TITLE: How to interpret the phrase "transforms under the irreducible representation"?
QUESTION [6 upvotes]: I'm reading Robert Gilmore's "Lie Groups, Physics, and Geometry," and trying to understand his brief presentation of Galois theory. I think I get the gist of the method, but would be grateful for help understanding some details. Let me first quote a bit from page 10, highlighting the terminology I don't understand, and after the quotes, I will focus on specifics. Please bear with me while I try to clarify my questions:

The general quadratic equation has the form
  $$(z-r_1)(z-r_2)=z^2-I_1 z+I_2=0$$
  $$\begin{equation}\tag{1.17}I_1=r_1 + r_2\end{equation}$$
  $$I_2=r_1 r_2$$
  The Galois group is $S_2$ with subgroup chain shown in Fig. 1.3. [elided]
  The character table for the commutative group $S_2$ is
  $$\begin{equation}\tag{1.18}\begin{array}{r|crc}
\text{Irreducible Rep's} & I & (12) & \text{Basis Functions} \\ \hline 
\Gamma^1 & 1 & 1 & u_1 = r_1 + r_2 \\
\Gamma^2 & 1 & -1 & u_2 = r_1 - r_2 
\end{array}\end{equation}$$
   Linear combinations of the roots that transform under the one-dimensional irreducible representations $\Gamma^1, \Gamma^2$ are
  $$\begin{equation}\tag{1.19}\begin{bmatrix} u_1 \\ u_2 \end{bmatrix}=\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} r_1 \\ r_2 \end{bmatrix}=\begin{bmatrix} r_1+r_2 \\ r_1-r_2 \end{bmatrix}\end{equation}$$

The equations (1.17) and (1.19) are trivial to understand. What's bugging me is (1.18) and how the group representations fit into the scheme. Please bear with me while I clarify my questions.
I understand that $\Gamma^1$ and $\Gamma^2$ are one-dimensional (scalar) representations of the group $S_2 = \{I, (12)\}$. That means that the group elements of $S_2$ map to the elements of $\Gamma^1$ in such a way that (group-theoretic) products of elements in $S_2$ map to (matrix) products of the corresponding elements in $\Gamma^1$; ditto $\Gamma^2$.  
I see that we want to apply the elements of $S_2$ to the sequence of roots $(r_1, r_2)$, permuting them, and then examine the corresponding actions on the linear combinations $r_1+r_2$ and $r_1-r_2$. 
Now arises the first question: how did we find these linear combinations? It looks like magic that the $\Gamma$'s, castrated one-dimensional group representations that they are, just copied into a square matrix and applied to a column vector of the roots, deliver up exactly the linear combinations that we need to investigate, the ones that will generate solutions to the quadratic equation. This gets even more spooky as we graduate up the line to the cubic and quartic, where the character tables feed us, similarly, exactly the linear combinations we need to solve the general equations. Of course, when we get to quintics, the game is over, and that's the whole point of (at least this elementary corner of) Galois theory.
My confusion deepens. Gilmore writes, on page 9, that these linear combinations are "basis functions" for the irreducible representations, the $\Gamma$s. I usually conceptualize archetypes for "basis functions" as sine waves for Fourier transforms, or Dirac deltas, or other orthonormal polynmomials, and such. I'm scratching my head wondering what on Earth Gilmore could mean. The irreducible representations are elements of a vector spaces, to be sure, and there should be various bases for these vector spaces, to be sure. But how are the linear combinations basis functions for the $\Gamma$? Functions of what, to what? What's the domain (the set of roots, $\{r_1, r_2\}$?)? What's the range? The linear combinations of roots don't map to the set of roots, and I can't see the sense in which they might be bases for the vector space in which the one-dimensional irreducible representations $\Gamma^1$ and $\Gamma^2$ live. 
I now sink completely when Gilmore says that the "linear combinations of roots transform under the one-dimensional irreducible representations...". Transform? To what? "Transform" is a transitive verb, and it needs a subject and an object, doesn't it? These one-dimensional irreducible representations are just scalar numbers, so using them to transform a linear combination of roots can only be to multiply the linear combinations by the scalars? Or did we magically back out from the representations to the group, $S_2$, where I can understand how to apply the group elements, $I$ and $(12)$, to the sequence of roots, permuting them. 
I can see the machinery at work, it's not difficult at all, but I can't connect what Gilmore writes with what I think I know. I am sure he uses abbreviated terminology, to save space, but I am not connecting. 
I apologize in advance if I haven't clarified my questions sufficiently or if it's too long, but I hoped that someone out there with mastery of the subject might clear the fog for me.

REPLY [4 votes]: Because the many different types of applications of repn theory are themselves important, and the comparisons among them shed light on aspects otherwise possibly overlooked... perhaps another "answer" here won't hurt: I think there is yet-another reasonable segue from more primitive notions to "repn theory"...
(Also, it is not clear to me that anyone needs to read a whole book about repns of finite groups in characteristic $0$, especially over $\mathbb C$. Many of these suffer from "completeness", so that critical ideas are swamped by compulsion to include all-possible...)
When we have a finite group $G$ with $n$ elements acting linearly on a vectorspace $V$ over a field $k$ of characteristic $0$ (so we can divide by positive integers such as $n$), it is easy to form $G$-invariant elements, by averaging: given $v\in V$, the element $\frac{1}{n}\sum_{g\in G} g\cdot v$ is easily checked to be invariant, by changing variables in the sum. Further, if $v$ was already $G$-invariant, this map just returns $v$. That is, this averaging is some kind of projector to the "trivial $G$-isotype" in $G$, that is, things un-moved by $G$. This is where expressions like $r_1+r_2$ come from. The application to Galois theory has the vector space $V$ be the larger field.
Slightly more generally, let $\chi:G\rightarrow k^\times$ be a group hom (maybe to ${\pm 1}\subset k^\times$). To say that a vector "transforms by/under/whatever-verb" is that $g\cdot v=\chi(g)v$. In the previous paragraph, $\chi=1$. It is a small step from the pure averaging as in the previous paragraph to try $\hbox{proj}_\chi v=(?) \frac{1}{n}\sum_{g\in G} \chi(g)\,g\cdot v$. In fact, the obvious change-of-variables argument shows that this weighted average transforms by $\chi^{-1}$, so, in fact, and inverse has to be inserted somewhere, for example, $\hbox{proj}_\chi v=\frac{1}{n}\sum_{g\in G} \chi^{-1}(g)\,g\cdot v$.
As far as Galois theory goes, to solve equations in radicals following Lagrange and Vandermonde (when possible), the previous paragraph explains why we'd want to adjoin sufficiently many roots of unity to the ground field, and how to form expressions varying by prescribed characters $\chi$ of the group. The expressions are Lagrange resolvents.
An interesting complication occurse for non-abelian groups $G$, that in an action of $G$ on $V$, there are typically more vectors than those that transform by a one-dimensional repn $\chi$ as above. Namely, $V$ may break up into "irreducible" pieces some of which are larger than one-dimensional. In this case, it requires more set-up to explain what "transforms by" could mean, as well as to write down the corresponding "projector". (There still is a projector, namely, averaging against the function $\Theta_\pi(g)=\hbox{trace}(\pi(g))$ where $\pi:G\rightarrow GL_n(k)$...) 
The "traditional" treatment of Galois theory occurs much prior to the traditional treatment of repn theory, and the traditional treatments of both ignore each other... for no mathematical reason... so there aren't so many available references.