Abstract Nonsense

Crushing one theorem at a time

The Connection Between the Square Root Function and the Number of Self-Conjugate Irreps (Cont.)


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

Ambivalent Conjugacy Classes

Let G be a finite group and let \mathcal{C} be a conjugacy class in G. We say that \mathcal{C} is ambivalent if \iota\left(\mathcal{C}\right)\subseteq\mathcal{C} where \iota:G\to G is the inversion map g\mapsto g^{-1}. Since G and consequently \mathcal{C} is finite and \iota:G\to G is a bijection, it follows from first principles that being ambivalent is equivalent to \mathcal{C}=\iota\left(\mathcal{C}\right). Now, for notational convenience for A\subseteq G we denote \iota\left(A\right) by A^{-1} and so ambivalence of \mathcal{C} takes either of the two equivalent forms \mathcal{C}^{-1}\subseteq\mathcal{C} or \mathcal{C}^{-1}=\mathcal{C}. We now show the relationship between ambivalent conjugacy classes of a finite group G and the number of self-conjugate \alpha\in\widehat{G}. Indeed:

 

Theorem: Let G be a finite group  with conjugacy classes \mathcal{C}_1,\cdots,\mathcal{C}_k and let \mathfrak{a} and \mathfrak{s} be the number of ambivalent conjugacy classes in G and the number of self-conjugate \alpha\in\widehat{G} respectively. Then, \mathfrak{a}=\mathfrak{s}.

 

Proof: Note first that

 

\displaystyle \frac{1}{|G|}\sum_{g\in G}\chi^{(\alpha)}(g)^2=|c_\alpha|\quad\quad\mathbf{(1)}

 

Indeed, note that if \text{Conj}^J_{\rho^{(\alpha)}} is any complex conjugate of any irrep \rho^{(\alpha)}\in\alpha then it’s clear that \chi_{\text{Conj}^J_{\rho^{(\alpha)}}}(g)=\overline{\chi^{(\alpha)}(g)} for every g\in G (this follows by considering our earlier characterization of complex conjugate maps) and since \chi^{(\alpha)}(g)=\overline{\overline{\chi^{(\alpha)}(g)}} it follows that the left-hand side of \mathbf{(1)} can be considered as \left\langle \chi^{(\overline{\alpha})},\chi^{(\alpha)}\right\rangle where \overline{\alpha} denotes the element of \widehat{G} containing \text{Conj}^J_{\rho^{(\alpha)}} from where the claim follows. Clearly then from this and the fact that each \chi^{(\alpha)}  is a class function we see that

 

\displaystyle \begin{aligned}\mathfrak{s} &= \frac{1}{|G|}\sum_{\alpha\in\widehat{G}}\sum_{g\in G}\chi^{(\alpha)}(g)^2\\ &=\sum_{j=1}^{k}\frac{\#\left(\mathcal{C}_j\right)}{|G|}\sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}\left(g_j\right)^2\end{aligned}

 

where g_j is any element of \mathcal{C}_j. Recall though that \chi^{(\alpha)}\left(g_j\right)=\overline{\chi^{(\alpha)}\left(g_j^{-1}\right)}. Thus, it follows from the second orthogonality relation that

 

\displaystyle \sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}(g_j)^2=\frac{|G|}{\#\left(\mathcal{C}_j\right)}c\left(g_j,g_j^{-1}\right)

 

But, by definition c\left(g_j,g_j^{-1}\right) is equal to 1 if and only if g_j is conjugate to g_j^{-1} which is true if and only if \mathcal{C}_j=\mathcal{C}_{g_j^{-1}} which is true if and only if \mathcal{C}_j=\mathcal{C}_j^{-1}. Thus, c\left(g_j,g_j^{-1}\right) is one if \mathcal{C}_j is ambivalent and zero otherwise. So,

 

\displaystyle \begin{aligned}\mathfrak{s} &= \sum_{j=1}^{k}\frac{\#\left(\mathcal{C}_j\right)}{|G|}\sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}\left(g_j\right)^2\\ &= \sum_{j=1}^{k}\frac{\#\left(\mathcal{C}_j\right)}{|G|}\frac{|G|}{\#\left(\mathcal{C}_j\right)}c\left(g_j,g_j^{-1}\right)\\ &= \sum_{j=1}^{k}c\left(g_j,g_j^{-1}\right)\\ &= \sum_{\mathcal{C}_j\text{ is ambivalent}}1\\ &=\mathfrak{a}\end{aligned}

 

from where the conclusion follows. \blacksquare

 

 

 References:

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

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

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March 25, 2011 - Posted by | Algebra, Group Theory, Representation Theory | , , , , , , , , ,

1 Comment »

  1. […] (where ambivalent conjugacy classes are defined as before) […]

    Pingback by Representation Theory: Irreducible R-characters of a Finite Group in Terms of its Irreducible C-characters « Abstract Nonsense | April 4, 2011 | Reply


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