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I am studying the book "Representation Theory" by Fulton and Harris. And I just can not understand the part where they prove the uniqueness of induced representation. If someone could explain it I'd greatly appreciate it! It's on page 33 and it goes:

Choose a representative $g_{\sigma} \in G$ for each coset $\sigma \in G/H,$ with $e$ representing the trivial coset $H$. To see the uniqueness, note that each element of $V$ has a unique expression $v = \sum g_{\sigma} w_{\sigma}$ for elements $w_{\sigma}$ in $W$. Given $g \in G$ write $g \centerdot g_{\sigma} = g_{\tau} \centerdot h$ for some $\tau \in G/H$ and $h \in H.$ Then we must have

$$g \centerdot (g_{\sigma} w_{\sigma})= g_{\tau}( h w_{\sigma})\;.$$

This proves The uniqueness ...

I understand everything until the end, but I just don't understand how this proves the uniqueness... If someone could give me a little more explanation, I would appreciate it! Thanks!

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up vote 1 down vote accepted

The question of uniqueness is whether we have any freedom in defining the action of an arbitrary element $g$ on $W$. The equation shows that we don't: The action of $g$ on each summand of $W$, and thus on $W$, is entirely determined by the action of $H$ on $V$. Writing out the action for a linear combination and denoting $g_\tau$ by $g_{\sigma g}$ and $h$ by $h_g$ to mark the dependencies, we have

$$ g\sum g_\sigma w_\sigma=\sum g(g_\sigma w_\sigma)=\sum g_{\sigma g}(h_gw_\sigma)\;, $$

and this fully determines the action of $g$ on any element of $V$.

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Thank you for your reply. I am still a little confused because you say this determines the action of g on any element of V and is entirely determined by the action of H on V. But what about the $g_{\sigma g}$ that appear in the summands? Isn't is dependent on them too? Sincerely. – J Kasahara Oct 16 '12 at 15:24

Given a group action $H \mapsto \mathrm{GL}(W)$, they want to extend to a group action $G \mapsto \mathrm{GL}\left( \sum_{\sigma \in G/H} \sigma W \right)$ acting on the direct sum over coset spaces.

Is the action of $g \in G$ well-defined on any element, $v = \sum g_{\sigma} w_{\sigma}$ ? Fulton's identity says we can always find a coset representative $g \centerdot g_{\sigma} = g_{\tau} h \in g_\tau H$. The coset is unique but the coset representative is only unique up to conjugacy by elements of $H$.

The action of $g$ on $g_\sigma w_\sigma$ decomposes the action of $h: W \mapsto W$ and the action of $g_\tau$ permuting the various coset spaces $W_\sigma$.

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