Nilpotent orbit varieties and the atomic decomposition of the $q$-Kostka polynomials

We study the coordinate rings $k[\Cmubar\cap\hbox{\Frakvii t}]$ of scheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here $\mu'$ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi \cite{deConcini&Procesi} proved a conjecture of Kraft \cite{Kraft} that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer \cite{Springer76,Springer78}. The famous $q$-Kostka polynomial $\Klmt(q)$ is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by $\lambda$ in the ring $k[\Cmubar\cap\hbox{\Frakvii t}]$. \LS \cite{L&S:Plaxique,Lascoux} gave combinatorially a decomposition of $\Klmt(q)$ as a sum of ``atomic'' polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model. Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen \cite{Mehta&vanderKallen} imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer \cite{Broer}. This gives a direct-sum decomposition of the ideals yielding the $k[\Cmubar\cap \hbox{\Frakvii t}]$, and a new proof of the atomic decomposition of the $q$-Kostka polynomials.