Foundations of Computational Mathematics

, Volume 16, Issue 5, pp 1241–1261 | Cite as

Plethysm and Lattice Point Counting

  • Thomas KahleEmail author
  • Mateusz Michałek


We apply lattice point counting methods to compute the multiplicities in the plethysm of \(\textit{GL}(n)\). Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition \(\mu \) of 3, 4, or 5, we obtain an explicit formula in \(\lambda \) and k for the multiplicity of \(S^\lambda \) in \(S^\mu (S^k)\).


Plethysm Ehrhart function Quasi-polynomial Lattice point counting 

Mathematics Subject Classification

Primary: 20G05 11P21 Secondary: 11H06 05A16 52B20 52B55 20C15 



The authors would like to thank Sven Verdoolaege for his prompt responses to issues raised on the isl -mailing list. The second author would like to thank Laurent Manivel for introducing him to the subject of plethysm. After the first posting of this paper on the arXiv Matthias Christandl, Laurent Manivel, and Michèle Vergne provided very insightful comments on how to apply Meinrenken-Sjamaar theory to the plethysm. We would like to thank them also for their suggestions on how to improve the paper. This project started, while Michałek was an Oberwolfach Leibniz fellow and invited Kahle for work at MFO. The project was finished at Freie Universität Berlin during Michałek’s DAAD PRIME fellowship.


