Quantum cohomology of Hilbn(C{double-struck}2) and the weighted hook walk on Young diagrams

Ionuţ Ciocan-Fontanine, Matjaž Konvalinka, Igor Pak

Research output: Contribution to journalArticle

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Abstract

Following the work of Okounkov and Pandharipande (2010) [OP1,OP2], Diaconescu [D], and the recent work of I. Ciocan-Fontanine et al. (in preparation) [CDKM] studying the equivariant quantum cohomology QH*(C{double-struck}*)2 (Hilbn) of the Hilbert scheme and the relative Donaldson-Thomas theory of P{double-struck}1×C{double-struck}2, we establish a connection between the J-function of the Hilbert scheme and a certain combinatorial identity in two variables. This identity is then generalized to a multivariate identity, which simultaneously generalizes the branching rule for the dimension of irreducible representations of the symmetric group in the staircase shape. We then establish this identity by a weighted generalization of the Greene-Nijenhuis-Wilf hook walk, which is of independent interest.

Original languageEnglish (US)
Pages (from-to)268-283
Number of pages16
JournalJournal of Algebra
Volume349
Issue number1
DOIs
StatePublished - Jan 1 2012

Keywords

  • Hilbert scheme of points
  • Hook walk
  • Hook-length formula
  • Quantum cohomology

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