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

Abstract

AbstractWe study the coordinate rings ofscheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here μ′ 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 famousq-Kostka polynomialis the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by λ in the ring. Lascoux and Schützenberger [15, 13] gave combinatorially a decomposition ofas 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, and a new proof of the atomic decomposition of theq-Kostka polynomials.