a question about basis functions of a group

I'm learning about group representation theory, and I got a little trouble in understanding basis functions (I have just learnt the basis functions of point groups).

My main questions is :

Can I consider basis functions of a point group to be a complete basis for constructing any functions that can be described by the point group? For instance, I have the set of basis functions of $D_{3}$ group, does it mean that I can use the set of functions to construct any objects whose symmetry could be described by $D_3$ group?

I learnt about quantum physics before, and I just took it for granted that any wave function can be represented as a linear combination of eigen vectors of a Hamiltonian. By my reckoning, those eigen functions just serve as x axis and y axis in a $2D$ plane, and can be used to decompose any vectors in the Hilbert space where they live.

But now, basis functions come to me as something related to the symmetry of some object. I have understood wonderful orthogonality theorem (WOT), and those theorems of orthogonality that could be proved by wonderful orthogonality theorem.

But I still cannot build a good relation between the two pictures

1. Eigen functions of a Hamiltonian spans a Hilbert space in a sense that we can use (x, y) to label a point in a $2D$ plane.
2. Due to WOT and other related theorems, for a point group, we can construct a set of basis functions. The number of basis functions is determined by the order of the group $h$ by the relation $\sum_{j} l_{j}^{2} = h$.

Why the dimension of a space is determined by the order of a group (a point group)? And how to make sense that there be a space of really high dimension (a really big point group) lives in a trivial $3D$ space?

• Why down vote? have I missed some points? Oct 23, 2017 at 6:08
• Can someone give me a quick answer for my main problem first? Oct 23, 2017 at 7:08
• There seem to be a lot of non-standard terms here that I have trouble decoding. For one thing, I assume that point group really just means group? And what is a basis function? Oct 23, 2017 at 9:23
• @TobiasKildetoft the group that I meant is a point group, I have just learnt point groups. And the basis functions that I'm referring to are basis of irreducible representations of a group. And the whole thing is about group representation theory, and I learnt the thing basically from link: web.mit.edu/course/6/6.734j/www/group-full02.pdf Oct 23, 2017 at 9:30
• So it seems from that link that there is some physics going on. Mathematically, there is no such thing as a point group, and I am still not sure what you mean by a basis function. Do you mean one element for each irreducible representation, or a basis for some specific representation? Oct 23, 2017 at 9:32