Abstract Nonsense

Crushing one theorem at a time

The Center of a Character (Pt. I)


Point of post: In this post we discuss the notion the ‘center of a character’ and its relations to other concepts we’ve previously learned.

Motivation

In previous posts we’ve seen how we can take concepts from plain group theory and attempt to make an analogy for characters. In this post we extend this further by defining the ‘center’ of a character, which, as we shall see are just those g\in G for which |\chi(g)|=\chi(e). In other words, its just those g\in G for which the modulus of \chi(g) is maximized. This shall prove useful in the future, in particular it shall help us devolop enough machinery to prove Burnside’s Theorem–the classic use of representation theory.

Center of a Character

Let G be a finite group and \chi some character of G. Then, we define the center of \chi, denoted \bold{Z}(\chi), to be

 

\bold{Z}(\chi)=\left\{g\in G:\left|\chi(g)\right|=\chi(e)\right\}

 

Our first claim is that, despite what is not immediately apparent, \bold{Z}(\chi)\leqslant G for every \chi. Indeed, this follows immediately from:

Theorem: Let G be a finite group and \chi the character of the representation \rho. Then,

 

\bold{Z}\left(\chi\right)=\left\{g\in G:\rho(g)=z\mathbf{1}\;\; z\in\mathbb{C}\right\}

 

Proof: It’s clear by the unitarity of \rho(g) for every g\in G that if \rho(g)=zI then |z|=1. Thus, \chi(g)=\text{tr}\left(zI\right)=z\chi(e) and so \left|\chi(g)\right|=\chi(e) as desired.

 

Conversely, suppose that g\in\bold{Z}\left(\chi\right). Recall from the spectral theorem that we have that \rho(g) is similar to \text{diag}(\lambda_1,\cdots,\lambda_n) where n=\deg\rho and \lambda_1,\cdots,\lambda_n are the eigenvalues of \rho(g) with multiplicity. But, by assumption we have that

\displaystyle \left|\sum_{j=1}^{n}\lambda_j\right|=n

 

from where it follows from the fact that |\lambda_j|=1 one must have that \lambda_1=\cdots=\lambda_j and so \rho(g) is similar to \lambda_1 \mathbf{1} and thus \rho(g)=\lambda_1\mathbf{1}. The conclusion follows. \blacksquare

 

Corollary:Let G be a finite group and \chi a character for G. Then, \bold{Z}(G)\leqslant G.

Proof: This clearly follows since if g,g'\in \bold{Z}(G) then \rho(g)=z\mathbf{1} and \rho(g')=z'\mathbf{1} and so evidently \rho(gg'^{-1})=\rho(g)\rho(g')^{-1}=z\mathbf{1}z'^{-1}\mathbf{1}=(zz'^{-1})\mathbf{1} and so gg'^{-1}\in\bold{Z}(G). The conclusion follows. \blacksquare

 

 

If we have a representation \rho of a finite group G one can easily verify that for any subgroup H of G the restriction of \rho to H is a representation of H. Moreover, the character of \rho restricted to H is also a character on H. Also, it’s apparent that if \rho_{\mid H} is irreducible then \rho is irreducible. Thus, if \chi_{\mid H}\cong \chi^{(\alpha)} for some \alpha\in\widehat{H} then \chi\cong \chi^{(\beta)} for some \beta\in\widehat{G}. With this in mind, we make the following claim:

 

Theorem: Let G be a finite group and \chi a character of G. Then, \chi_{\mid \bold{Z}(\chi)}=\chi(e)\chi_\lambda for some degree-one irrep \lambda.

Proof: We know by definition that for every g\in \bold{Z}(G) there exists some z_g\in\mathbb{C} for which \rho(g)=z_g\mathbf{1}. Define \lambda:\bold{Z}(G)\to\mathbb{C}^{\times} by \lambda(g)=z_g. This is clearly a degree-one irrep and moreover it’s clear that \rho_{\bold{Z}(G)}=\lambda(g)\mathbf{1}. Thus,

\chi_{\rho_{\mid\bold{Z}(G)}}(g)=\text{tr}\left(\rho_{\bold{Z}(G)}(g)\right)=\text{tr}\left(\lambda(g)\mathbf{1}\right)=\chi(e)\lambda(g)

 

from where the conclusion follows. \blacksquare

References:

1. Isaacs, I. Martin. Character Theory of Finite Groups. New York: Academic, 1976. Print.

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March 8, 2011 - Posted by | Algebra, Representation Theory | , , , ,

4 Comments »

  1. […] Point of post: This is a continuation of this post. […]

    Pingback by Representation Theory: The Center of a Character (Pt. II) « Abstract Nonsense | March 9, 2011 | Reply

  2. […] that there existed a conjugacy class of such that then for every either or where is the center of the character […]

    Pingback by Representation Theory: Burnside’s Theorem « Abstract Nonsense | March 11, 2011 | Reply

  3. […] our last post we saw how every character on a finite group naturally creates a host of subgroups of , namely […]

    Pingback by Representation Theory: The Index of the Center of a Character « Abstract Nonsense | March 21, 2011 | Reply

  4. […] of numbers (a matrix really) which using our past information about the kernel of a character, the center of a character, etc. to shall tell us most of things one may initally like to know about a group. The interesting […]

    Pingback by Representation Theory: Character Tables « Abstract Nonsense | March 21, 2011 | Reply


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