Abstract Nonsense

Crushing one theorem at a time

Schur’s Lemma (*-representation Form)


Point of post: In this post we discuss the notion of Schur’s lemma as it applies to irreducible \ast-representations, and with this show that every irrep of an abelian group must have degree one.

Motivation

In our last post we saw how the existence of a non-zero intertwinor for an irrep guarantees the equivalence of the two irreps, since any such intertwinor must be an isomorphism. We also saw how if the two irreps and representation spaces coincided that the intertwinor must, in fact, be a multiple of the identity map. In this post we describe the notion of an irreducible \ast-representation and then give a version of Schur’s second lemma for such representations. This will allow us to conclude, after some work, that every irrep of an abelian group must have degree one.

Irreducible \ast-representations and the \ast-representation Form of Schur’s Lemma

Let G be a finite group and \mathcal{A}\left(G\right) the group algebra. Then, if \mathscr{V} is a pre-Hilbert space and \varrho:\mathcal{A}\left(G\right)\to\text{End}\left(\mathscr{V}\right)  a  \ast-representation we call \mathscr{W}\leqslant\mathscr{V} \varrho-invariant (or invariant when \varrho is evident) if \mathscr{W} is invariant under \varrho_a for each a\in\mathcal{A}\left(G\right). We then, unsurprisingly, define a \ast-representation \varrho:\mathcal{A}\left(G\right)\to\text{End}\left(\mathscr{V}\right) to be irreducible if the only \varrho-invariant subspaces of \mathscr{V} are \{\bold{0}\} and \mathscr{V}.

Our first result result about irreducible \ast-representations is a fundamental one. Namely:

 

Theorem: Let G be a finite group, \mathcal{A}\left(G\right) the group algebra, and \mathscr{V} a pre-Hilbert space. Then if \varrho:\mathcal{A}\left(G\right)\to\text{End}\left(\mathscr{V}\right) is an irreducible \ast-representation then the induced representation \rho:G\to\mathcal{U}\left(G\right):g\mapsto \varrho(\delta_g) is irreducible. Conversely, if \rho:G\to\mathcal{U}\left(\mathscr{V}\right) is an irrep then the induced \ast-representation

 

\displaystyle\varrho:\mathcal{A}\left(G\right)\to\text{End}\left(\mathscr{V}\right):a\mapsto \sum_{g\in G}a(g)\rho_g

 

Proof: Suppose first that \varrho is an irrep and \mathscr{W}\leqslant\mathscr{V} is invariant under \rho_g for each g\in G. We note then that for every a\in\mathcal{A}\left(G\right) one has

 

\displaystyle \varrho_a\left(\mathscr{W}\right)=\sum_{g\in G}a(g)\rho_g\left(\mathscr{W}\right)\subseteq\sum_{g\in G}a(g)\mathscr{W}\subseteq\mathscr{W}

 

and thus \mathscr{W} is \varrho-invariant, but by assumption this implies that \mathscr{W} is either \mathscr{V} or \{\bold{0}\}. From where it follows that \mathscr{W} is \rho-invariant implies that \mathscr{W} is either \{\bold{0}\} or \mathscr{V} and thus \rho is an irrep.

Conversely, suppose that \rho:G\to\mathcal{U}\left(G\right) is an irrep and let \varrho be the induced irrep. Assume then that \mathscr{W} is invariant under \varrho then we have in particular that \mathscr{W} is invariant under \varrho(\delta_g)=\rho_g for each g\in G and thus \mathscr{W} is \rho-invariant and thus by assumption \mathscr{W} is either \mathscr{V} or \{\bold{0}\}, and by applying similar logic as last time we may conclude that \mathscr{W} is \varrho-invariant implies that \mathscr{W} is \{\bold{0}\} or \mathscr{V} and thus \varrho is irreducible. \blacksquare

 

 

With this we now have that third form of Schur’s lemma, the \ast-representation form:

Theorem(Schur’s Lemma Third Form-\ast-representation Form): Let G be a finite group, \mathcal{A}\left(G\right) the group algebra, and \mathscr{V},\mathscr{W} a pre-Hilbert spaces. Then if \varrho':\mathcal{A}\left(G\right)\to\text{End}\left(\mathscr{V}\right) and \varrho':G\to\text{End}\left(\mathscr{W}\right) are two irreducible \ast-representations and T:\mathscr{V}\to\mathscr{W} such that T\varrho_g=\varrho'_gT for every g\in G then either T=\bold{0} or T is an isomorphism. Moreover, if \varrho=\varrho' and \mathscr{V}=\mathscr{W} then T=\lambda\mathbf{1} for some \lambda\in\mathbb{C}.

Proof: We note first that since \varrho is irreducible the induced representations \rho:G\to\to\mathcal{U}\left(\mathscr{V}\right) and \rho':G\to\mathcal{U}\left(\mathscr{W}\right) $ must be irreps, but by definition T\rho_g=T\varrho(\delta_g)=\varrho'(\delta_g)T=\rho'_gT from where it follows by the first form of Schur’s lemma that T is either zero or an isomorphism.

Assume now that \varrho=\varrho' and \mathscr{V}=\mathscr{W} then using the fact that the induced representation \rho is irreducible and T\rho_g=\rho_gT for each g\in G implies by the second form of Schur’s lemma that T=\lambda\mathbf{1} for some \lambda\in\mathbb{C}. \blacksquare

 

As a quick application of Schur’s Lemmas we prove the extremely interesting following theorem:

Theorem: Let G be a finite abelian group, \mathscr{V} a pre-Hilbert space, and \rho:G\to\mathcal{U}\left(\mathscr{V}\right) an irreducible representation. Then, \deg\rho=1.

Proof: Consider the induced \ast-representation \varrho:\mathcal{A}\left(G\right)\to\text{End}\left(\mathscr{V}\right). Since \rho is an irrep we know that \varrho is an irreducible \ast representation. Note though that for any g\in G and any a\in\mathcal{A}\left(G\right) one has that

 

\displaystyle \begin{aligned}\rho_g\varrho(a) &=\rho_g\sum_{h\in G}a(h)\rho_h\\ &=\sum_{h\in G}a(h)\rho_g\rho_h\\ &=\sum_{g\in G}a(h)\rho_{gh}\\ &=\sum_{h\in G}a(h)\rho_{hg}\\ &=\sum_{h\in G}a(h)\rho_{h}\rho_g\\ &=\varrho(a)\rho_g\end{aligned}

 

and thus by the third form of Schur’s lemma we may conclude that there exists \{\lambda_g\}_{g\in G}\subseteq\mathbb{C} such that \rho_g=\lambda_g\mathbf{1} for each g\in G. But, this clearly implies that every subspace of \mathscr{V} is invariant under every \rho_g and thus \rho-invariant. Thus, if \dim\mathscr{V}>1 we could choose \mathscr{W}\leqslant\mathscr{V} with 0<\dim\mathscr{W}<\dim\mathscr{V} and by earlier remark this would be \rho-invariant contradicting the irreducibility of \rho. It follows that \dim\mathscr{V}=\deg\rho=1.

 

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 27, 2011 - Posted by | Algebra, Representation Theory | , , ,

1 Comment »

  1. […] The necessity of this theorem was a previous theorem and so it suffices to show sufficiency. To do this we merely note that if each irrep has degree one […]

    Pingback by Representation Theory Irreducible Characters (Pt. II) « Abstract Nonsense | February 25, 2011 | Reply


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