Abstract
Given a finite group \(G\) and a subgroup \(H\le G\), we develop a Fourier analysis for \(H\)-conjugacy invariant functions on \(G\), without the assumption that \(H\) is a multiplicity-free subgroup of \(G\). We also study the Fourier transform for functions in the center of the algebra of \(H\)-conjugacy invariant functions on \(G\). We show that a recent calculation of Cesi is indeed a Fourier transform of a function in the center of the algebra of functions on the symmetric group that are conjugacy invariant with respect to a Young subgroup.
Similar content being viewed by others
References
Brender, M.: Spherical functions on the symmetric groups. J. Algebra 42(2), 302–314 (1976)
Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Harmonic analysis on finite groups: representation theory, Gelfand pairs and Markov chains. Cambridge Studies in Advanced Mathematics, vol. 108. Cambridge University Press, Cambridge (2008)
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, vol. 121. Cambridge University Press, Cambridge (2010)
Cesi, F.: On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions. J. Algebraic Combin. 32(2), 155–185 (2010)
Greenhalgh, A.S.: Measures on groups with subgroups invariance properties, Technical report No. 321. Department of Statistics, Stanfors University (1989)
Karlof, J.: The subclass algebra associated with a finite group and subgroup. Trans. Am. Math. Soc. 207, 329–341 (1975)
Scarabotti, F.: Time to reach stationarity in the Bernoulli–Laplace diffusion model with many urns. Adv. Appl. Math. 18(3), 351–371 (1997)
Scarabotti, F.: The Stanley-Féray-Śniady formula for the generalized characters of the symmetric group. Colloq. Math. 124(2), 285–291 (2011)
Scarabotti, F., Tolli, F.: Harmonic analysis on a finite homogeneous space. Proc. Lond. Math. Soc.(3) 100(2), 348–376 (2010)
Scarabotti, F., Tolli, F.: Harmonic analysis on a finite homogeneous space II: the Gelfand–Tsetlin decomposition. Forum Mathematicum 22(5), 879–911 (2010)
Strahov, E.: Generalized characters of the symmetric group. Adv. Math. 212(1), 109–142 (2007)
Travis, D.: Spherical functions on finite groups. J. Algebra 29, 65–76 (1974)
Vershik, A.M., Okun’kov, A.Yu.: A new approach to representation theory of symmetric groups. II. J. Math. Sci. (NY) 131(2), 5471–5494 (2005)
Wigner, E.P.: Restriction of irreducible representations of groups to a subgroup. Proc. Roy. Soc. Lond. Ser. A 322(1549), 181–189 (1971)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by K. Gröchenig.
Rights and permissions
About this article
Cite this article
Scarabotti, F., Tolli, F. Fourier analysis of subgroup conjugacy invariant functions on finite groups. Monatsh Math 170, 465–479 (2013). https://doi.org/10.1007/s00605-012-0445-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-012-0445-2