  1. 1.
    Abdelmalek Abdesselam and Jaydeep Chipalkatti, Brill–Gordan loci, transvectants and an analogue of the Foulkes conjecture, Advances in Mathematics 208 (2007), no. 2, 491–520.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Yoshio Agaoka, Decomposition formulas of the plethysm \(\{m\}\otimes \{\mu \}\), Hiroshima University, 2002.Google Scholar
  3. 3.
    M Welleda Baldoni, Matthias Beck, Charles Cochet, and Michèle Vergne, Volume computation for polytopes and partition functions for classical root systems, Discrete & Computational Geometry 35 (2006), no. 4, 551–595.Google Scholar
  4. 4.
    Michel L Balinski and Fred J Rispoli, Signature classes of transportation polytopes, Mathematical programming 60 (1993), no. 1-3, 127–144.Google Scholar
  5. 5.
    Leonid Bedratyuk, Analogue of the Cayley–Sylvester formula and the Poincaré series for the algebra of invariants of n-ary form, Linear and Multilinear Algebra 59 (2011), no. 11, 1189–1199.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Arkady D Berenstein and Andrei V Zelevinsky, Triple multiplicities for \(sl(r+1)\) and the spectrum of the exterior algebra of the adjoint representation, Journal of Algebraic Combinatorics 1 (1992), no. 1, 7–22.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ethan D Bolker, Transportation polytopes, Journal of Combinatorial Theory, Series B 13 (1972), no. 3, 251–262.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Emmanuel Briand, Rosa Orellana, and Mercedes Rosas, Quasipolynomial formulas for the Kronecker coefficients indexed by two two–row shapes, DMTCS Proceedings (2009), no. 01, 241–252.Google Scholar
  9. 9.
    Emmanuel Briand, Rosa Orellana, and Mercedes Rosas, Reduced Kronecker coefficients and counter–examples to Mulmuley’s strong saturation conjecture SH, Computational Complexity 18 (2009), no. 4, 577–600.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Emmanuel Briand, Rosa Orellana, and Mercedes Rosas, The stability of the Kronecker product of Schur functions, Journal of Algebra 331 (2011), no. 1, 11–27.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Michel Brion, Stable properties of plethysm: on two conjectures of Foulkes, Manuscripta Mathematica 80 (1993), no. 1, 347–371.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Michel Brion and Michèle Vergne, Residue formulae, vector partition functions and lattice points in rational polytopes, Journal of the American Mathematical Society 10 (1997), no. 4, 797–833.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Winfried Bruns, Aldo Conca, and Matteo Varbaro, Relations between the minors of a generic matrix, Advances in Mathematics 244 (2013), 171–206.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Peter Bürgisser, Matthias Christandl, and Christian Ikenmeyer, Even partitions in plethysms, J. Algebra 328 (2011), 322–329.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Peter Bürgisser, Joseph M Landsberg, Laurent Manivel, and Jerzy Weyman, An overview of mathematical issues arising in the geometric complexity theory approach to \(P\ne \) NP, SIAM Journal on Computing 40 (2011), no. 4, 1179–1209.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Christophe Carré, Plethysm of elementary functions, Bayreuther Mathematische Schriften (1990), no. 31, 1–18.Google Scholar
  17. 17.
    Christophe Carre and Jean-Yves Thibon, Plethysm and vertex operators, Advances in Applied Mathematics 13 (1992), no. 4, 390–403.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Y. M. Chen, A. M. Garsia, and J. Remmel, Algorithms for plethysm, Combinatorics and algebra (Boulder, Colo., 1983), Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 109–153.Google Scholar
  19. 19.
    Matthias Christandl, Brent Doran, Stavros Kousidis, and Michael Walter, Eigenvalue distributions of reduced density matrices, Communications in Mathematical Physics 332 (2014), no. 1, 1–52.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Matthias Christandl, Brent Doran, and Michael Walter, Computing multiplicities of lie group representations, Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on, IEEE, 2012, pp. 639–648.Google Scholar
  21. 21.
    Jesús A De Loera and Edward D Kim, Combinatorics and geometry on transportation polytopes: An update, Discrete Geometry and Algebraic Combinatorics 625 (2014), 37.Google Scholar
  22. 22.
    Suzie C. Dent and Johannes Siemons, On a conjecture of Foulkes, Journal of Algebra 226 (2000), no. 1, 236 – 249.Google Scholar
  23. 23.
    D. G. Duncan, On D. E. Littlewood’s algebra of \(S\) -functions, Canadian J. Math. 4 (1952), 504–512.Google Scholar
  24. 24.
    H. O. Foulkes, Plethysm of \(S\) -functions, Philos. Trans. Roy. Soc. London. Ser. A. 246 (1954), 555–591.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Mihai Fulger and Xin Zhou, Schur asymptotics of Veronese syzygies, Mathematische Annalen 362 (2015), no. 1–2, 529–540.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    William Fulton and Joe Harris, Representation theory. a first course, Springer, Berlin, 1991.Google Scholar
  27. 27.
    Roger Howe, \((GL_n,GL_m)\) -duality and symmetric plethysm, Proc. Indian Acad. Sci. Math. Sci. 97 (1987), no. 1-3, 85–109.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Thomas Kahle and Mateusz Michalek, Obstructions to combinatorial formulas for plethysm, preprint, arXiv:1507.07131 (2015).
  29. 29.
    Ronald King and Trevor Welsh, Some remarks on characters of symmetric groups, Schur functions, Littlewood-Richardson and Kronecker coefficients, work in progress
