Se p 19 98 Science Fiction and Macdonald ’ s Polynomials

This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules M µ and corresponding elements of the Macdonald basis. We recall that in [10], M µ is defined for a partition µ ⊢ n, as the linear span of derivatives of a certain bihomogeneous polynomial ∆ µ y n. It has been conjectured in [6], [10] that M µ has n! dimensions and that its bigraded Frobenius characteristic is given by the symmetric polynomial H µ (x; q, t) = λ⊢n S λ (X) K λµ (q, t) where the K λµ (q, t) are related to the Macdonald q, t-Kostka coefficients K λµ (q, t) by the identity K λµ (q, t) = K λµ (q, 1/t)t n(µ) with n(µ) the x-degree of ∆ µ (x; y). Using this conjectured relation we can translate observed or proved properties of the modules M µ into identities for Macdonald polynomials. Computer data has suggested that as ν varies among the immediate predecessors of a partition µ, the spaces M ν behave like a boolean lattice. The same appears to holds true when ν varies among the immediate successors of µ. Combining this property with a number of observed facts and some " heuristics " we have been led to formulate a number of remarkable conjectures about the Macdonald polynomials. In particular we obtain a representation theoretical interpretation for some of the symmetries that can be found in the computed tables of q, t-Kostka coefficients. The expression " Science Fiction " here refers to a package of " heuristics " that we use to describe relations amongst the modules M ν. These heuristics are purely speculative assertions that are used as a convenient guide to the construction of identities relating the corresponding bigraded Frobenius characteristics. Nevertheless computer experimentation reveals that these assertions are " generically " correct. Moreover, the evidence in support of the symmetric function identities we have derived from them are overwhelming. In particular, we show that various independent consequences of our heuristics lead to the same final identities.