MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 added 510 characters in body

It's important here to work over a field of characteristic 0, since the traditional notion of character gets too complicated otherwise (and is better replaced by the notion of Brauer character). Moreover, the field should at least initially be a splitting field for the group, say algebraically closed.

In the highlighted part of the question, you say: Suppose we know only the cardinals of conjugacy classes of G, together with the table of multiplication of these classes. But we don't know the table of characters. But knowing how the sums over classes multiply would allow you to compute the character table, following ideas which go back to Burnside. See for example Section 33 in the old Curtis-Reiner book Representation Theory of Finite Groups and Associative Algebras (1962). It's rare to know so much in advance anyway when the group is at all complicated, which makes the question look a bit artificial to me. On the other hand, one can certainly study the isotypic components of the regular representation in the spirit of Wedderburn structure theory. For an arbitrary given representation, there is a similar isotypic decomposition. But for character theory the most natural question is how to obtain the full decomposition into irreducibles.

By now there are sophisticated computational methods available for working with characters of fairly large finite groups, but this viewpoint gets farther from the theoretical question here.

ADDED: Though I'm still uncertain what the actual question here is, it's worth pointing to the discussion in Sections 2.6-2.7 of Linear Representations of Finite Groups by the other Serre. There one finds a "canonical" decomposition of a given representation space corresponding to the way its character would decompose into multiples of irreducible characters. How effectively one can find such a decomposition depends of course on precisely what is already known about the finite group in question.

1

It's important here to work over a field of characteristic 0, since the traditional notion of character gets too complicated otherwise (and is better replaced by the notion of Brauer character). Moreover, the field should at least initially be a splitting field for the group, say algebraically closed.

In the highlighted part of the question, you say: Suppose we know only the cardinals of conjugacy classes of G, together with the table of multiplication of these classes. But we don't know the table of characters. But knowing how the sums over classes multiply would allow you to compute the character table, following ideas which go back to Burnside. See for example Section 33 in the old Curtis-Reiner book Representation Theory of Finite Groups and Associative Algebras (1962). It's rare to know so much in advance anyway when the group is at all complicated, which makes the question look a bit artificial to me. On the other hand, one can certainly study the isotypic components of the regular representation in the spirit of Wedderburn structure theory. For an arbitrary given representation, there is a similar isotypic decomposition. But for character theory the most natural question is how to obtain the full decomposition into irreducibles.

By now there are sophisticated computational methods available for working with characters of fairly large finite groups, but this viewpoint gets farther from the theoretical question here.