This work studies the intersection of certain k-tuples of Garsia-Haiman S-n-modules M-mu. We recall that in A. Garsia, M. Haiman, Electronic J. Combin. 3(2) Foata Festschrift (1996) R24, 60 for mu proves n, M-mu is defined as the linear span of all partial derivatives of a certain bihomogeneous polynomial Delta(mu)(X,Y) in the variables x(1),x(2),...,x(n), y(1),y(2),...,y(n) It has been conjectured that M-mu has n! dimensions and that its bigraded Frobenius characteristic is given by a renormalized version of Macdonald's polynomials F. Bergeron, A. Garsia, Science fiction and Macdonald's polynomials, in: R. Floreanini, L. Vinet (Eds.), Algebraic Methods and q-Special Functions, CRM Proceedings & Lecture Notes, American Mathematical Society, Providence, RI, 48 pp. (Computer data have suggested a precise presentation for certain irreducible representations of Frobenius characteristic S-2k1j appearing in M-mu. This allows an explicit description of the intersection of M-nu's, as nu varies among immediate predecessors of a partition mu. We present here explicit results about the space boolean AND(nu-->mu) M-nu and its Frobenius characteristic, as well as a conjecture for the general form of this intersection. We give an explicit proof for hook shapes. (C) 2000 Elsevier Science B.V. All rights reserved.
