1. CJM Online first
2. CJM 1998 (vol 50 pp. 525)
 Brockman, William; Haiman, Mark

Nilpotent orbit varieties and the atomic decomposition of the $q$Kostka polynomials
We study the coordinate rings~$k[\Cmubar\cap\hbox{\Frakvii t}]$ of
schemetheoretic
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
nonnegative integer coefficients, and Lascoux proposed a
corresponding decomposition in the cohomology model.
Our work provides a geometric interpretation of the atomic
decomposition. The Frobeniussplitting results of Mehta and van der
Kallen~\cite{Mehta&vanderKallen} imply a directsum 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 directsum
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.
Keywords:$q$Kostka polynomials, atomic decomposition, nilpotent conjugacy classes, nilpotent orbit varieties Categories:05E10, 14M99, 20G05, 05E15 
