Tableau complexes

Allen Knutson, Ezra Miller, Alexander Yong

Research output: Contribution to journalArticlepeer-review

Abstract

Let X, Y be finite sets and T a set of functions X → Y which we will call " tableaux". We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such tableau complexes have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions [BP79]. In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are labeled by Buch's set-valued semistandard tableaux. One vertex decomposition of this "Young tableau complex" parallels Lascoux's transition formula for vexillary double Grothendieck polynomials [La01, La03]. Consequently, we obtain formulae (both old and new) for these polynomials. In particular, we present a common generalization of the formulae of Wachs [Wa85] and Buch [Bu02], each of which implies the classical tableau formula for Schur polynomials.

Original languageEnglish (US)
Pages (from-to)317-343
Number of pages27
JournalIsrael Journal of Mathematics
Volume163
DOIs
StatePublished - Jan 2008
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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