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A [[Homomorphism|homomorphism]] of the group into the group of all invertible transformations of a set $V$. A representation $\rho$ of a group $G$ is called linear if $V$ is a vector space over a field $k$ and if the transformations $\rho(g)$, $g\in G$, are linear. Often, linear representations are, for shortness, simply termed representations (cf. [[Representation theory|Representation theory]]). In the theory of representations of abstract groups the theory of finite-dimensional representations of finite groups is best developed (cf. [[Finite group, representation of a|Finite group, representation of a]]; [[Representation of the symmetric groups|Representation of the symmetric groups]]).
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A [[homomorphism]] of the group into the group of all invertible transformations of a set $V$.  
  
If $G$ is a topological group, then one considers continuous linear representations of $G$ on a topological vector space $V$ (cf. [[Continuous representation|Continuous representation]]; [[Representation of a topological group|Representation of a topological group]]). If $G$ is a Lie group and $V$ is a finite-dimensional space over $\mathbf R$ or $\mathbf C$, then a continuous linear representation is automatically real analytic. Analytic and differentiable representations of a Lie group are defined also in the infinite-dimensional case (cf. [[Analytic representation|Analytic representation]]; [[Infinite-dimensional representation|Infinite-dimensional representation]]). To each differentiable representation $\rho$ of a Lie group $G$ corresponds some linear representation of its Lie algebra — the differential representation of $\rho$ (cf. [[Representation of a Lie algebra|Representation of a Lie algebra]]). If $G$ is moreover connected, then its finite-dimensional representations are completely determined by their differentials. The most developed branch of the representation theory of topological groups is the theory of finite-dimensional linear representations of semi-simple Lie groups, which is often formulated in the language of Lie algebras (cf. [[Finite-dimensional representation|Finite-dimensional representation]]; [[Representation of the classical groups|Representation of the classical groups]]; [[Cartan theorem|Cartan theorem]] on the highest weight vector), the representation theory of compact groups, and the theory of unitary representations (cf. [[Representation of a compact group|Representation of a compact group]]; [[Unitary representation|Unitary representation]]).
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A ''permutation representation'' is a homomorphism to the [[symmetric group]] $S_V$: a [[group action]] of $G$ on $V$: cf. [[Permutation group]].
  
For algebraic groups one has the theory of rational representations (cf. [[Rational representation|Rational representation]]), which is in many aspects analogous to the theory of finite-dimensional representations of Lie groups.
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A representation $\rho$ of a group $G$ is called ''linear'' if $V$ is a vector space over a field $k$ and if the transformations $\rho(g)$, $g\in G$, are linear. Often, linear representations are, for shortness, simply termed representations (cf. [[Representation theory]]). In the theory of representations of abstract groups the theory of finite-dimensional representations of finite groups is best developed (cf. [[Finite group, representation of a|Finite group, representation of a]]; [[Representation of the symmetric groups|Representation of the symmetric groups]]).
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If $G$ is a topological group, then one considers continuous linear representations of $G$ on a topological vector space $V$ (cf. [[Continuous representation]]; [[Representation of a topological group]]). If $G$ is a Lie group and $V$ is a finite-dimensional space over $\mathbf R$ or $\mathbf C$, then a continuous linear representation is automatically real analytic. Analytic and differentiable representations of a Lie group are defined also in the infinite-dimensional case (cf. [[Analytic representation]]; [[Infinite-dimensional representation]]). To each differentiable representation $\rho$ of a Lie group $G$ corresponds some linear representation of its Lie algebra — the differential representation of $\rho$ (cf. [[Representation of a Lie algebra]]). If $G$ is moreover connected, then its finite-dimensional representations are completely determined by their differentials. The most developed branch of the representation theory of topological groups is the theory of finite-dimensional linear representations of semi-simple Lie groups, which is often formulated in the language of Lie algebras (cf. [[Finite-dimensional representation]]; [[Representation of the classical groups]]; [[Cartan theorem]] on the highest weight vector), the representation theory of compact groups, and the theory of unitary representations (cf. [[Representation of a compact group]]; [[Unitary representation]]).
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For algebraic groups one has the theory of rational representations (cf. [[Rational representation]]), which is in many aspects analogous to the theory of finite-dimensional representations of Lie groups.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) {{MR|0412321}} {{ZBL|0342.22001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) {{MR|0793377}} {{ZBL|0484.22018}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D.P. Zhelobenko, A.I. Shtern, "Representations of Lie groups" , Moscow (1981) (In Russian) {{MR|1104272}} {{MR|0709598}} {{ZBL|0581.22016}} {{ZBL|0521.22006}} </TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) {{MR|0412321}} {{ZBL|0342.22001}} </TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) {{MR|0793377}} {{ZBL|0484.22018}} </TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top"> D.P. Zhelobenko, A.I. Shtern, "Representations of Lie groups" , Moscow (1981) (In Russian) {{MR|1104272}} {{MR|0709598}} {{ZBL|0581.22016}} {{ZBL|0521.22006}} </TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.J. Benson, "Modular representation theory: New trends and methods" , ''Lect. notes in math.'' , '''1081''' , Springer (1984) {{MR|0765858}} {{ZBL|0564.20004}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C.W. Curtis, I. Reiner, "Methods of representation theory" , '''1–2''' , Wiley (Interscience) (1981–1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Feit, "The representation theory of finite groups" , North-Holland (1982) {{MR|0661045}} {{ZBL|0493.20007}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.-P. Serre, "Linear representations of finite groups" , Springer (1977) (Translated from French) {{MR|0450380}} {{ZBL|0355.20006}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> B. Huppert, "Endliche Gruppen" , '''1''' , Springer (1967) pp. 64 {{MR|0224703}} {{ZBL|0217.07201}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) {{MR|0855239}} {{ZBL|0604.22001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> J. Tits, "Tabellen zu den einfachen Lie Grupppen und ihren Darstellungen" , ''Lect. notes in math.'' , '''40''' , Springer (1967) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> G. Warner, "Harmonic analysis on semisimple Lie groups" , '''1–2''' , Springer (1972) {{MR|}} {{ZBL|}} </TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> D.J. Benson, "Modular representation theory: New trends and methods" , ''Lect. notes in math.'' , '''1081''' , Springer (1984) {{MR|0765858}} {{ZBL|0564.20004}} </TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> C.W. Curtis, I. Reiner, "Methods of representation theory" , '''1–2''' , Wiley (Interscience) (1981–1987)</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Feit, "The representation theory of finite groups" , North-Holland (1982) {{MR|0661045}} {{ZBL|0493.20007}} </TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top"> J.-P. Serre, "Linear representations of finite groups" , Springer (1977) (Translated from French) {{MR|0450380}} {{ZBL|0355.20006}} </TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top"> B. Huppert, "Endliche Gruppen" , '''1''' , Springer (1967) pp. 64 {{MR|0224703}} {{ZBL|0217.07201}} </TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top"> A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) {{MR|0855239}} {{ZBL|0604.22001}} </TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top"> J. Tits, "Tabellen zu den einfachen Lie Grupppen und ihren Darstellungen" , ''Lect. notes in math.'' , '''40''' , Springer (1967) {{MR|}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a8]</TD> <TD valign="top"> G. Warner, "Harmonic analysis on semisimple Lie groups" , '''1–2''' , Springer (1972) {{MR|}} {{ZBL|}} </TD></TR>
 +
</table>

