Abstract
Glider representations can be defined for a finite algebra filtration FKG determined by a chain of subgroups 1 ⊂ G1 ⊂… ⊂ Gd = G. In this paper we develop the generalized character theory for such glider representations. We give the generalization of Artin’s theorem and define a generalized inproduct. For finite abelian groups G with chain 1 ⊂ G, we explicitly calculate the generalized character ring and compute its semisimple quotient. The papers ends with a discussion of the quaternion group as a first non-abelian example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Caenepeel, F., Van Oystaeyen, F.: Clifford theory for glider representations. Algebr. Represent. Theory 19(6), 1477–1493 (2016)
Caenepeel, F., Van Oystaeyen, F.: Glider representations of group algebra filtrations of nilpotent groups. Algebr. Represent. Theory, https://doi.org/10.1007/s10468-017-9725-9
El Baroudy, M., Van Oystaeyen, F.: Fragments with finiteness condtions in particular over group rings. Comm. Algebra 28(1), 321–336 (2000)
James, G., Liebeck, M.: Representations and Characters of Groups, 2nd edn. Cambridge University Press, New York (2001). viii+ 458 pp.
Jespers, E., Leal, G., Paques, A.: Central idempotents in the rational group algebra of a finite nilpotent group. J. Algebra Appl. 2(1), 57–62 (2003)
Nawal, S., Van Oystaeyen, F.: An introduction of fragmented structures over filtered rings. Comm. Algebra 23, 975–993 (1995)
Okniński, J.: Semigroup Algebras. Marcel Dekker (1998)
Serre, J.-P.: Linear Representations of Finite Groups, Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, vol. 42. Springer, New York (1977). x + 170 pp.
Acknowledgements
The first author expresses his gratitude to Geoffrey Janssens and Eric Jespers for fruitful discussions on the calculation of the Jacobson radical and the primitive central idempotents of semigroup algebras appearing in the theory of glider representations.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Radha Kessar
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Frederik Caenepeel was partly Aspirant PhD Fellow of FWO and partly postdoctoral researcher at the Shanghai Center for Mathematical Sciences.
Rights and permissions
About this article
Cite this article
Caenepeel, F., Van Oystaeyen, F. Generalized Characters for Glider Representations of Groups. Algebr Represent Theor 23, 303–326 (2020). https://doi.org/10.1007/s10468-018-09850-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-018-09850-8