Annals of Combinatorics

, Volume 22, Issue 3, pp 543–562 | Cite as

From Partition Identities to a Combinatorial Approach to Explicit Satake Inversion

  • Heekyoung HahnEmail author
  • JiSun Huh
  • EunSung Lim
  • Jaebum Sohn


In this paper, we provide combinatorial proofs for certain partition identities which arise naturally in the context of Langlands’ beyond endoscopy proposal. These partition identities motivate an explicit plethysm expansion of \({{\rm Sym}^j}\) \({{{\rm Sym}^{k}}V}\) for \({{\rm GL}_2}\) in the case k = 3. We compute the plethysm explicitly for the cases k = 3, 4. Moreover, we use these expansions to explicitly compute the basic function attached to the symmetric power L-function of \({{\rm GL}_2}\) for these two cases.

Mathematics Subject Classification

11P84 11S40 05E05 05E10 


partition identities multiplicities in the plethysm expansion explicit Satake inversions 


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  1. 1.
    Andrews G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  2. 2.
    Bressoud D.M.: Unimodality of Gaussian polynomials. Discrete Math. 99(1-3), 17–24 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kronholm B., Larsen A.: Symmetry and prime divisibility properties of partitions of n into exactly m parts. Ann. Combin. 19(4), 735–747 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bump, D.: Automorphic Forms and Representations. Cambridge Stud. Adv. Math., Vol. 55. Cambridge University Press, Cambridge (1998)Google Scholar
  5. 5.
    Casselman B.: Symmetric powers and the Satake transform. Bull. Iranian Math. Soc. 43(4), 17–54 (2017)MathSciNetGoogle Scholar
  6. 6.
  7. 7.
    Dolgachev, I.: Lectures on Invariant Theory. London Math. Soc. Lecture Note Ser., Vol. 296. Cambridge University Press, Cambridge (2003)Google Scholar
  8. 8.
    Getz J.R.: Nonabelian Fourier transforms for spherical representations. Pacific J. Math. 294(2), 351–373 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guerreiro, J.: An explicit inversion formula for the p-adic Whittaker transform on \({GL_n(\mathbb{Q}_p)}\). Submitted. arXiv:1702.08271v1 (2017)
  10. 10.
    Hahn H.: On tensor third L-functions of automorphic representations of \({GL_n(\mathbb{A}_F)}\). Proc. Amer. Math. Soc. 144(12), 5061–5069 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hahn, H.: On classical groups detected by the triple tensor products and the Littlewood- Richardson semigroup. Res. Number Theory 2, Art. 19 (2016)Google Scholar
  12. 12.
    Langlands, R.P.: Beyond endoscopy. In: Hida, H., Ramakrishnan, D., Shahidi, F. (eds.) Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 611–697. Johns Hopkins University Press, Baltimore, MD (2004)Google Scholar
  13. 13.
    Li, W.-W.: Basic functions and unramified local L-factors for split groups. Sci. China Math. (2016) 60(5), 777–812 (2017)Google Scholar
  14. 14.
    Rademacher H.: Lectures on Elementary Number Theory. Robert E. Krieger Publishing Co., Huntingdon, NY (1977)zbMATHGoogle Scholar
  15. 15.
    Sakellaridis, Y.: Inverse Satake transforms. Proceedings of the Simons Symposium on “Geometric Aspects of the Trace Formula” (to appear)Google Scholar
  16. 16.
    Stanley R.P.: Enumerative Combinatorics, Vol. I.. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Heekyoung Hahn
    • 1
    Email author
  • JiSun Huh
    • 2
  • EunSung Lim
    • 3
  • Jaebum Sohn
    • 3
  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Department of MathematicsAjou UniversitySuwonRepublic of Korea
  3. 3.Department of MathematicsYonsei UniversitySeoulRepublic of Korea

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