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

  Published:1998-06-01
 Printed: Jun 1998
  • William Brockman
  • Mark Haiman
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Abstract

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.
Keywords: $q$-Kostka polynomials, atomic decomposition, nilpotent conjugacy classes, nilpotent orbit varieties $q$-Kostka polynomials, atomic decomposition, nilpotent conjugacy classes, nilpotent orbit varieties
MSC Classifications: 05E10, 14M99, 20G05, 05E15 show english descriptions Combinatorial aspects of representation theory [See also 20C30]
None of the above, but in this section
Representation theory
Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]
05E10 - Combinatorial aspects of representation theory [See also 20C30]
14M99 - None of the above, but in this section
20G05 - Representation theory
05E15 - Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]
 

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