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- W2783626036 abstract "Since the early days of representation theory of finite groups in the 19th century, it was known that complex linear representations of finite groups live over number fields, that is, over finite extensions of the field of rational numbers.While the related question of integrality of representations was answered negatively by the work of Cliff, Ritter and Weiss as well as by Serre and Feit, it was not known how to decide integrality of a given representation.In this thesis we show that there exists an algorithm that given a representation of a finite group over a number field decides whether this representation can be made integral.Moreover, we provide theoretical and numerical evidence for a conjecture, which predicts the existence of splitting fields of irreducible characters with integrality properties.In the first part, we describe two algorithms for the pseudo-Hermite normal form, which is crucial when handling modules over ring of integers.Using a newly developed computational model for ideal and element arithmetic in number fields, we show that our pseudo-Hermite normal form algorithms have polynomial running time.Furthermore, we address a range of algorithmic questions related to orders and lattices over Dedekind domains, including computation of genera, testing local isomorphism, computation of various homomorphism rings and computation of Solomon zeta functions.In the second part we turn to the integrality of representations of finite groups and show that an important ingredient is a thorough understanding of the reduction of lattices at almost all prime ideals.By employing class field theory and tools from representation theory we solve this problem and eventually describe an algorithm for testing integrality.After running the algorithm on a large set of examples we are led to a conjecture on the existence of integral and nonintegral splitting fields of characters.By extending techniques of Serre we prove the conjecture for characters with rational character field and Schur index two." @default.
- W2783626036 created "2018-01-26" @default.
- W2783626036 creator A5000020334 @default.
- W2783626036 date "2016-01-01" @default.
- W2783626036 modified "2023-09-27" @default.
- W2783626036 title "Integrality of representations of finite groups" @default.
- W2783626036 hasPublicationYear "2016" @default.
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