Arbitrary functions can be expanded by basis functions of invariant subspaces of a group

On page 96 of M. S. Dresselhaus's Applications of Group Theory to the Physics of Solids, it is said that

Any arbitrary function $$F$$ can be written as a linear combination of a complete set of basis functions $$f^{\Gamma_{n'}}_{j'}$$
$$F= \sum_{n'}\sum_{j'} f^{\Gamma_{n'}}_{j'} |\Gamma_{n'}, j'\rangle$$

in which $$|\Gamma_{n'}, j'\rangle$$ represents the $$j'\text{-th}$$ basis function of the $$n'\text{-th}$$ irreducible representation of some finite group.

I don't understand why an arbitrary function can be expanded.

Here is my understanding:

Given a finite group $$G$$, by studying its character table, we know the invariant subspaces of the group. Finding the set of basis functions of each of the invariant subspaces of $$G$$ gives us the set of basis functions {$$|\Gamma_{n'}, j'\rangle$$}.

What is so special about this set of functions? Clearly, the size of the group determines the size of the set {$$|\Gamma_{n'}, j'\rangle$$}, does it mean that a smaller set of {$$|\Gamma_{n'}, j'\rangle$$} has the same 'expressive power' as a larger set?

EDIT:

Inspired by ZeroTheHero's answer, here is what I understood:

A real-valued function defined on space $$X$$ is a map $$\text{A point in } X \rightarrow \Bbb R$$ Given a finite group $$G$$, we construct a linear representation of which in space $$X$$. Since the representation can be decomposed into the irreducible representations of $$G$$, space $$X$$ can be decomposed into the invariant subspaces of $$G$$. So we say any arbitrary function F can be written as a linear combination of a complete set of basis functions.

Given a space $$X$$, there are many sets of basis functions that we can find. The invariant subspaces of a group (as long as a linear representation of the group can be constructed in $$X$$) just provides a way to partition the space and simultaneously leads us to those basis functions.

• Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. – Qmechanic Oct 2 '18 at 14:17
• I voted to migrate this to Math.SE. – AccidentalFourierTransform Oct 2 '18 at 14:40

The simplest example would be the function $$f=x_1+y_2$$, which is clearly not invariant under permutation since $$P_{12}f= x_2+y_1$$. However, the functions $$f_+= x_1+y_2+ x_2+y_1\, ,\qquad f_-= x_1+y_2 -(x_2+y_1)$$ transform irreducibly under permutation (here up to a sign) of the indices, and indeed $$x_1+y_2 = \frac{1}{2}(f_++f_-)$$ Another obvious example would be the two-particle spin state $$\vert +\rangle_1\vert -\rangle_2 = \frac{1}{2} \left(\vert +\rangle_1\vert -\rangle_2+ \vert +\rangle_1\vert -\rangle_2\right)+ \frac{1}{2} \left(\vert +\rangle_1\vert -\rangle_2- \vert +\rangle_1\vert -\rangle_2\right)$$ where $$\frac{1}{2} \left(\vert +\rangle_1\vert -\rangle_2 \pm \vert +\rangle_1\vert -\rangle_2\right)$$ are symmetric/antisymmetric w/r to permutations.
This holds quite generally. Maybe the better known example would be how an generic function $$f(\theta,\varphi)$$ can be expanded as a sum of spherical harmonics $$Y_{\ell,m}(\theta,\varphi)$$, keeping in mind the spherical harmonics are basis functions for $$SO(3)$$ irreps.