Intersection of modules related to Macdonald's polynomials
Let \(\mu=(\mu_1,\ldots,\mu_k)\), \(\mu_k>0\), be a partition of \(n\) and let \[ \Delta_{\mu}(X,Y)=\det\left(x_i^{p^j}y_i^{q_j}/p_j!q_j!\right)_{i,j=1,\dots,n} \] be the \(n\times n\) determinant in \(X=\{x_1,\ldots,x_n\}\), \(Y=\{y_1,\ldots,y_n\}\), where \((p_1,q_1),\ldots,(p_n,q_n)\) are all points of the Ferrers diagram \[ \{(a,b)\mid 0\leq a\leq\mu_{j+1}-1\text{ and }0\leq b\leq k-1\} \] associated with \(\mu\). The Garsia-Haiman \({\mathcal S}_n\)-module \({\mathbf M}_{\mu}\) is spanned by all partial derivatives of \(\Delta_{\mu}(X,Y)\). There is a conjecture that the bigraded Frobenius characteristic of \({\mathbf M}_{\mu}\) is given by a renormalized version of Macdonald's polynomials. The purpose of the paper under review is to explore properties of intersections of modules \({\mathbf M}_{\nu}\) for partitions lying below a given partition \(\mu\) of \(n+1\). The authors present explicit results on the intersection and its Frobenius characteristic and a conjecture for the general form of this intersection. Although the conjecture is the result of extensive computer algebra experiments involving explicit computations with the \({\mathcal S}_n\)-modules \({\mathbf M}_{\mu}\), it has received a support from many sources. In particular, the authors have given a proof of the conjecture in the case of hook shapes.
scientific article; zbMATH DE number 1453344