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Nilpotent Orbit Varieties and the Atomic Decomposition of the Q-Kostka Polynomials

Published online by Cambridge University Press:  20 November 2018

William Brockman  and
Mark Haiman
William Brockman
Affiliation:
Dept. of Mathematics, Wells Hall Michigan State Univ. East Lansing, MI 48824 USA, e-mail: [email protected]
Mark Haiman
Affiliation:
Dept. of Mathematics Univ. California San Diego La Jolla, CA, 92093-0112 USA, e-mail: [email protected]
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Abstract

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We study the coordinate rings of $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$ 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 [5] proved a conjecture of Kraft [12] that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer [22, 23]. The famous $q$-Kostka polynomial ${{\tilde{K}}_{\lambda \mu }}(q)$ is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by $\lambda $ in the ring $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$. Lascoux and Schützenberger [15, 13] gave combinatorially a decomposition of ${{\tilde{K}}_{\lambda \mu }}(q)$ 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 [19] 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 [3]. This gives a direct-sum decomposition of the ideals yielding the $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$, and a new proof of the atomic decomposition of the $q$-Kostka polynomials.

Keywords

05E1014M9920G0505E15q-Kostka polynomialsatomic decompositionnilpotent conjugacy classesnilpotent orbit varieties
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 3 , 01 June 1998 , pp. 525 - 537
Copyright
Copyright © Canadian Mathematical Society 1998

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