Realizability of two-dimensional linear groups over rings of integers of algebraic number fields

Dmitry Malinin; Freddy Van Oystaeyen

doi:10.1007/s10468-009-9184-z

Realizability of two-dimensional linear groups over rings of integers of algebraic number fields

Dmitry Malinin, Freddy Van Oystaeyen

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

Given the ring of integers O_K of an algebraic number field K, for which natural numbers n there exists a finite group G ⊂ GL(n, O _K) such that O_KG, the O_K-span of G, coincides with M(n, O^K), the ring of (n×n)-matrices over O_K? The answer is known if n is an odd prime. In this paper we study the case n = 2; in the cases when the answer is positive for n = 2, for n = 2m there is also a finite group G ⊂ GL(2m, O_K) such that O_KG = M(2m, O_K).

Original language	English
Pages (from-to)	201-211
Number of pages	11
Journal	Algebras and Representation Theory
Volume	14
Issue number	2
DOIs	https://doi.org/10.1007/s10468-009-9184-z
Publication status	Published - Apr 2011

Keywords

Algebraic number fields
Brauer reduction
Globally irreducible representations
Rings of integers
Schur ring

ASJC Scopus subject areas

Mathematics(all)

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10.1007/s10468-009-9184-z

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title = "Realizability of two-dimensional linear groups over rings of integers of algebraic number fields",

abstract = "Given the ring of integers OK of an algebraic number field K, for which natural numbers n there exists a finite group G ⊂ GL(n, O K) such that OKG, the OK-span of G, coincides with M(n, OK), the ring of (n×n)-matrices over OK? The answer is known if n is an odd prime. In this paper we study the case n = 2; in the cases when the answer is positive for n = 2, for n = 2m there is also a finite group G ⊂ GL(2m, OK) such that OKG = M(2m, OK).",

keywords = "Algebraic number fields, Brauer reduction, Globally irreducible representations, Rings of integers, Schur ring",

author = "Dmitry Malinin and {Van Oystaeyen}, Freddy",

year = "2011",

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doi = "10.1007/s10468-009-9184-z",

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T1 - Realizability of two-dimensional linear groups over rings of integers of algebraic number fields

AU - Malinin, Dmitry

AU - Van Oystaeyen, Freddy

PY - 2011/4

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N2 - Given the ring of integers OK of an algebraic number field K, for which natural numbers n there exists a finite group G ⊂ GL(n, O K) such that OKG, the OK-span of G, coincides with M(n, OK), the ring of (n×n)-matrices over OK? The answer is known if n is an odd prime. In this paper we study the case n = 2; in the cases when the answer is positive for n = 2, for n = 2m there is also a finite group G ⊂ GL(2m, OK) such that OKG = M(2m, OK).

AB - Given the ring of integers OK of an algebraic number field K, for which natural numbers n there exists a finite group G ⊂ GL(n, O K) such that OKG, the OK-span of G, coincides with M(n, OK), the ring of (n×n)-matrices over OK? The answer is known if n is an odd prime. In this paper we study the case n = 2; in the cases when the answer is positive for n = 2, for n = 2m there is also a finite group G ⊂ GL(2m, OK) such that OKG = M(2m, OK).

KW - Algebraic number fields

KW - Brauer reduction

KW - Globally irreducible representations

KW - Rings of integers

KW - Schur ring

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Realizability of two-dimensional linear groups over rings of integers of algebraic number fields

Abstract

Keywords

ASJC Scopus subject areas

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