TY - JOUR

T1 - Tableau complexes

AU - Knutson, Allen

AU - Miller, Ezra

AU - Yong, Alexander

N1 - Funding Information:
Acknowledgments. We would like to thank Eric Babson, Nantel Bergeron, Christian Krattenthaler, Carsten Lange and Ravi Vakil for helpful comments. We would also like to thank the two anonymous referees—the one from this journal, of course, but also a different one who mistakenly refereed this paper when asked to review [KMY05]. AK was supported by NSF grant DMS-0303523. EM was partially supported by NSF grant DMS-0304789, NSF CAREER grant DMS-0449102 and a University of Minnesota McKnight Land-Grant Professorship. This work was partially completed while AY was an NSERC visitor to the Fields Institute during the 2005 semester on “The geometry of string theory”, and while an NSF supported visitor at the Mittag-Leﬄer institute, during the 2005 semester on “Algebraic combinatorics”. All of us would like to thank the 2005 AMS Summer Research Institute on Algebraic Geometry in Seattle, where this work was also carried out.

PY - 2008/1

Y1 - 2008/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s11856-008-0014-5

DO - 10.1007/s11856-008-0014-5

M3 - Article

AN - SCOPUS:58449126309

VL - 163

SP - 317

EP - 343

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -