Abstract Nonsense

Crushing one theorem at a time

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


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

\text{ }

Thus,

\text{ }

\displaystyle \begin{aligned}\text{tr}\left(\varrho\left(\sum_{x\in C}\right)\right) &= \left(G:H\right)\sum_{j=1}^{m}\left\langle\varrho\left(\sum_{x\in C}\delta_x\right)(f_j),f_j \right\rangle\\ &=\left(G:H\right)\sum_{j=1}^{m}\left\langle \varrho\left(\sum_{x\in C\cap H}\delta_x\right)(f_j),f_j\right\rangle\\ &= \left(G:H\right)\sum_{j=1}^{m}\left\langle \varrho\left(\sum_{\ell=1}^{n}\sum_{x\in C_\ell}\delta_x\right)(f_j),f_j\right\rangle\\ &=\left(G:H\right)\sum_{\ell=1}^{n}\sum_{x\in C_\ell}\sum_{j=1}^{m}\left\langle \varrho\left(\delta_x\right)(f_j),f_j\right\rangle\end{aligned}

\text{ }

That said, note that for k\in H one has

\text{ }

\displaystyle \begin{aligned}\left\langle \varrho(\delta_k)(f_j),f_j\right\rangle &= \left\langle \text{Ind}^G_H(\rho)_k(f_j),f_j\right\rangle\\ &= \frac{1}{|G|}\sum_{g\in G}\left\langle f_j\left(k^{-1}g\right),f_j(g)\right\rangle_\mathscr{V}\\ &= \frac{1}{|G|}\sum_{h\in H}\left\langle f_j\left(k^{-1}h\right),f_j(h)\right\rangle_\mathscr{V}\\ &=\frac{1}{|H|}\sum_{h\in H}\left\langle \rho_{k^{-1}h}^{-1}(v_j),\rho_h^{-1}(v_j)\right\rangle_\mathscr{V}\\ &= \frac{1}{|H|}\sum_{h\in H}\left\langle \rho_h^{-1}\rho_k(v_j),\rho_h^{-1}(v_j)\right\rangle_\mathscr{V}\\ &= \frac{1}{|H|}\sum_{h\in H}\left\langle \rho_k(v_j),v_j\right\rangle\\ &= \left\langle \rho_k(v_j),v_j\right\rangle\end{aligned}

\text{ }

Thus, \left\langle \varrho(\delta_k)(f_j),f_j\right\rangle is equal the diagonal entry of the matrix corresponding to v_j for \rho_k with respect to the basis \{v_1,\cdots,v_m\}. Thus,

\text{ }

\displaystyle \sum_{j=1}^{m}\left\langle \varrho(\delta_k)(f_j),f_j\right\rangle=\text{tr}\left(\rho_h\right)=\chi(h)

\text{ }

And thus, finally putting it all together

\text{ }

\displaystyle \begin{aligned}\text{Ind}^G_H(\chi)(C) &=\frac{1}{\#(C)}\text{tr}\left(\varrho\left(\sum_{x\in C}\right)\right)\\ &=\frac{\left(G:H\right)}{\#(C)}\sum_{\ell=1}^{n}\sum_{x\in C_\ell}\sum_{j=1}^{m}\left\langle \varrho(\delta_x)(f_j),f_j\right\rangle\\ &=\frac{\left(G:H\right)}{\#(C)}\sum_{\ell=1}^{n}\sum_{x\in C_\ell}\chi\left(C_\ell\right)\\ &= \frac{(G:H)}{\#(C)}\sum_{\ell=1}^{n}\#\left(C_\ell\right)\chi\left(C_\ell\right)\end{aligned}

\text{ }

\text{ }

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 | , , , ,

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