Let us "take" a finite group G. Here "take" I mean any type of group-theoretic description you prefer: e.g. as an explicit subset of GL (or other group) or Cayley table, whatever.
Question: How should we compute representation theoretic information about the group ?
By "representation theoretic information" I mean
a) number of irreps over complex numbers
b) their dimensions
c) their characters
d) matrices defining the representations
Any other kind of comments on further questions like modular or rational representations is also welcome.
What complexity should we expect from these algorithms ? And, are the current algorithms best possible or developing group theory may lead to more effective algorithms ?
Let me clarify my question by example.
For example - number of irreps = number of conjugacy classes. So if I would be asked question a) (i.e. compute the number of irreps) I would do the following. Just take element - compute its conjugacy class, take element, out this conjugacy class - compute its conjugacy class and so on. Finally I get the number of conjugacy classes and hence I get number of irreps.
So pay attention a) I assume that all operations like "take element", "are two elements equal", "take inverse of an element" has same complexity say "1".
b) this algorithm is based on the not so trivial fact that number of irreps = number of conjugacy classes.
Subquestion are there less complexity algorithms ?