Journal of Algebraic Combinatorics

, Volume 48, Issue 2, pp 179–225 | Cite as

The conjugacy action of \(S_n\) and modules induced from centralisers

  • Sheila SundaramEmail author


We establish, for the character table of the symmetric group, the positivity of the row sums indexed by irreducible characters, when restricted to various subsets of the conjugacy classes. A notable example is that of partitions with all parts odd. More generally, we study representations related to the conjugacy action of the symmetric group. These arise as sums of submodules induced from centraliser subgroups, and their Frobenius characteristics have elegant descriptions, often as a multiplicity-free sum of power-sum symmetric functions. We describe a general framework in which such representations, and consequently such linear combinations of power sums, can be analysed. The conjugacy action for the symmetric group, and more generally for a large class of groups, is known to contain every irreducible. We find other representations of dimension n! with this property, including a twisted analogue of the conjugacy action.


Conjugacy action Character tables Symmetric power Exterior power Plethysm Ramanujan sum 

Mathematics Subject Classification

20C05 20C15 20C30 05E18 06A07 


  1. 1.
    Cadogan, C.: The Möbius function and connected graphs. J. Comb. Theory B 11(3), 193–200 (1971)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ceccherini-Silberstein, T., Scarabotti, T., Tolli, F.: Clifford theory and applications, functional analysis. J. Math. Sci. N.Y. 156(1), 29–43 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Frumkin, A.: Theorem about the conjugacy representation of \(S_n\). Israel J. Math. 55, 121–128 (1986)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Foulkes, H.O.: Characters of symmetric groups induced by characters of cyclic subgroups. In: Welsh, D.J.A., Woodall, D.R. (eds.) Proceedings Conference on Combinatorial Mathematics, Oxford, The Institute for Mathematics and its Applications, Southend-on-Sea, Essex, pp. 141–154 (1972)Google Scholar
  5. 5.
    Frame, J.S.: On the reduction of the conjugating representation of a finite group. Bull. Am. Math. Soc. 53, 584–589 (1947)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fulton, W., Harris, J.: Representation Theory, A First Course, Graduate Texts in Mathematics, vol. 129. Springer, Berlin (1991)zbMATHGoogle Scholar
  7. 7.
    Hölder, O.: Zur Theorie der Kreisteilungsgleichung \(K_m(x)=0,\). Prace Matematyczno Fizyczne 43, 13–23 (1936)zbMATHGoogle Scholar
  8. 8.
    Hanlon, P.: The fixed-point partition lattices. Pac. J. Math. 96, 319–341 (1981)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hersh, P.L., Reiner, V.: Representation stability for cohomology of configuration spaces in \({ R}^d\). Int. Math. Res. Not. IMRN 2017(5), 1433–1486 (2017)MathSciNetGoogle Scholar
  10. 10.
    Heide, G., Saxl, J., Tiep, P.H., Zalesski, A.E.: Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type. Proc. Lond. Math. Soc. (3) 106(4), 908–930 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford Science Publications, Oxford University Press, Oxford (1979)zbMATHGoogle Scholar
  12. 12.
    James, G.D., Kerber, A.: The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications. Addison-Wesley, Boston (1981)Google Scholar
  13. 13.
    Kráskiewicz, W., Weyman, J.: Algebra of coinvariants and the action of a Coxeter element. Bayreuth Math. Schr. 63, 265–284 (2001)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  15. 15.
    Nagura, J.: On the interval containing at least one prime number. Proc. Japan Acad. 28(4), 177–181 (1952). doi: 10.3792/pja/1195570997 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    ShreevatsaR. Primes between \(n\) and \(2n\) (version: 2013-08-13).
  17. 17.
    Reutenauer, C.: Free Lie Algebras. London Mathematical Society Monographs. Oxford Science Publications, Oxford (1993)Google Scholar
  18. 18.
    Rosen, K.H.: Elementary Number Theory and its Applications, 4th edn. Addison-Wesley, Boston (1999)Google Scholar
  19. 19.
    Scharf, T.: Ein weiterer Beweis, daß die konjugierende Darstellung der symmetrischen Gruppe jede irreduzible Darstellung enthält. Arch. Math. 54, 427–429 (1990)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Simon, B.: Representations of Finite and Compact Groups, Graduate Studies in Mathematics, vol. 10. American Mathematical Society, Providence (1996)Google Scholar
  21. 21.
    Solomon, L.: On the sum of the elements in the character table of a finite group. Proc. Am. Math. Soc. 12(6), 962–963 (1961)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Stanley, R.: Some aspects of groups acting on finite posets. J. Comb. Theory (A) 32(2), 132–161 (1982)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Stanley, R.: Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics 62, vol. 2. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  24. 24.
    Sundaram, S.: The homology representations of the symmetric group on Cohen–Macaulay subposets of the partition lattice. Adv. Math. 104(2), 225–296 (1994)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Swanson, J.: On the existence of tableaux with given modular major index. Algebraic Comb. (to appear). arXiv:1701.04963
  26. 26.
    von Sterneck, R.D.: Ein Analogon zur additiven Zahlentheorie. Sitzber. Akad. Wiss. Wien Math-Naturw. Klasse 111(Abt. IIa), 1567–1601 (1902)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Pierrepont SchoolWestportUSA

Personalised recommendations