Science Fiction and Macdonald's Polynomials

TL;DR

This paper explores the deep relationships between Garsia-Haiman modules and Macdonald polynomials, proposing conjectures and providing representation-theoretic insights into their structure and symmetries.

Contribution

It introduces new conjectures about the structure of Garsia-Haiman modules and their connection to Macdonald polynomials, including a representation-theoretic interpretation of symmetries.

Findings

Garsia-Haiman modules have dimensions n!

The bigraded Frobenius characteristic relates to Macdonald polynomials

Conjectures about the boolean lattice behavior of modules

Abstract

This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules $ {\bf M}_\mu$ and corresponding elements of the Macdonald basis. We recall that ${\bf M}_\mu$ is defined for a partition $\mu\part n$, as the linear span of 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 by Garsia and Haiman that ${\bf M}_\mu$ has $n!$ dimensions and that its bigraded Frobenius characteristic is given by the symmetric polynomial ${\widetilde{H}}_\mu(x;q,t)=\sum_{\lambda\part n} S_\lambda(X) {\widetilde{K}}_{\lambda\mu}(q,t)$ where the ${\widetilde{K}}_{\lambda\mu}(q,t)$ are related to the Macdonald $q,t$-Kostka coefficients $ K_{\lambda\mu}(q,t)$ by the identity ${\widetilde{K}}_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu)}$ with $n(\mu)$ the x-degree of $\Delta_ \mu(x;y)$. Computer data has suggested that as $\nu$ varies among the immediate predecessors of a partition $\mu$, the spaces ${\bf M}_\nu$ behave like a boolean lattice. We 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.