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Representations of finite groups

1990 ChoiceReviews  
Introduction A representation (π, V ) of G on a finite-dimensional complex vector space V is a homomorphism π from the group G to the group GL(V ) of invertible complex linear maps from V to itself.  ...  Because any finite abelian group is a product of cyclic groups, the group G * is always isomorphic to G, but not canonically.  ...  Let G be the 'Heisenberg' group in SL 3 (Z/p) of all unipotent matrices The center Z of this group is made up of the matrices with x = y = 0, hence isomorphic to Z/p.  ... 
doi:10.5860/choice.27-3345 fatcat:f4kyra5h5rdsvm3flrwitfbhh4

Representations of finite groups [article]

Ruslan Sharipov
2006 arXiv   pre-print
This book is an introduction to a fast developing branch of mathematics - the theory of representations of groups. It presents classical results of this theory concerning finite groups.  ...  of a finite group G.  ...  As for the finite-dimensional representations of finite groups, we have proved their equivalence to some unitary representations. § 3. Characters of group representations.  ... 
arXiv:math/0612104v1 fatcat:pbzdulunsnfgbbusahgkbezfxm

Representations of finite groups [chapter]

2010 Spaces of Constant Curvature  
Introduction A representation (π, V ) of G on a finite-dimensional complex vector space V is a homomorphism π from the group G to the group GL(V ) of invertible complex linear maps from V to itself.  ...  Because any finite abelian group is a product of cyclic groups, the group G * is always isomorphic to G, but not canonically.  ...  Let G be the 'Heisenberg' group in SL 3 (Z/p) of all unipotent matrices The center Z of this group is made up of the matrices with x = y = 0, hence isomorphic to Z/p.  ... 
doi:10.1090/chel/372/04 fatcat:qp5qddceo5cydfl3nksqv3t7ha

Representations of Finite Groups [chapter]

2019 Group Theory for Physicists  
Typical structures amenable to representation theory are groups, associative algebras and Lie algebras. In this thesis we study linear representations of finite groups.  ...  Representation theory is concerned with the ways of writing elements of abstract algebraic structures as linear transformations of vector spaces.  ...  Introduction Representations of finite groups provide a way of describing an abstract group as a group of matrices. We will only consider linear representations.  ... 
doi:10.1017/9781108554862.004 fatcat:tipbzfgwwfe33eqm5fa2zy6tla

Representations of Finite Groups [chapter]

1999 Lectures on Representation Theory  
Because any finite abelian group is a product of cyclic groups, the group G * is always isomorphic to G, but not canonically.  ...  Introduction A representation (π, V ) of G on a finite-dimensional complex vector space V is a homomorphism π from the group G to the group GL(V ) of invertible complex linear maps from V to itself.  ...  The classification of irreducible representations of G is reduced, to some extent, to that of classifying those of the smaller groups H and H\G.  ... 
doi:10.1142/9789812815743_0002 fatcat:pk44aqwqxze47n3lozeiwlcopq

Graph representations of finite Abelian groups

Bohdan Zelinka
1981 Czechoslovak Mathematical Journal  
of finite Abelian groups.  ...  Let (^ be a non-trivial finite Abelian group which can be expressed as a direct product of cyclic groups of pairwise different orders. Then the group © has a directed graph representation. Proof.  ... 
doi:10.21136/cmj.1981.101725 fatcat:r2oqa5avnrc7dprgy7yon6m5ci

Homotopy representations of finite groups

Tammo Dieck, Ted Petrie
1982 Publications mathématiques (Bures-sur-Yvette)  
The proof of (10.17) is a consequence of the following lemmas. Lemma (10.18). -dimQV(G)^o) < i.  ...  TAMMOTOMDIECKANDTEDPETRIE Homotopy representation groups The homotopy representation groups now to be defined are the analogues of the representation ring.  ...  It is quite straightforward to show that v{G, X) is a finite group using the degree function. We note that (o(X, X) is a ring for any homotopy representation X of G.  ... 
doi:10.1007/bf02700464 fatcat:26coqdtwvrgvjeqbzijdigg23a

Representation Theory of Finite Groups [article]

