Abstract
The Burnside ring \(\mathcal {B}(G)\) of a finite group G, a classical tool in the representation theory of finite groups, is studied from the point of view of computational algebra. In the first part (cf. Kreuzer and Patil, Beitr Algebra Geom 58:427–452, 2017) we examined the ring theoretic properties of \(\mathcal {B}(G)\) using the methods of computer algebra. In this part we shift our focus to important maps between two Burnside rings and make several well-known maps of representation theory explicitly computable. The inputs of all our algorithms are the tables of marks of the two groups, and the outputs are matrices of integers representing the maps via their image in the ghost ring. All algorithms have been implemented in the computer algebra system ApCoCoA and are illustrated by applying them to explicit examples. Especially, we study the restriction and induction maps, the projection and inflation maps, the conjugation isomorphism, and the Frobenius-Wielandt homomorphism.
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Acknowledgements
The first author thanks the Indian Institute of Science, Bangalore, and in particular project MATRICS DSTO-1983 of the second author for the hospitality and support he enjoyed during the writing of this paper. The second author thanks the Fakultät für Informatik und Mathematik, Universität Passau, Germany for the warm hospitality and partial support during his visit in May 2019. The authors are indebted to the referee for his very careful reading of the paper and helpful suggestions.
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Kreuzer, M., Patil, D.P. Computational aspects of Burnside rings part II: important maps. Beitr Algebra Geom 62, 475–494 (2021). https://doi.org/10.1007/s13366-020-00520-9
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DOI: https://doi.org/10.1007/s13366-020-00520-9
Keywords
- Burnside ring
- Table of marks
- Restriction map
- Induction map
- Projection map
- Inflation map
- Frobenius-Wielandt homomorphism