Intersection of modules related to Macdonaldʹs polynomials Original Research Article

This work studies the intersection of certain k-tuples of Garsia-Haiman Sn-modules Mμ. We recall that in A. Garsia, M. Haiman, Electronic J. Combin. 3(2) Foata Festschrift (1996) R24, 60 for μ⊢n, Mμ is defined as the linear span of all partial derivatives of a certain bihomogeneous polynomial Δμ(X,Y) in the variables x1,x2,…,xn, y1,y2,…,yn. It has been conjectured that Mμ has n! dimensions and that its bigraded Frobenius characteristic is given by a renormalized version of Macdonaldʹs polynomials F. Bergeron, A. Garsia, Science fiction and Macdonaldʹs 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 S2k1j appearing in Mμ. This allows an explicit description of the intersection of Mνʹs, as ν varies among immediate predecessors of a partition μ. We present here explicit results about the space ⋂ν→μMν and its Frobenius characteristic, as well as a conjecture for the general form of this intersection. We give an explicit proof for hook shapes.