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The Equivariant Volumes of the Permutahedron

  • Federico ArdilaEmail author
  • Anna Schindler
  • Andrés R. Vindas-Meléndez
Article

Abstract

We prove that if \(\sigma \) is a permutation of \(S_n\) with m cycles of lengths \(l_1, \ldots , l_m\), the subset of the permutahedron \(\Pi _n\) fixed by the natural action of \(\sigma \) is a polytope with volume \(n^{m-2} \gcd (l_1, \ldots , l_m)\).

Notes

Acknowledgements

Some of the results of this paper are part of the Master’s theses of AS (under the supervision of FA) and ARVM (under the supervision of FA and Matthias Beck) at San Francisco State University [9, 15]. We are grateful to Anastasia Chavez, John Guo, Andrés Rodríguez, and Nicole Yamzon for their valuable feedback during our group research meetings, and the Mathematics Department at SFSU for providing a wonderful environment to produce this work. We are also thankful to the referees, whose suggestions helped us improve the presentation. In particular, one of the referees pointed out the connection with equivariant secondary polytopes. Part of this project was carried out while FA was a Simons Research Professor at the Mathematical Sciences Research Institute; he thanks the Simons Foundation and MSRI for their support. ARVM thanks Matthias Beck and Benjamin Braun for the support and fruitful conversations.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Federico Ardila
    • 1
    • 2
    Email author
  • Anna Schindler
    • 1
    • 3
  • Andrés R. Vindas-Meléndez
    • 1
    • 4
  1. 1.San Francisco State UniversitySan FranciscoUSA
  2. 2.Universidad de Los AndesBogotáColombia
  3. 3.North Seattle CollegeSeattleUSA
  4. 4.University of KentuckyLexingtonUSA

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