Subrepresentation

In <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a>, a subrepresentation of a <a href="/facts/Group_representation/FsuuDagQ">representation</a> 
 
 
 
 (
 π
 ,
 V
 )
 
 
 {\displaystyle (\pi ,V)}
 
 of a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G is a representation 
 
 
 
 (
 π
 
 
 |
 
 
 W
 
 
 ,
 W
 )
 
 
 {\displaystyle (\pi |_{W},W)}
 
 such that W is a <a href="/facts/Vector_subspace/2wtKTgHL">vector subspace</a> of V and 
 
 
 
 π
 
 
 |
 
 
 W
 
 
 (
 g
 )
 =
 π
 (
 g
 )
 
 
 |
 
 
 W
 
 
 
 
 {\displaystyle \pi |_{W}(g)=\pi (g)|_{W}}
 
.
A nonzero finite-dimensional representation always contains a nonzero subrepresentation that is <a href="/facts/Irreducible_representation/Hu5geBgk">irreducible</a>, the fact seen by <a href="/facts/Mathematical_induction/KxAPqNrH">induction</a> on dimension. This fact is generally false for infinite-dimensional representations.
If 
 
 
 
 (
 π
 ,
 V
 )
 
 
 {\displaystyle (\pi ,V)}
 
 is a representation of G, then there is the trivial subrepresentation:

V
          
            G
          
        
        =
        {
        v
        ∈
        V
        ∣
        π
        (
        g
        )
        v
        =
        v
        ,
        
        g
        ∈
        G
        }
        .
      
    
    {\displaystyle V^{G}=\{v\in V\mid \pi (g)v=v,\,g\in G\}.}

If 
 
 
 
 f
 :
 V
 →
 W
 
 
 {\displaystyle f:V\to W}
 
 is an <a href="/facts/Equivariant_map/KkIkDwIH">equivariant map</a> between two representations, then its kernel is a subrepresentation of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 and its image is a subrepresentation of 
 
 
 
 W
 
 
 {\displaystyle W}
 
.

<ul><li><a href="/facts/William_Fulton_(mathematician)/8ubLHaPy">Fulton, William</a>; <a href="/facts/Joe_Harris_(mathematician)/XPzjGwRf">Harris, Joe</a> (1991). Representation theory. A first course. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-0979-9">10.1007/978-1-4612-0979-9</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-97495-8. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1153249">1153249</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/246650103">246650103</a>.</li></ul>

Subrepresentation open-in-new

Subrepresentation