Abstract Nonsense

Crushing one theorem at a time

Different Formula For the Character of an Induced Representation (Pt. I)


Point of post: In this post we derive a formula for the induced character of a representation different than the one derived in the previous post.

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Motivation In our last post we derived a formula for the induced character of a representation. The interesting thing is that we derived this formula using the ‘type’ of induced representation we derived second. A natural question is that if we can use our first equivalent form of induced representations to derive another formula for induced characters. In this post we do precisely that.

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Second Formula

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Let G be a finite group, H\leqslant G, and \rho:H\to\mathcal{U}\left(\mathscr{V}\right) a representation. Consider then the first form of the induced representation: \text{Ind}^G_H(\rho):G\to\mathcal{U}\left(\mathscr{X}\right). If \chi is the character for \rho we define, analogously to the previous post, the induced character \text{Ind}^G_H(\chi) to be the character of \text{Ind}^G_H(\rho). We now compute \text{Ind}^G_H(\chi) in terms of \chi. But first we make an observation:

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Lemma: Let G be a group and C a conjugacy class in G. Then, for any H\leqslant G there exists distinct conjugacy classes C_1,\cdots,C_n in H such that C\cap H=C_1\cup\cdots\cup C_n.

Proof: Indeed we let C_1,\cdots,C_n be the elements of \left\{C_h:h\in C\cap H\right\} where C_h is the conjugacy class in H of h. Clearly then for each h\in C\cap H we have that C_h\subseteq C\cap H (since C\cap H=\left\{ghg^{-1}:g\in G\right\}\cap H\supseteq\left\{khk^{-1}:k\in H\right\}=C_h) and so C_1\cup\cdots\cup C_n\subseteq C\cap H. Conversely, if h\in C\cap H then h\in C_h and so \displaystyle C\subseteq \bigcup_{h\in H}C_h=C_1\cup\cdots\cup C_n. \blacksquare

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Moreover, it’s clear that this decomposition is unique. Thus, we may state unambiguously:

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Theorem: Let G be a finite group, H\leqslant G, and \rho:H\to\mathcal{U}\left(\mathscr{V}\right) a representation of H with character \chi. Consider then the first construction of the induced representation \text{Ind}^G_H(\rho):G\to\mathcal{U}\left(\mathscr{X}\right). Then, if C is any conjugacy class and C\cap H has the unique decomposition (as described in the lemma) as C\cap H=C_1\cup\cdots\cup C_n where C_j is a conjugacy class inH then

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\displaystyle \text{Ind}^G_H(\chi)(C)=\frac{(G:H)}{\#(C)}\sum_{j=1}^{n}\#(C_j)\chi(C_j)

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(where we recall that since a character is a class function by \text{Ind}^G_H(\chi)(C) and \chi(C_j) we mean the unique value each of those characters takes on the conjugacy class).

Proof: We begin by considering the usual extension of \text{Ind}^G_H(\rho) to \mathcal{A}(G) given by

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\displaystyle \varrho:\mathcal{A}(G)\to\text{End}\left(\mathscr{X}\right):a\mapsto \sum_{g\in G}a(g)\text{Ind}^G_H(\rho)_g

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and recall that \varrho is in fact a homomorphism of \ast-algebras. Then we see that

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\displaystyle \begin{aligned}\text{tr}\left(\varrho\left(\sum_{x\in C}\delta_ x\right)\right) &=\sum_{x\in C}\sum_{g\in G}\text{tr}\left(\delta_x(g)\text{Ind}^G_H(\rho)_x\right)\\ &=\sum_{x\in C}\text{tr}\left(\text{Ind}^G_H(\rho)_x\right)\\ &=\sum_{x\in C}\text{Ind}^G_H(\chi)(C)\\ &=\#(C)\text{Ind}^G_H(\chi)(C)\end{aligned}

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Let then v_1,\cdots,v_m be a basis for \mathscr{V} and \{t_1,\cdots,t_r\} a transversal for G/H. Define then f_j for j=1,\cdots,m by

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\displaystyle f_j(g)=\begin{cases}\sqrt{(G:H)}\rho_g^{-1}(v_j) & \mbox{if}\quad g\in H\\ \bold{0} & \mbox{if}\quad g\notin H\end{cases}

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what we claim then is that if f_{i,j}=\text{Ind}^G_H(\rho)_{t_i}\left(f_j\right) for i\in[r] and j\in[m] then \left\{f_{i,j}\right\}_{i\in[r],j\in[m]}=\mathcal{B} is an orthonormal basis for \mathscr{X}. Since (G:H)\text{dim}(\mathscr{V})=\#\left(\mathcal{B}\right) which we know is equal to \dim\left(\mathscr{X}\right) it clearly suffices to prove orthonormality. To do this we note that

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\displaystyle \begin{aligned}\left\langle f_{i,j},f_{p,q}\right\rangle &= \frac{1}{|G|}\sum_{g\in G}\left\langle f_{i,j}(g),f_{p,q}(g)\right\rangle_\mathscr{V}\\ &= \frac{1}{|G|}\sum_{g\in G}\left\langle \text{Ind}^G_H(\rho)_{t_i}(f_j)(g),\text{Ind}^G_H(\rho)_{t_p}(f_q)(g)\right\rangle_{\mathscr{V}}\\ &= \frac{1}{|G|}\sum_{g\in G}\left\langle f_j\left(t_i^{-1}g\right),f_q\left(t_p^{-1}g\right)\right\rangle_{\mathscr{V}}\end{aligned}\quad\quad\mathbf{(1)}

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References:

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

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April 26, 2011 - Posted by | Algebra, Representation Theory | , , , ,

1 Comment »

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

    Pingback by Different Formula For the Character of an Induced Representation (Pt. II) « Abstract Nonsense | April 26, 2011 | Reply


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