Abstract Nonsense

Crushing one theorem at a time

Subrepresentations, Direct Sum of Representations, and Irreducible Representations (Pt. III)


Point of post: This post is a continuation of this one.

Irreducible Representations

 

Let \mathscr{V} be a finite dimensional pre-Hilbert space and G a finite group. A representation \rho:G\to\mathcal{U}\left(\mathscr{V}\right) is called irreducible if the only subspaces of \mathscr{V} invariant are \mathscr{V} and \{\bold{0}\}. It is often the case, and we will follow this here, that an irreducible representation is shortened to irrep. It’s clear from the last two theorems in our last section that \rho is irrep if and only if \rho is not equivalent to the direct sum of two non-trivial representations. (a non-trivial representation \omega is a representation such that \deg\omega>1).

 

We now make clear why irreps are the “building” blocks of representations, we do this via the next theorem:

 

Theorem: Let G be a finite group. Then, any non-trivial representation on \rho is equivalent to the direct sum of irreducible representations.

Proof: We proceed by induction on \deg\rho. If \deg\rho=1 this is trivial since every representation of degree one is an irrep (it can’t have a non-trivial invariant subspace since it can’t have any non-trivial subspace!). Assume that it’s true for all \rho such that \deg\rho<n and let \omega be some representation with \deg\omega=n. If \omega is an irrep we’re done, and if not then \omega\simeq \rho\oplus\rho' for some representations \rho and \rho' with 0<\deg\rho,\deg\rho'<n thus by assumption we have that \displaystyle \rho\simeq \nu_1\oplus\cdots\oplus \nu_k and \rho'\simeq \upsilon_1\oplus\cdots\oplus\upsilon_\ell and thus evidently (it’s fairly obvious, and easy to prove) we may conclude that

 

\omega\simeq \nu_1\oplus\cdots\oplus\nu_k\oplus\upsilon_1\oplus\cdots\oplus\upsilon_\ell

 

from where the conclusion follows. \blacksquare

 

Clearly the relation \simeq is an equivalence relation on the set of irreps on a group G. We denote the set of all such equivalence classes as \widehat{G}. This will be of the utmost importance in the theory to come.

 

References:

1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print.


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January 18, 2011 - Posted by | Algebra, Representation Theory | , , , ,

2 Comments »

  1. […] seen that every finite group gives rise naturally to a set of equivalence classes of irreps . Suppose […]

    Pingback by Representation Theory: Matrix Entry Functions « Abstract Nonsense | February 22, 2011 | Reply

  2. […] following the same line of reasoning as before we may conclude […]

    Pingback by Representation Theory: Representations on Real and Quaternionic Vector Spaces (Examples and Basics) « Abstract Nonsense | March 28, 2011 | Reply


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