Representation Confusion

On Friday, my inorganic professor has succeeded in confusing me completely when discussing the representation of groups (as matrices). I am going to try explaining it to myself here; feel free to correct me.

Well, ok. Somewhat loosely defined, a symmetry operation g is an action performed on an object (e.g. a molecule) that leaves the object's position unchanged. If we define the binary operation * as composition of operations (e.g. for any two symmetry operations g and h, g*h means perform h on the object first, then g, successively) The set of all such operations that can be done on a particular object constitutes a group, called the point group of the object (if the operations are done about a point), because the set with operation * satisfies all the properties needed for a group (closure, associativity, existence of an identity and an inverse for each element in the set).

Groups, apparently, are a little too abstract. Since we usually apply symmetry operations on (well, ok, n = 3), on a molecule with several identical atoms at certain positions, it's often useful to represent each symmetry operation as something else. In particular, matrices are nice and easy to work with - problems in groups are now problems in linear algebra.

So we pick a basis in (the canonical one is perfectly good) and write each symmetry operation as a matrix with respect to that basis. Why we can write each symmetry operation as a matrix: we can write operations C_n and S_n in terms of matrices (using completely geometric arguments) and suitable compositions of these two operations will create any other symmetry operation desired. It also follows that all symmetry operations are linear maps from , because matrices are linear maps.

Incidentally, all the maps are bijective because they are invertible (by definition of a group, every element must have an inverse).

Somewhat confusingly, we can also view a symmetry operation as a map that permutes a set of points (especially with rotation). For example, we might view C₄ as an operation that permutes (1, 2, 3, 4) to (4, 1, 2, 3). Confusing, because a "permutation space" isn't a vector space (what does it mean to add two elements in the set?); it doesn't have a basis or anything like that. But one can easily abuse notation and write the ordered set (1, 2, 3, 4) as if it's a column vector, and apply a matrix operation to it (in simple cases like C₄, it even gives correct answers). That's what happened in class.

Anyway. One nice thing about these matrices is that we can block-diagonalize them, as well as calculate their trace (which, for some reason, people call the "character" of the matrix). Each block is a matrix (fill in zeros where the other blocks were to get a 3 by 3 matrix), called an "irreducible representation". It turns out that there are a finite number of these blocks per set of matrices in a point group, and that these blocks form an orthogonal basis for the set (...proof?). So now we have: the set of linear maps (matrices) associated with a point group is a finite-dimensional vector space too.

Then there were character tables and a whole bunch of unproved equations, and I remain confused. Need to find Cotton's book and look it up there.