Multiple left regular representations generated by alternants
For a lattice diagram, i.e. a collection of lattice cells \(L=\{(p_1,q_1),\dots,(p_n,q_n)\}\), \(p_i,q_i\in{\mathbb{N}}\cup\{0\}\), let \(\Delta_L(x;y)=\det\left(x_i^{p_j}y_i^{q_j}/p_j!q_j!\right)_{i,j=1}^n\) and let \({\mathbb{M}}_L\) be the \(S_n\)-module spanned by all partial derivatives of \(\Delta_L(x;y)\). The Garsia-Haiman conjecture states that, for \(L=\mu\) being a partition, the bigraded Frobenius characteristic of \({\mathbb{M}}_{\mu}\) is given by a renormalized version of the Macdonald polynomial. Given this conjecture, \({\mathbb{M}}_{\mu}\) is a graded version of the left regular representation of \(S_n\), and \(\text{ dim}{\mathbb{M}}_{\mu}=n!\) (where \(\mu\vdash n\)) and this is known as the \(n!\) conjecture. In the paper under review the authors consider a family of lattice diagrams \(L\) with the so-called MLRR-property which states that the module \({\mathbb{M}}_L\) decomposes into a direct sum of left regular representations of \(S_n\). They study both proven and conjectured families of diagrams with this property. The authors establish that the one-dimensional lattice diagrams have the MLRR-property. Then they generalize this result for any complex reflection group \(G\) and a \(G\)-alternant \(\Delta(x)\) and consider examples involving the wreath product \(C_m\wr S_n\). Further, the authors consider the two-dimensional diagrams consisting of a partition plus an external cell. Finally, they conclude with additional conjectures for general diagrams that unify properties of all considered cases.
scientific article; zbMATH DE number 1525302