Some Natural Bigraded $S_n$-Modules

Abstract

We construct for each $\mu\vdash n $ a bigraded $S_n$-module $\mathbf{H}_\mu$ and conjecture that its Frobenius characteristic $C_{\mu}(x;q,t)$ yields the Macdonald coefficients $K_{\lambda\mu}(q,t)$. To be precise, we conjecture that the expansion of $C_{\mu}(x;q,t)$ in terms of the Schur basis yields coefficients $C_{\lambda\mu}(q,t)$ which are related to the $K_{\lambda\mu}(q,t)$ by the identity $C_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$. The validity of this would give a representation theoretical setting for the Macdonald basis $\{ P_\mu(x;q,t)\}_\mu$ and establish the Macdonald conjecture that the $K_{\lambda\mu}(q,t)$ are polynomials with positive integer coefficients. The space $\mathbf{H}_\mu$ is defined as the linear span of derivatives of a certain bihomogeneous polynomial $\Delta_\mu(x,y)$ in the variables $x_1,x_2,\ldots ,x_n$, $y_1,y_2,\ldots ,y_n$. On the validity of our conjecture $\mathbf{H}_\mu$ 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 $C_{\lambda\mu}(q,t)$ have a number of basic properties in common with the coefficients $\tilde{K}_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$. For instance, we show that $C_{\lambda\mu}(0,t)=\tilde{K}_{\lambda\mu}(0,t)$, $C_{\lambda\mu}(q,0)=\tilde{K}_{\lambda\mu}(q,0)$ and show that on the $n!$ conjecture we must also have the equalities $C_{\lambda\mu}(1,t)=\tilde{K}_{\lambda\mu}(1,t)$ and $C_{\lambda\mu}(q,1)=\tilde{K}_{\lambda\mu}(q,1)$. The conjectured equality $C_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$ will be shown here to hold true when $\lambda$ or $\mu$ is a hook. It has also been shown (see [9]) when $\mu$ is a $2$-row or $2$-column partition and in [18] when $\mu$ is an augmented hook.