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Let $H$ be a subgroup of a finite group $G$.Given an irreducible representation $\pi$ of $G$,we may decompose its restriction to $H$ into irreducible $H$- representations.Show that every irreducible representation of $H$ can be obtained in this way. My initial idea was to use induced representations,but later I wanted prove this result without appealing to that concept.I am stuck with this problem for quite some time.Please help.Thanks.

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1 Answer 1

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I'm not really sure why you'd insist you don't want to use induced representations here; Frobenius reciprocity means it's the natural approach, and this is an immediate corollary of Frobenius reciprocity...

But OK. Suppose there is an irreducible representation $\sigma$ of $H$ which isn't contained in the restriction of an irreducible representation of $G$. Then $\sigma$ doesn't occur in the restriction to $H$ of the regular representation $\Bbb{C}[G]$ of $G$, which contains a copy of $\Bbb{C}[H]$, and hence of $\sigma$, so you're done.

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  • $\begingroup$ Can you please elaborate your last line a bit? $\endgroup$
    – Ester
    Commented Nov 10, 2015 at 17:22
  • $\begingroup$ There's a canonical embedding $\Bbb{C}[H]\subset\Bbb{C}[G]$ (given by taking coefficients to be zero for basis elements in $G$ but not in $H$), and restriction of $\Bbb{C}[G]$ to $H$ corresponds simply to restricting the action from $G$ to $H$, so $\Bbb{C}[H]$ is a $H$-submodule of $\Bbb{C}[G]$. $\endgroup$
    – PL.
    Commented Nov 10, 2015 at 19:31
  • $\begingroup$ Where have you used the fact that the given representation is irreducible? $\endgroup$
    – Ester
    Commented Nov 10, 2015 at 19:41

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