Submodules
Fancy words: a submodule is a subobject in the category of group representations. What this means is that if 
and 
are 
-modules, and if we have an injective morphism of modules 
, then we say that 
is a “submodule” of 
. And, just to be clear, a -morphism is injective if and only if it’s injective as a linear map; its kernel is zero. We call 
the “inclusion map” of the submodule.
In practice, we often identify a -submodule with the image of its inclusion map. We know from general principles that since is injective, then is isomorphic to its image, so this isn’t really a big difference. What we can tell, though, is that the action of 
sends the image back into itself.
That is, let’s say that 
is the image of some vector 
. I say that for any group element , acting by on gives us some other vector that’s also in the image of . Indeed, we check that
)=\iota\left(\left[\rho_W(g)\right](w)\right)](https://s-ssl.wordpress.com/latex.php?latex=%5Cdisplaystyle%5Cleft%5B%5Crho_V%28g%29%5Cright%5D%28%5Ciota%28w%29%29%3D%5Ciota%5Cleft%28%5Cleft%5B%5Crho_W%28g%29%5Cright%5D%28w%29%5Cright%29&bg=e6e6e6&fg=333333&s=0 "\displaystyle\left[\rho_V(g)\right](\iota(w))=\iota\left(\left[\rho_W(g)\right](w)\right)")
which is again in the image of , as asserted. We say that the image of is “-invariant”.
The flip side of this is that any time we find such a -invariant subspace of , it gives us a submodule. That is, if is a -module, and 
is a -invariant subspace, then we can define a new representation on by restriction: 
. The inclusion map that takes any vector 
and considers it as a vector in clearly intertwines the original action 
and the restricted action 
, and its kernel is trivial. Thus constitutes a -submodule.
As an example, let be any finite group, and let ![\mathbb{C}[G]](https://s-ssl.wordpress.com/latex.php?latex=%5Cmathbb%7BC%7D%5BG%5D&bg=e6e6e6&fg=333333&s=0 "\mathbb{C}[G]")
be its group algebra, which carries the left regular representation 
. Now, consider the subspace spanned by the vector
That is, consists of all vectors for which all the coefficients 
are equal. I say that this subspace ![V\subseteq\mathbb{C}[G]](https://s-ssl.wordpress.com/latex.php?latex=V%5Csubseteq%5Cmathbb%7BC%7D%5BG%5D&bg=e6e6e6&fg=333333&s=0 "V\subseteq\mathbb{C}[G]")
is -invariant. Indeed, we calculate
=c\left[\rho(g)\right]\left(\sum\limits_{g'\in G}\mathbf{g'}\right)=c\sum\limits_{g'\in G}\left[\rho(g)\right](\mathbf{g'})=c\sum\limits_{g'\in G}\mathbf{gg'}](https://s-ssl.wordpress.com/latex.php?latex=%5Cdisplaystyle%5Cleft%5B%5Crho%28g%29%5Cright%5D%28cv%29%3Dc%5Cleft%5B%5Crho%28g%29%5Cright%5D%5Cleft%28%5Csum%5Climits_%7Bg%27%5Cin+G%7D%5Cmathbf%7Bg%27%7D%5Cright%29%3Dc%5Csum%5Climits_%7Bg%27%5Cin+G%7D%5Cleft%5B%5Crho%28g%29%5Cright%5D%28%5Cmathbf%7Bg%27%7D%29%3Dc%5Csum%5Climits_%7Bg%27%5Cin+G%7D%5Cmathbf%7Bgg%27%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\left[\rho(g)\right](cv)=c\left[\rho(g)\right]\left(\sum\limits_{g'\in G}\mathbf{g'}\right)=c\sum\limits_{g'\in G}\left[\rho(g)\right](\mathbf{g'})=c\sum\limits_{g'\in G}\mathbf{gg'}")
But this last sum runs through all the elements of , just in a different order. That is, =cv](https://s-ssl.wordpress.com/latex.php?latex=%5Cdisplaystyle%5Cleft%5B%5Crho%28g%29%5Cright%5D%28cv%29%3Dcv&bg=e6e6e6&fg=333333&s=0 "\displaystyle\left[\rho(g)\right](cv)=cv")
, and so carries the one-dimensional trivial representation of . That is, we’ve found a copy of the trivial representation of as a submodule of the left regular representation.
As another example, let 
be one of the symmetric groups. Again, let carry the left regular representation, but now let be the one-dimensional space spanned by
It’s a straightforward exercise to show that is a one-dimensional submodule carrying a copy of the signum representation.
Every -module contains two obvious submodules: the zero subspace 
and the entire space itself are both clearly -invariant. We call these submodules “trivial”, and all others “nontrivial”.
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September 22, 2010 -
Posted by John Armstrong |
Algebra, Group theory, Representation Theory
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