  30. 30.
    Victor Klee and Christoph Witzgall, Facets and vertices of transportation polytopes, Mathematics of the decision sciences 3 (1968), 257–282.MathSciNetGoogle Scholar
  31. 31.
    Alexander A Klyachko, Stable bundles, representation theory and hermitian operators, Selecta Mathematica, New Series 4 (1998), no. 3, 419–445.Google Scholar
  32. 32.
    Allen Knutson and Terence Tao, The honeycomb model of \(GL_n({\mathbb{C}})\) tensor products I: Proof of the saturation conjecture, Journal of the American Mathematical Society 12 (1999), no. 4, 1055–1090.Google Scholar
  33. 33.
    Allen Knutson, Terence Tao, and Christopher Woodward, The honeycomb model of \(GL_n({\mathbb{C}})\) tensor products II: Puzzles determine facets of the littlewood-richardson cone, Journal of the American Mathematical Society 17 (2004), no. 1, 19–48.Google Scholar
  34. 34.
    Joseph M Landsberg, Geometric complexity theory: an introduction for geometers, Annali dell’universita’di Ferrara 61 (2015), no. 1, 65–117.Google Scholar
  35. 35.
    Dudley E. Littlewood, Polynomial concomitants and invariant matrices, Journal of the London Mathematical Society 1 (1936), no. 1, 49–55.Google Scholar
  36. 36.
    Fu Liu, Perturbation of transportation polytopes, Journal of Combinatorial Theory, Series A 120 (2013), no. 7, 1539–1561.MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Nicholas A Loehr and Jeffrey B Remmel, A computational and combinatorial exposé of plethystic calculus, Journal of Algebraic Combinatorics 33 (2011), no. 2, 163–198.MathSciNetCrossRefGoogle Scholar
  38. 38.
    Ian Grant Macdonald, Symmetric functions and hall polynomials,Oxford university press, 1998.Google Scholar
  39. 39.
    Laurent Manivel, Gaussian maps and plethysm, Lecture Notes in Pure and Applied Mathematics (1998), 91–118.Google Scholar
  40. 40.
    Laurent Manivel, On the asymptotics of Kronecker coefficients, preprint, arXiv:1411.3498 (2014).
  41. 41.
    Laurent Manivel and Mateusz Michałek, Effective constructions in plethysms and Weintraub’s conjecture, Algebras and Representation Theory (2012), 1–11.Google Scholar
  42. 42.
    Laurent Manivel and Mateusz Michałek, Secants of minuscule and cominuscule minimal orbits, Linear Algebra and its Applications 481 (2015), 288–312.MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Tom McKay, On plethysm conjectures of Stanley and Foulkes, Journal of Algebra 319 (2008), no. 5, 2050 – 2071.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Eckhard Meinrenken, On Riemann-Roch formulas for multiplicities, Journal of the American Mathematical Society 9 (1996), 373–390.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Eckhard Meinrenken and Reyer Sjamaar, Singular reduction and quantization, Topology 38 (1999), no. 4, 699–762.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Ketan D Mulmuley and Milind Sohoni, Geometric complexity theory I: An approach to the P vs. NP and related problems, SIAM Journal on Computing 31 (2001), no. 2, 496–526.Google Scholar
  47. 47.
    S. P. O. Plunkett, On the plethysm of \(S\) -functions, Canad. J. Math. 24 (1972), 541–552.MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Claudiu Raicu, Secant varieties of Segre-Veronese varieties, Algebra and Number Theory 6 (2012), no. 8, 1817–1868.MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    David B Rush, Cyclic sieving and plethysm coefficients, preprint, arXiv:1408.6484 (2014).
  50. 50.
    Reyer Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Annals of Mathematics (1995), 87–129.Google Scholar
  51. 51.
    Richard P. Stanley, Positivity problems and conjectures in algebraic combinatorics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295–319.Google Scholar
  52. 52.
    R. M. Thrall, On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math. 64 (1942), 371–388.MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Marc A. A. van Leeuwen, Arjeh M. Cohen, and Bert Lisser, LiE: A package for lie group computations, CAN (Computer Algebra Nederland), 1992.Google Scholar
  54. 54.
    Sven Verdoolaege, isl: An integer set library for the polyhedral model, Mathematical Software—ICMS 2010 (Komei Fukuda, Joris Hoeven, Michael Joswig, and Nobuki Takayama, eds.), Lecture Notes in Computer Science, vol. 6327, Springer, 2010, pp. 299–302.Google Scholar
  55. 55.
    Sven Verdoolaege, Rachid Seghir, Kristof Beyls, Vincent Loechner, and Maurice Bruynooghe, Counting integer points in parametric polytopes using Barvinok’s rational functions, Algorithmica 48 (2007), no. 1, 37–66.MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Steven H. Weintraub, Some observations on plethysms, Journal of Algebra 129 (1990), no. 1, 103–114.MathSciNetGoogle Scholar
  57. 57.
    Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
  58. 58.
    Mei Yang, An algorithm for computing plethysm coefficients, Discrete Mathematics 180 (1998), no. 1, 391–402.MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Mei Yang, The first term in the expansion of plethysm of Schur functions, Discrete Mathematics 246 (2002), no. 1, 331–341.MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993, Translated from the Russian manuscript by the author.Google Scholar

Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Fakultät für MathematikOtto-von-Guericke UniversitätMagdeburgGermany
  2. 2.Polish Academy of SciencesWarsawPoland
  3. 3.Simons Institute for the Theory of ComputingUC BerkeleyBerkeleyUSA

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