Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Matching polytopes and Specht modules

Author: Ricky Ini Liu
Journal: Trans. Amer. Math. Soc. 364 (2012), 1089-1107
MSC (2010): Primary 05E10
Published electronically: October 4, 2011
MathSciNet review: 2846364
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the dimension of the Specht module of a forest $ G$ is the same as the normalized volume of the matching polytope of $ G$. We also associate to $ G$ a symmetric function $ s_G$ (analogous to the Schur symmetric function $ s_\lambda $ for a partition $ \lambda $) and investigate its combinatorial and representation-theoretic properties in relation to the Specht module and Schur module of $ G$. We then use this to define notions of standard and semistandard tableaux for forests.

References [Enhancements On Off] (What's this?)

  • 1. Jack Edmonds.
    Maximum matching and a polyhedron with $ 0,1$-vertices.
    J. Res. Nat. Bur. Standards Sect. B, 69B:125-130, 1965. MR 0183532 (32:1012)
  • 2. William Fulton and Joe Harris.
    Representation theory, volume 129 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, 1991.
    A first course, Readings in Mathematics. MR 1153249 (93a:20069)
  • 3. G. D. James.
    A characteristic-free approach to the representation theory of $ {\mathfrak{S}}\sb {n}$.
    J. Algebra, 46(2):430-450, 1977. MR 0439924 (55:12805)
  • 4. Bernhard Korte and Jens Vygen.
    Combinatorial optimization, volume 21 of Algorithms and Combinatorics.
    Springer-Verlag, Berlin, fourth edition, 2008.
    Theory and algorithms. MR 2369759 (2008j:90002)
  • 5. L. Lovász and M. D. Plummer.
    Matching theory, volume 121 of North-Holland Mathematics Studies.
    North-Holland Publishing Co., Amsterdam, 1986.
    Annals of Discrete Mathematics, 29. MR 859549 (88b:90087)
  • 6. Peter Magyar.
    Borel-Weil theorem for configuration varieties and Schur modules.
    Adv. Math., 134(2):328-366, 1998. MR 1617793 (2000e:14087)
  • 7. M. H. Peel and G. D. James.
    Specht series for skew representations of symmetric groups.
    J. Algebra, 56(2):343-364, 1979. MR 528580 (80h:20021)
  • 8. Victor Reiner and Mark Shimozono.
    Specht series for column-convex diagrams.
    J. Algebra, 174(2):489-522, 1995. MR 1334221 (96m:20020)
  • 9. Victor Reiner and Mark Shimozono.
    Percentage-avoiding, northwest shapes and peelable tableaux.
    J. Combin. Theory Ser. A, 82(1):1-73, 1998. MR 1616579 (2000a:05220)
  • 10. Bruce E. Sagan.
    The symmetric group: representations, combinatorial algorithms, and symmetric functions, volume 203 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, second edition, 2001. MR 1824028 (2001m:05261)
  • 11. Richard P. Stanley.
    Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics.
    Cambridge University Press, Cambridge, 1999.
    With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 05E10

Retrieve articles in all journals with MSC (2010): 05E10

Additional Information

Ricky Ini Liu
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Received by editor(s): October 7, 2010
Received by editor(s) in revised form: November 29, 2010
Published electronically: October 4, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society