 11, 2005.
Accepted: September 14, 2006.
Final Version: January 5, 2007.
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