Anupam Singh
2010 arXiv   pre-print
The point of view of these notes on the topic is to bring out the flavor that Representation Theory is an extension of the first course on Group Theory.  ...  A large number of worked out examples are the main feature of these notes. The prerequisite for this note is basic group theory and linear algebra.  ...  CHAPTER 5 Representation Theory of Finite Abelian Groups over C Throughout this chapter G denotes a finite Abelian group. Proposition 5.1. Let k = C and G be a finite Abelian group.  ... 
arXiv:1001.0462v1 fatcat:vd3gybd36vbc3of5aasuzzclxy

Representations of Finite Polyadic Groups [article]

Mohammad Shahryari
2010 arXiv   pre-print
Using this correspondence, we generalize some well-known properties of irreducible characters in finite groups to the case of polyadic groups.  ...  We prove that there is a one-one correspondence between sets of irreducible representations of a polyadic group and its Post's cover.  ...  Especially, the number of irreducible representations of any finite polyadic group is finite.  ... 
arXiv:1011.0954v1 fatcat:ydpxvhykrvf6vc2dm7wmyrrwxu

Representations of locally finite groups

David J. Winter
1968 Bulletin of the American Mathematical Society  
The purpose of this paper is to give a brief general account of the completely reducible finite-dimensional representations of a locally finite group G over a given algebraically closed field K.  ...  If G is finite, the assertion follows (upon passing to the group algebra of G over F) from the fact that if A is a finite-dimensional associa* tive algebra over F, then an irreducible (finite-dimensional  ... 
doi:10.1090/s0002-9904-1968-11913-5 fatcat:zrz4qlxwtvewxmyzqbjhm53lka

Tensor representation of finite groups

Oliver Aberth
1972 Journal of combinatorial theory. Series A  
Employing the tensor terminology introduced, a proof that the degrees of the irreducible representations of a finite group divide the group order is given which is somewhat different from the usual proofs  ...  A cubic array of O's and l's, a 34mensional analog of the 2dimensional array of O's and l's defined by a permutation matrix, can be employed to represent a finite group.  ...  The orthogonality relations for group representations and group characters are of fundamental importance in the study of finite groups.  ... 
doi:10.1016/0097-3165(72)90082-9 fatcat:mszx37dwvzgdnlgaieq72wowbe

Small Representations of Finite Classical Groups [article]

Shamgar Gurevich, Roger Howe
2016 arXiv   pre-print
Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group.  ...  Despite the classification by Lusztig of the irreducible representations of finite groups of Lie type, it seems that this aspect remains obscure.  ...  Finally, S.G. thanks the UW-Madison for choosing him as a Vilas Associate for the period of working on this project.  ... 
arXiv:1609.01276v3 fatcat:h3smzfjbqbhr7ftbwokmmudadi

On Abelian Group Representability of Finite Groups [article]

Eldho K. Thomas, Nadya Markin, Frédérique Oggier
2012 arXiv   pre-print
A set of quasi-uniform random variables X_1,...,X_n may be generated from a finite group G and n of its subgroups, with the corresponding entropic vector depending on the subgroup structure of G.  ...  It is known that the set of entropic vectors obtained by considering arbitrary finite groups is much richer than the one provided just by abelian groups.  ...  abelian Group Representability of Nilpotent Groups A finite nilpotent group is a direct product of its Sylow p subgroups.  ... 
arXiv:1212.1298v2 fatcat:pwwypthqk5gn7ev4p45zmi6x2y

On Abelian group representability of finite groups

Eldho K. Thomas, Nadya Markin, Frédérique Oggier
2014 Advances in Mathematics of Communications  
A set of quasi-uniform random variables X 1 , .  ...  abelian Group Representability of Nilpotent Groups A finite nilpotent group is a direct product of its Sylow p subgroups.  ...  The contributions of this paper are to introduce the notion of abelian group representability for an arbitrary finite group G, and to characterize the abelian group representability of several classes  ... 
doi:10.3934/amc.2014.8.139 fatcat:ke5ovyi6m5habmwxinnmdynmnm

Convergence and limits of linear representations of finite groups [article]

Gabor Elek
2015 arXiv   pre-print
Motivated by the theory of graph limits, we introduce and study the convergence and limits of linear representations of finite groups over finite fields.  ...  The limit objects are infinite dimensional representations of free groups in continuous algebras. We show that under a certain integrality condition, the algebras above are skew fields.  ...  By a finite dimensional unitary representation (of degree r), we mean a homomorphism κ : F r → U (n) of the free group on r generators into the unitary group U (n).  ... 
arXiv:1406.5032v5 fatcat:sowcowk7enbhfghebw7u6dvq5a
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