A Proof of then! Conjecture for Generalized Hooks

In [4], Garsia and Haiman [Electronic J. of Combinatorics3, No. 2 (1996)] pose a conjecture central to their study of the Macdonald polynomialsHμ(x; q, t). For eachμ⊢none defines a certain determinantΔμ(Xn, Yn) in two sets of variables. Then! conjecture asserts that the vector space given by the linear span of derivatives ofΔμ, written L[∂px ∂qy Δμ(Xn, Yn)], has dimensionn!. The conjecture reduces to a well-known result about the Vandermonde determinant whenμ=(1n) orμ=(n) (see [1], for example). Garsia and Haiman (see [Proc. Natl. Acad. Sci.90(1993), 3607–3610]) have demonstrated the conjecture for two-rowed shapesμ=(a, b), two-columned shapesμ=(2a, 1b) and hook shapesμ=(a, 1b). In this paper, we give an overview of the methods used by Reiner in his thesis to prove then! conjecture for generalized hooks, that is, forμ=(a, 2, 1b).