Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question

2 Answers 2

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 $V$.

share|improve this answer
    
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 $W_\sigma$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.