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''of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m0647801.png" />''
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A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m0647802.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m0647803.png" /> in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m0647804.png" /> (cf. [[Representation of a group|Representation of a group]]) such that in some basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m0647805.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m0647806.png" /> the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m0647807.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m0647808.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m0647809.png" />, has only one non-zero element in each row and each column. Sometimes, by a monomial representation is meant the regular matrix representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478010.png" /> (cf. also [[Regular representation|Regular representation]]). A monomial representation is a special case of an imprimitive representation (see [[Imprimitive group|Imprimitive group]]). Namely, the set of one-dimensional subspaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478011.png" /> generated by the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478012.png" /> is a system of imprimitivity for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478013.png" />. Conversely, if for some representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478015.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478016.png" /> there is a system of imprimitivity consisting of one-dimensional subspaces, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478017.png" /> is a monomial representation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478018.png" /> be any subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478019.png" />; an example of a monomial representation is the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478020.png" /> induced from a one-dimensional representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478021.png" /> (see [[Induced representation|Induced representation]]). Such a representation is also called an induced monomial representation (see [[#References|[1]]]). Not every monomial representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478022.png" /> is induced (however, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478023.png" /> is irreducible, then it will be induced). The definition of a monomial representation given above arose in the classical theory of representations of finite groups. Often, however, this definition can be modified — any representation of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478024.png" /> induced from a one-dimensional representation of some subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478025.png" /> of it being called a monomial representation. In this form the definition of a monomial representation makes sense not just for finite groups and their finite-dimensional representations, but also, for example, for Lie groups and their representations in Hilbert spaces (the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478026.png" /> is then assumed to be closed). For a fairly large class of groups the construction of the monomial representations turns out to be sufficient for the description of all irreducible unitary representations. Namely, groups all irreducible unitary representations (cf. [[Irreducible representation|Irreducible representation]]; [[Unitary representation|Unitary representation]]) of which are monomial are called monomial groups, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478028.png" />-groups. They include all finite nilpotent groups and all connected nilpotent Lie groups. All finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478029.png" />-groups, and also all monomial Lie groups, are solvable (see [[#References|[2]]]).
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''of a finite group  $  G $''
 +
 
 +
A representation $  \rho $
 +
of $  G $
 +
in a finite-dimensional vector space $  V $(
 +
cf. [[Representation of a group|Representation of a group]]) such that in some basis $  ( e) = \{ e _ {1} \dots e _ {n} \} $
 +
of $  V $
 +
the matrix $  M _ {\rho ( g) }  ^ {(} e) $
 +
of $  \rho ( g) $,  
 +
for any $  g \in G $,  
 +
has only one non-zero element in each row and each column. Sometimes, by a monomial representation is meant the regular matrix representation $  g \rightarrow M _ {\rho ( g) }  ^ {(} e) $(
 +
cf. also [[Regular representation|Regular representation]]). A monomial representation is a special case of an imprimitive representation (see [[Imprimitive group|Imprimitive group]]). Namely, the set of one-dimensional subspaces in $  V $
 +
generated by the vectors $  e _ {1} \dots e _ {n} $
 +
is a system of imprimitivity for $  \rho $.  
 +
Conversely, if for some representation $  \phi $
 +
of $  G $
 +
in a vector space $  W $
 +
there is a system of imprimitivity consisting of one-dimensional subspaces, then $  \phi $
 +
is a monomial representation. Let $  H $
 +
be any subgroup of $  G $;  
 +
an example of a monomial representation is the representation of $  G $
 +
induced from a one-dimensional representation of $  H $(
 +
see [[Induced representation|Induced representation]]). Such a representation is also called an induced monomial representation (see [[#References|[1]]]). Not every monomial representation $  \rho $
 +
is induced (however, if $  \rho $
 +
is irreducible, then it will be induced). The definition of a monomial representation given above arose in the classical theory of representations of finite groups. Often, however, this definition can be modified — any representation of a group $  G $
 +
induced from a one-dimensional representation of some subgroup $  H $
 +
of it being called a monomial representation. In this form the definition of a monomial representation makes sense not just for finite groups and their finite-dimensional representations, but also, for example, for Lie groups and their representations in Hilbert spaces (the subgroup $  H $
 +
is then assumed to be closed). For a fairly large class of groups the construction of the monomial representations turns out to be sufficient for the description of all irreducible unitary representations. Namely, groups all irreducible unitary representations (cf. [[Irreducible representation|Irreducible representation]]; [[Unitary representation|Unitary representation]]) of which are monomial are called monomial groups, or $  M $-
 +
groups. They include all finite nilpotent groups and all connected nilpotent Lie groups. All finite $  M $-
 +
groups, and also all monomial Lie groups, are solvable (see [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</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)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</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)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For a fuller account on the state of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064780/m06478030.png" />-groups as known 20 years ago see [[#References|[a1]]].
+
For a fuller account on the state of $  M $-
 +
groups as known 20 years ago see [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)  pp. Chapt. V, §18</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Feit,  "Characters of finite groups" , Benjamin  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)  pp. Chapt. V, §18</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Feit,  "Characters of finite groups" , Benjamin  (1967)</TD></TR></table>

Revision as of 08:01, 6 June 2020


of a finite group $ G $

A representation $ \rho $ of $ G $ in a finite-dimensional vector space $ V $( cf. Representation of a group) such that in some basis $ ( e) = \{ e _ {1} \dots e _ {n} \} $ of $ V $ the matrix $ M _ {\rho ( g) } ^ {(} e) $ of $ \rho ( g) $, for any $ g \in G $, has only one non-zero element in each row and each column. Sometimes, by a monomial representation is meant the regular matrix representation $ g \rightarrow M _ {\rho ( g) } ^ {(} e) $( cf. also Regular representation). A monomial representation is a special case of an imprimitive representation (see Imprimitive group). Namely, the set of one-dimensional subspaces in $ V $ generated by the vectors $ e _ {1} \dots e _ {n} $ is a system of imprimitivity for $ \rho $. Conversely, if for some representation $ \phi $ of $ G $ in a vector space $ W $ there is a system of imprimitivity consisting of one-dimensional subspaces, then $ \phi $ is a monomial representation. Let $ H $ be any subgroup of $ G $; an example of a monomial representation is the representation of $ G $ induced from a one-dimensional representation of $ H $( see Induced representation). Such a representation is also called an induced monomial representation (see [1]). Not every monomial representation $ \rho $ is induced (however, if $ \rho $ is irreducible, then it will be induced). The definition of a monomial representation given above arose in the classical theory of representations of finite groups. Often, however, this definition can be modified — any representation of a group $ G $ induced from a one-dimensional representation of some subgroup $ H $ of it being called a monomial representation. In this form the definition of a monomial representation makes sense not just for finite groups and their finite-dimensional representations, but also, for example, for Lie groups and their representations in Hilbert spaces (the subgroup $ H $ is then assumed to be closed). For a fairly large class of groups the construction of the monomial representations turns out to be sufficient for the description of all irreducible unitary representations. Namely, groups all irreducible unitary representations (cf. Irreducible representation; Unitary representation) of which are monomial are called monomial groups, or $ M $- groups. They include all finite nilpotent groups and all connected nilpotent Lie groups. All finite $ M $- groups, and also all monomial Lie groups, are solvable (see [2]).

References

[1] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)
[2] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)

Comments

For a fuller account on the state of $ M $- groups as known 20 years ago see [a1].

References

[a1] B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) pp. Chapt. V, §18
[a2] W. Feit, "Characters of finite groups" , Benjamin (1967)
How to Cite This Entry:
Monomial representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monomial_representation&oldid=47891
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article