Advertisement

Specht Polytopes and Specht Matroids

  • John D. Wiltshire-Gordon
  • Alexander Woo
  • Magdalena ZajaczkowskaEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 80)

Abstract

The generators of the classical Specht module satisfy intricate relations. We introduce the Specht matroid, which keeps track of these relations, and the Specht polytope, which also keeps track of convexity relations. We establish basic facts about the Specht polytope: the symmetric group acts transitively on its vertices and irreducibly on its ambient real vector space. A similar construction builds a matroid and polytope for a tensor product of Specht modules, giving Kronecker matroids and Kronecker polytopes instead of the usual Kronecker coefficients. We call this process of upgrading from numbers to matroids and polytopes “matroidification”. In the course of describing these objects, we also give an elementary account of the construction of Specht modules. Finally, we provide code to compute with Specht matroids and their Chow rings.

MSC 2010 codes:

05E10 

Notes

Acknowledgements

This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. The authors wish to thank Bernd Sturmfels, Diane Maclagan, Gregory G. Smith for their leadership and encouragement, and all the participants of the Combinatorial Algebraic Geometry thematic program at the Fields Institute, at which this work was conceived. They also thank the Fields Institute and the Clay Mathematics Institute for hospitality and support. Finally, thanks to Bernd Sturmfels also for suggesting the term “matroidification” and to anonymous referees for their helpful suggestions.

References

  1. 1.
    Karim Adiprasito, June Huh, and Eric Katz: Hodge Theory for Combinatorial Geometries, arXiv:1511.02888 [math.CO].Google Scholar
  2. 2.
    Federico Ardila, Matthias Beck, Serkan Hoşten, Julian Pfeifle, and Kim Seashore: Root polytopes and growth series of root lattices, SIAM J. Discrete Math. 25 (2011) 360–378.Google Scholar
  3. 3.
    Corrado de Concini and Claudio Procesi: Wonderful models of subspace arrangements, Selecta Math. (N.S.) 1 (1995) 459–494.Google Scholar
  4. 4.
    Corrado de Concini and Claudio Procesi: Hyperplane arrangements and holonomy equations, Selecta Math. (N.S.) 1 (1995) 495–535.Google Scholar
  5. 5.
    René Birkner: Polyhedra,a package for computations with convex polyhedral objects, J. Softw. Algebra Geom. 1 (2009) 11–15.Google Scholar
  6. 6.
    Eva Maria Feichtner and Sergey Yuzvinsky: Chow rings of toric varieties defined by atomic lattices, Invent. Math. 155 (2004) 515–536.Google Scholar
  7. 7.
    J. Sutherland Frame, Gilbert de Beauregard Robinson, and Robert M. Thrall: The hook graphs of the symmetric groups Canadian J. Math. 6 (1954) 316–324.Google Scholar
  8. 8.
    Henri Garnir: Théorie de la représentation linéaire des groupes symétriques, Mémoires de la Société Royale des Sciences de Liège, Ser. 4, Vol. 10, 1950.Google Scholar
  9. 9.
    Israel Gelfand, Mark Graev, and Alexander Postnikov: Combinatorics of hypergeometric functions associated with positive roots, in The Arnold–Gelfand mathematical seminars, 205–221, Birkhäuser Boston, Boston, MA, 1997.Google Scholar
  10. 10.
    Gordon James and Adalbert Kerber: The representation theory of the symmetric group, Mathematics and its Applications 16, Addison-Wesley Publishing Co., Reading, MA, 1981.Google Scholar
  11. 11.
    Daniel R. Grayson and Michael E. Stillman: Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.
  12. 12.
    Leonid Monin and Julie Rana: Equations of \(\overline{M}_{0,n}\), in Combinatorial Algebraic Geometry,113–132, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar
  13. 13.
    Jean-Christophe Novelli, Igor Pak, and Alexander Stoyanovskii: A direct bijective proof of the hook-length formula, Discrete Math. Theor. Comput. Sci. 1 (1997) 53–67.Google Scholar
  14. 14.
    The On-Line Encyclopedia of Integer Sequences, published electronically at oeis.org, 2016.
  15. 15.
    James Oxley: Matroid Theory, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1992.Google Scholar
  16. 16.
    Bruce E. Sagan: The symmetric group. Representations, combinatorial algorithms, and symmetric functions, Second edition, Graduate Texts in Mathematics 203, Springer-Verlag, New York, 2001.Google Scholar
  17. 17.
    The Sage Developers: SageMath, the Sage Mathematics Software System (Version 7.3), 2016, available at www.sagemath.org.
  18. 18.
    Wilhelm Specht: Die irreduziblen Darstellungen der symmetrischen Gruppe, Math. Z. 39 (1935) 696–711.Google Scholar
  19. 19.
    Bernd Sturmfels: Fitness, apprenticeship, and polynomials, in Combinatorial Algebraic Geometry,1–19, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar
  20. 20.
    Alfred Young: The Collected Papers of Alfred Young, University of Toronto Press, 1977. Representations are described in the eight articles titled On Quantitative Substitutional Analysis, published in Proc. London Math. Soc. We refer more specially to QS1, 33 (1900) 97–146; QS3, 28 (1928) 255–292; QS4, 31 (1930) 253–272.Google Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • John D. Wiltshire-Gordon
    • 1
  • Alexander Woo
    • 2
  • Magdalena Zajaczkowska
    • 3
    Email author
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.Department of MathematicsUniversity of IdahoMoscowUSA
  3. 3.Mathematics Institute, University of WarwickCoventryUK

Personalised recommendations