Some natural bigraded S_n-modules

This work is gratefully dedicated to Dominique Foata for his in- spiring and pioneering work in algebraic combinatorics. We hope that he will flnd it to be in harmony with the Lotharingian spirit which he has nurtured for so many years. ABSTRACT. We construct for each "' n ab igradedSn-module H" and conjecture that its Frobenius characteristic C"(x;q;t) yields the Macdonald coe-cients K‚"(q;t) . To be precise, we conjecture that the expansion of C"(x;q;t) in terms of the Schur basis yields coe-cients C‚"(q;t) which are related to the K‚"(q;t) by the identity C‚"(q;t )= K‚"(q;1=t)tn("). The validity of this would give a representation theoretical setting for the Macdonald basisfP"(x;q;t)g" and establish the Macdonald conjecture that the K‚"(q;t) are polynomials with positive integer coe-cients. The space H" is deflned as the linear span of derivatives of a certain bihomogeneous polynomial ¢"(x;y) in the variables x1;x2;:::;xn, y1;y2;:::;yn. On the validity of our conjecture H" would necessarily have n! dimension. We refer to the latter assertion as the n!-conjecture. Several equivalent forms of this conjecture will be discussed here together with some of their consequences. In particular, we derive that the polynomials C‚"(q;t) have a number of basic properties in common with the coe-cients ~ K‚"(q;t )= K‚"(q;1=t)tn("). For instance, we show that