Can I get help understanding representations and subrepresentations? This is in light of the problem posted here.
I think I understand the overall idea; we want to essentially equip special vector spaces with groups to gain more insight on the group and what it can do. In the text I'm using, a subrepresentation is defined as: 
"A subrepresentation $\rho$ of a representation  $\pi: G \rightarrow GL(V)$ means
that $\rho:G\rightarrow GL(W)$, where $W$ is a subspace of $V$ such that $\pi(g)W \subset W$ for all $g \in G$ and $\rho(g)$ is the restriction of $\pi(g)$ to $W$, that is,
$$\pi(g)|_w=\rho(g) \forall g \in G.$$ "


*

*What does $\pi(g)W \subset W$ mean? Does that mean take some element in $W$ and multiply $\pi(g)$ by that element and check to see if it's in $W$?

*What exactly is the restriction? I talked about it once in another course, but I don't really understand the idea, or how it applies to subrepresentations. Is there an intuitive way to think of it?

*For the problem in the link posted above, how would I start in ensuring that it is a subrepresentation of $\mathbb{C}^3$? Thanks in advance!
 A: *

*$\pi(g)W \subset W$ means, quite literally, that $\pi(g)W = \{ \pi(g)(w) \mid w \in W\}$ is a subset of $W$. If you view $\pi(g)$ as a linear operator on $V$, this says that the operator maps all of $W$ into itself. In the language of linear algebra, we say that $W$ is a $\pi(g)$-invariant subspace.

*The restriction of a linear map is the exact same linear map, except its domain is restricted to a possibly smaller subspace. If $W$ is $\pi(g)$-invariant, then the linear operator $\pi(g):V \to V$ restricts to an operator $\rho(g):W \to W$ by the formula $\rho(g)(w) := \pi(g)(w)$. In other words, $\rho(g)$ does exactly the same thing as $\pi(g)$, but its domain and codomain are smaller. The fact that $W$ is $\pi(g)$-invariant ensures that one may restrict the codomain of $\rho(g)$ to $W$.

*In that setting $V = \mathbb{C}^3$ and $\pi$ is the permutation representation of $S_3$. Let $W = \{(z_1,z_2,z_3) \in \mathbb{C}^3 \mid \sum z_i = 0\}$. For any $w = (z_1,z_2,z_3) \in W$ and any $g \in S_3$ we have $\pi(g)(w) = (z_{g(1)},z_{g(2)},z_{g(3)})$. Since the coordinates of $w$ sum to zero, the coordinates of $\pi(g)(w)$, which are the same three numbers but permuted in some fashion, must also sum to zero. This shows that $W$ is $\pi(g)$-invariant and thus that $\pi$ restricts to a two-dimensional subrepresentation on $W$.
