1
$\begingroup$

Ex 2.8 of Serre's book "Linear Representations of Finite Groups" says: Let $\rho:G\to V$ be a representation ($G$ finite and $V$ is complex, finite dimensional) and $V=W_1\oplus W_1 \oplus \dotsb \oplus W_2 \oplus \dotsb W_2\oplus \dotsb \oplus W_k$ be its explicit decomposition into irreducible subrepresentations. We know $W_i$ is only determined up to isomorphism, but $V_i := W_i\oplus \dotsb \oplus W_i$ is uniquely determined.

The question asks: Let $H_i$ be the vector space of linear mappings $h:W_i\to V_i$ such that $\rho_s h = h\rho_s$ for all $s\in G$. Show that $\dim H_i = \dim V_i / \dim W_i$ etc...

But I don't even understand what $\rho_s h = h\rho_s$ means in this case: to make sense of $h\rho_s$, don't we need to first fix some decomposition $W_i\oplus \dotsb \oplus W_i$, and consider $\rho$ restricted to one of these $W_i$? Is that what the question wants? But then how do I make sense of $\rho_s h$?

$\endgroup$
1
$\begingroup$

I'm not sure I understand what is worrying you, but each $W_{i}$ is a $G$-submodule, for any $w \in W_{i}, w\rho_{s} \in W_{i}.$ Then we can apply the map $h,$ s $(w\rho_{s})h$ is an element of $V_{i}.$ On the other hand, $wh \in V_{i}.$ We know that $V_{i}$ is also a $G$-submodule, so $(wh)\rho_{s}$ is an element of $V_{i}.$ The question asks you to consider those maps $h$ such that these two resulting elements of $V_{i}$ coincide for each $w \in W_{i}$ and each $s \in G.$ ( I am assuming that $\rho_{s}$ means the linear transformation associated to $s \in G$ by the representation $\rho).$

Clarification: Suppose that $V_{i}$ is a direct sum of $n_{i}$ isomorphic irreducible submodules. Strictly speaking, these could be labelled $W_{i_{1}},\ldots ,W_{i_{n_{i}}}.$ The intention is to look at maps from $W_{i_{1}}$ to $V_{i},$ but the resulting dimensions would be the same if any $W_{i_{j}}$ was used.

$\endgroup$
  • $\begingroup$ Does that mean $h$ is a map from each $W_i$ into $V_i$? I wasn't used to this because I thought in order to make sense we must specify a fixed domain for $h$, i.e. just one $W_i$ out of the (possibly) many. $\endgroup$ – user71815 Jun 9 '13 at 9:26
  • $\begingroup$ But then, say if $h(W_i)$ is in another (different) copy of $W_i \subseteq V_i$, how would we make sense of this for all $W_i$, simultaneously? $\endgroup$ – user71815 Jun 9 '13 at 9:29
  • $\begingroup$ I see now what worries you. I think the intention is that you fix one choice of a summand $W_{i}.$ They are all isomorphic. You don't have to worry about doing this for several $W_{i}$ simultaneously, although whichever single one you choose, the result is isomorphic. $\endgroup$ – Geoff Robinson Jun 9 '13 at 10:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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