A PROOF OF THE N - CONJECTURE FOR GENERALIZED HOOKS

In [4], Garsia and Haiman [Electronic J. of Combinatorics 3, No. 2 (19 96)] pose a conjecture central to their study of the Macdonald polynom ials H-mu(x; q, t). For each mu proves n one defines a certain determi nant Delta(mu)(X(n), Y-n) in two sets of variables. The n! conjecture asserts that the vector space given by the linear span of derivatives of Delta(mu), written L[partial derivative(x)(p) partial derivative(y) (q) Delta(mu)(X(n), Y-n)], has dimension it n!. The conjecture reduces to a well-known result about the Vandermonde determinant when mu = (1 (n)) or mu = (n) (see [1], for example). Garsia and Haiman (see [Proc. Natl. Acad. Sci. 90 (1993), 3607-3610]) have demonstrated the conject ure for two-rowed shapes mu = (a, b), two-columned shapes mu = (2(a), 1(b)) and hook shapes mu = (a, 1(b)). In this paper. we give an overvi ew of the methods used by Reiner in his thesis to prove the n! conject ure for generalized hooks, that is, for mu = (a, 2, 1(b)). (C) 1996 Ac ademic Press, Inc.