Abstract Nonsense

Crushing one theorem at a time

The Kernel of a Character


Point of post: In this post we discuss the notion of the kernel of a character in general preparation to prove some pretty serious theorems in pure group theory.

Motivation

Despite all the cool things we’ve done with representations, we must not forget that at the end of the day every representation is just a homomorphism between groups. It makes sense then that one of the most natural things to do with a representation is to look at its kernel with respect to its homomorphism qualities. It turns out though that there is a natural way to translate this notion of the kernel of a representation into the notion of the kernel of a character. This notion shall be paramount in the theory to come.

Kernel of a Character

Let G be a finite group and \rho:G\to\mathcal{U}\left(\mathscr{V}\right) a representation. We define as usual, pursuant to the usual definitions of the kernel of a homomorphism the kernel of \rho, denoted \ker\rho as

\text{ }

\ker\rho=\left\{g\in G:\rho(g)=\mathbf{1}\right\}

What is interesting about this whole concept is that there is way to describe \ker\rho in terms of its induced character \chi_\rho. Indeed:

Theorem: Let G be a finite group and \rho:G\to\mathcal{U}\left(\mathscr{V}\right) is a representation with induced character \chi_\rho. Then,

\text{ }

\displaystyle \ker\rho=\left\{g\in G:\chi_\rho(g)=\chi_\rho(e)\right\}

 \text{ }

Proof: Evidently if g\in\ker\rho then \rho(g)=\mathbf{1}_{\mathscr{V}} and so \chi_\rho(g)=\text{tr}(\rho(g))=\dim\mathscr{V}=\chi_\rho(e) from where the conclusion follows.

Conversely, suppose that \chi_\rho(g)=\chi_\rho(e). Note that since \rho(g) is unitary we know by the spectral theorem that \rho(g) is similar to \text{diag}(\lambda_1,\cdots,\lambda_n) (where n=\dim \mathscr{V} and the \lambda_j‘s are the eigenvalues) with respect to any basis. But, we know by the assumption that

$laetx \text{ }$

\displaystyle \sum_{j=1}^{n}\lambda_j=n

 \text{ }

and since \rho(g) is unitary we know that \lambda_j\in\mathbb{S}^1 for j=1,\cdots,n and so it follows then that \lambda_1=\cdots=\lambda_n=1. Thus, \rho(g) is similar to \mathbf{1} and thus \rho(g)=\mathbf{1}. Thus, g\in\ker\rho as desired. The conclusion follows. \blacksquare

It makes sense then to define the kernel of the character \chi, denoted \ker\chi, by

\text{ }

\ker\chi=\left\{g\in G:\chi(g)=\chi(e)\right\}

 \text{ }

In particular, we know that for every character \chi one has that \ker\chi\unlhd G. For the irreducible characters \chi^{(\alpha)}\;\; \alpha\in\widehat{G} we give the special symbols N^{(\alpha)} for \ker\chi^{(\alpha)}. What we now show is that knowing N^{(\alpha)} for every \alpha\in\widehat{G} enables one to know \ker\chi for every character \chi. Indeed:

Theorem: Let \chi be a character with representation as a linear combination of the irreducible characters \displaystyle \chi=\sum_{\alpha\in\widehat{G}}m^{(\alpha)}\chi^{(\alpha)}. Then,

\text{ }

\displaystyle \ker\chi=\bigcap\left\{N^{(\alpha)}:m^{(\alpha)}>0\right\}

 \text{ }

Proof: Clearly if \chi^{(\alpha)}(g)=d_\alpha for every \alpha such that m^{(\alpha)}>0 one sees that

\text{ }

\displaystyle \begin{aligned}\chi(g) &=\sum_{\alpha\in\widehat{G}\text{ s.t. }m^{(\alpha)}>0}m^{(\alpha)}\chi^{(\alpha)}(g)\\ &=\sum_{\alpha\in\widehat{G}\text{ s.t. }m^{(\alpha)}>0}m^{(\alpha)}\chi^{(\alpha)}(e)\\ &=\chi(e)\end{aligned}

\text{ }

and so g\in\ker\chi.

Conversely, since one evidently has that \left|\chi^{(\alpha)}(g)\right|\leqslant d_\alpha for every \alpha\in\widehat{G} we see that for any g\in \ker\chi one has that

\text{ }

\displaystyle \begin{aligned}\left|\chi(g)\right| &=\left|\sum_{\alpha\in\widehat{G}\text{ s.t. }m^{(\alpha)}>0}m^{(\alpha)}\chi^{(\alpha)})(g)\right|\\ &\leqslant\sum_{\alpha\in\widehat{G}\text{ st. }m^{(\alpha)}>0}m^{(\alpha)}\left|\chi^{(\alpha)}(g)\right|\\ &\leqslant\sum_{\alpha\in\widehat{G}\text{ s.t. }m^{(\alpha)}>0}m^{(\alpha)}d_\alpha\\ &= \chi(1)\\ &= \chi(g)\end{aligned}

\text{ }

from where it follows from this that \chi^{(\alpha)}(g) must be real and so if \chi^{(\alpha)}(g)<d_\alpha for any \alpha\in\widehat{G} then this would induce a strict inequality for \chi(g) and \chi(1). It follows that \chi^{(\alpha)}(g)=d_\alpha for every \alpha\in\widehat{G} such that m^{(\alpha)}>0. The conclusion follows. \blacksquare

\text{ }

\text{ }

References:

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

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March 7, 2011 - Posted by | Algebra, Representation Theory | , ,

5 Comments »

  1. […] saw in our last post that there is a particularly fruitful way to produce normal subgroups ofa finite group . Namely, […]

    Pingback by Representation Theory: Relation Between the Kernels of Characters and Normal Subgroups « Abstract Nonsense | March 7, 2011 | Reply

  2. […] previous posts we’ve seen how we can take concepts from plain group theory and attempt to make an analogy […]

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

  3. […] shall be a numerical array 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 […]

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

  4. Hi, Din’t you mean |Re(\lambda)| \le 1? The norm of the eigenvalues of a unitary matrix is always equal to 1.

    Comment by Soumyashant Nayak | April 24, 2011 | Reply

    • Soumyashant Nayak,

      Indeed. Thank you for pointing that out!

      Best,
      Alex Youcis

      Comment by Alex Youcis | April 24, 2011 | Reply


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