Monatshefte für Mathematik

, Volume 181, Issue 4, pp 937–965 | Cite as

Induced representations and harmonic analysis on finite groups

Article
First Online:
Received:
Accepted:
  • 107 Downloads

Abstract

The aim of the present paper is to develop a theory of spherical functions for noncommutative Hecke algebras on finite groups. Let G be a finite group, K a subgroup and \((\theta ,V)\) an irreducible, unitary K-representation. After a careful analysis of Frobenius reciprocity, we are able to introduce an orthogonal basis in the commutant of \(\text {Ind}_K^GV\), and an associated Fourier transform. Then we translate our results in the corresponding Hecke algebra, an isomorphic algebra in the group algebra of G. Again a complete Fourier analysis is developed. As particular cases, we obtain some classical results of Curtis and Fossum on the irreducible characters. Finally, we develop a theory of Gelfand–Tsetlin bases for Hecke algebras.

Keywords

Induced representation Frobenius reciprocity Fourier transform Hecke algebra Spherical function Gelfand–Tsetlin basis 

Mathematics Subject Classification

Primary 20C15l Secondary 20C08 43A90 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Bump, D.: Lie groups. Graduate Texts in Mathematics, p. 225. Springer-Verlag, New York (2004)Google Scholar
  2. 2.
    Ceccherini-Silberstein, T., Machì, A., Scarabotti, F., Tolli, F.: Induced representations and Mackey theory, functional analysis. J. Math. Sci. (N. Y.) 156(1), 11–28 (2009)Google Scholar
  3. 3.
    Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Trees, wreath products and finite Gelfand pairs. Adv. Math. 206(2), 503–537 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Harmonic analysis on finite groups: representation theory, Gelfand pairs and Markov chains. Cambridge Studies in Advanced Mathematics, p. 108. Cambridge University Press (2008)Google Scholar
  5. 5.
    Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Clifford theory and applications, functional analysis. J. Math. Sci. (N. Y.) 156(1), 29–43 (2009)Google Scholar
  6. 6.
    Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Representation theory of wreath products of finite groups, functional analysis. J. Math. Sci. (N. Y.) 156(1), 44–55 (2009)Google Scholar
  7. 7.
    Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Representation theory of the symmetric groups: the Okounkov-Vershik approach, character formulas, and partition algebras. Cambridge Studies in Advanced Mathematics, p. 121. Cambridge University Press (2010)Google Scholar
  8. 8.
    Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Representation theory and harmonic analysis of wreath products of finite groups. London Mathematical Society Lecture Note Series, p. 410. Cambridge University Press (2014)Google Scholar
  9. 9.
    Curtis, C.W., Fossum, T.V.: On centralizer rings and characters of representations of finite groups. Math. Z. 107, 402–406 (1968)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Curtis, Ch.W., Reiner, I.: Methods of Representation Theory. With Applications to Finite Groups and Orders. vol. I, Pure Appl. Math. Wiley, New York (1981)Google Scholar
  11. 11.
    D’Angeli, D., Donno, A.: Crested products of Markov chains. Ann. Appl. Probab. 19(1), 414–453 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    D’Angeli, D., Donno, A.: Markov chains on orthogonal block structures. Eur. J. Combin. 31(1), 34–46 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    D’Angeli, D., Donno, A.: Generalized crested products. Eur. J. Combin. 32(2), 243–257 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Diaconis, P.: Groups Representations in Probability and Statistics. IMS Hayward, CA (1988)MATHGoogle Scholar
  15. 15.
    Howlett, R., Lehrer, G.: Induced cuspidal representations and generalised Hecke rings. Invent. Math. 58(1), 37–74 (1980)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mizukawa, H.: Twisted Gelfand pairs of complex reflection groups and \( r\)-congruence properties of Schur functions. Ann. Comb. 15(1), 109–125 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nebe, G.: Orthogonal Frobenius reciprocity. J. Algebra 225(1), 250–260 (2000)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Okounkov, A., Vershik, A.M.: A new approach to the representation theory of symmetric groups. II (Russian), Zap.Nauchn. Sem. S.Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 307 (2004), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 10, 57–98, 281; translation in J. Math. Sci. (N.Y.) 131(2), 5471–5494 (2005)Google Scholar
  19. 19.
    Piatetski-Shapiro, I.: Complex Representations of GL(2,K) for Finite Fields K. Contemporary Mathematics, 16. American Mathematical Society, Providence, R.I. (1983)Google Scholar
  20. 20.
    Scarabotti, F.: Time to reach stationarity in the Bernoulli-Laplace diffusion model with many urns. Adv. Appl. Math. 18(3), 351–371 (1997)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Scarabotti, F., Tolli, F.: Harmonic analysis on a finite homogeneous space. Proc. Lond. Math. Soc. (3) 100(2), 348–376 (2010)Google Scholar
  22. 22.
    Scarabotti, F., Tolli, F.: Harmonic analysis on a finite homogeneous space II: the Gelfand Tsetlin decomposition. Forum Math. 22, 897–911 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Scarabotti, F., Tolli, F.: Fourier analysis of subgroup-conjugacy invariant functions on finite groups. Monatsh. Math. 170(3–4), 465–479 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Scarabotti, F., Tolli, F.: Hecke algebras and harmonic analysis on finite groups. Rend. Mat. Appl. VII 33, 27–52 (2013)MathSciNetMATHGoogle Scholar
  25. 25.
    Stembridge, J.R.: On Schur’s Q-functions and the primitive idempotents of a commutative Hecke algebra. J. Algebr. Comb. 1, 71–95 (1992)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Terras, A.: Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts, p. 43. Cambridge University Press, Cambridge (1999)Google Scholar

Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.Dipartimento SBAIUniversità degli Studi di Roma “La Sapienza”RomeItaly
  2. 2.Dipartimento di Matematica e FisicaUniversità Roma TRERomeItaly

Personalised recommendations