Complex group algebras of the double covers of the symmetric and alternating group

Christine Bessenrodt; Hung Ngoc Nguyen; Jørn Børling Olsson; Hung Tong-Viet

doi:10.2140/ant.2015.9.601

Complex group algebras of the double covers of the symmetric and alternating group

Christine Bessenrodt, Hung Ngoc Nguyen, Jørn Børling Olsson, Hung Tong-Viet

Department of Mathematical Sciences

13 Citations (Scopus)

Abstract

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let n ≥ 5 be an integer, G a finite group, and let Â_n and Ŝ^±_ndenote the double covers of A_n and Ŝ_n, respectively. We prove that (formula presented) if and only if (formula presented), and (formula presented) if and only if (formula presented). This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasisimple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.

Translated title of the contribution	Komplekse gruppealgebraer for de dobblete overdækningsgrupper af de symmetriske og alternerende grupper
Original language	English
Journal	Algebra & Number Theory
Volume	9
Issue number	3
Pages (from-to)	601-628
Number of pages	28
ISSN	1937-0652
DOIs	https://doi.org/10.2140/ant.2015.9.601
Publication status	Published - 11 Jul 2015

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title = "Complex group algebras of the double covers of the symmetric and alternating group",

abstract = "We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let n ≥ 5 be an integer, G a finite group, and let {\^A}n and Ŝ±ndenote the double covers of An and Ŝn, respectively. We prove that (formula presented) if and only if (formula presented), and (formula presented) if and only if (formula presented). This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasisimple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.",

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T1 - Complex group algebras of the double covers of the symmetric and alternating group

AU - Bessenrodt, Christine

AU - Nguyen, Hung Ngoc

AU - Olsson, Jørn Børling

AU - Tong-Viet, Hung

PY - 2015/7/11

Y1 - 2015/7/11

N2 - We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let n ≥ 5 be an integer, G a finite group, and let Ân and Ŝ±ndenote the double covers of An and Ŝn, respectively. We prove that (formula presented) if and only if (formula presented), and (formula presented) if and only if (formula presented). This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasisimple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.

AB - We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let n ≥ 5 be an integer, G a finite group, and let Ân and Ŝ±ndenote the double covers of An and Ŝn, respectively. We prove that (formula presented) if and only if (formula presented), and (formula presented) if and only if (formula presented). This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasisimple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.

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DO - 10.2140/ant.2015.9.601

M3 - Journal article

SN - 1937-0652

VL - 9

SP - 601

EP - 628

JO - Algebra and Number Theory

JF - Algebra and Number Theory

IS - 3

ER -

Complex group algebras of the double covers of the symmetric and alternating group

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