Revision as of 19:39, 1 December 2014

A homomorphism of the group into the group of all invertible transformations of a set $V$.

A permutation representation is a homomorphism to the symmetric group $S_V$: a group action of $G$ on $V$: cf. Permutation group.

A representation $\rho$ of a group $G$ is called linear if $V$ is a vector space over a field $k$ and if the transformations $\rho(g)$, $g\in G$, are linear. Often, linear representations are, for shortness, simply termed representations (cf. Representation theory). In the theory of representations of abstract groups the theory of finite-dimensional representations of finite groups is best developed (cf. Finite group, representation of a; Representation of the symmetric groups).

If $G$ is a topological group, then one considers continuous linear representations of $G$ on a topological vector space $V$ (cf. Continuous representation; Representation of a topological group). If $G$ is a Lie group and $V$ is a finite-dimensional space over $\mathbf R$ or $\mathbf C$, then a continuous linear representation is automatically real analytic. Analytic and differentiable representations of a Lie group are defined also in the infinite-dimensional case (cf. Analytic representation; Infinite-dimensional representation). To each differentiable representation $\rho$ of a Lie group $G$ corresponds some linear representation of its Lie algebra — the differential representation of $\rho$ (cf. Representation of a Lie algebra). If $G$ is moreover connected, then its finite-dimensional representations are completely determined by their differentials. The most developed branch of the representation theory of topological groups is the theory of finite-dimensional linear representations of semi-simple Lie groups, which is often formulated in the language of Lie algebras (cf. Finite-dimensional representation; Representation of the classical groups; Cartan theorem on the highest weight vector), the representation theory of compact groups, and the theory of unitary representations (cf. Representation of a compact group; Unitary representation).

For algebraic groups one has the theory of rational representations (cf. Rational representation), which is in many aspects analogous to the theory of finite-dimensional representations of Lie groups.

References

[1] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013
[2] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) MR0412321 Zbl 0342.22001
[3] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[4] D.P. Zhelobenko, A.I. Shtern, "Representations of Lie groups" , Moscow (1981) (In Russian) MR1104272 MR0709598 Zbl 0581.22016 Zbl 0521.22006


Comments

References

[a1] D.J. Benson, "Modular representation theory: New trends and methods" , Lect. notes in math. , 1081 , Springer (1984) MR0765858 Zbl 0564.20004
[a2] C.W. Curtis, I. Reiner, "Methods of representation theory" , 1–2 , Wiley (Interscience) (1981–1987)
[a3] W. Feit, "The representation theory of finite groups" , North-Holland (1982) MR0661045 Zbl 0493.20007
[a4] J.-P. Serre, "Linear representations of finite groups" , Springer (1977) (Translated from French) MR0450380 Zbl 0355.20006
[a5] B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) pp. 64 MR0224703 Zbl 0217.07201
[a6] A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) MR0855239 Zbl 0604.22001
[a7] J. Tits, "Tabellen zu den einfachen Lie Grupppen und ihren Darstellungen" , Lect. notes in math. , 40 , Springer (1967)
[a8] G. Warner, "Harmonic analysis on semisimple Lie groups" , 1–2 , Springer (1972)
How to Cite This Entry:
Representation of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_of_a_group&oldid=33023
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article