Multiple factors of the character of a representation

2010-11-18T12:52:45Z

In algebra, various objects admit a unique decomposition into irreducible elements. For instance integers $n\ge1$, univariate polynomials $p\in k[X]$ (even multivariate ones), or characters in representation theory of finite groups. In each situation, an irreducible occurs with a multiplicity. It is interesting, from a theoretical point of view, to have a reduction to the situation where every multiplicity is $1$ (or $0$ if you insist to write the product/sum with all irreducibles of the structure). This can be done explicitly in the case of polynomials, by dividing $p$ by the g.c.d. of $p$ and $p'$, the latter being calculated with the help of the Euclid algorithm.

Is there something similar for characters in representation theory of finite groups ? 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. Given a character $\chi$, is it possible for instance to split it as a sum $\chi_1+\cdots+\chi_r$, where $\chi_\ell$ gathers the irreducible characters entering in $\chi$ with multiplicity $\ell$ ?

Perhaps the question should be restricted to complex characters; who knows ? Even a weaker property could be interesting, provided it is associated with a finite-time algorithm.

Of course, I have in mind to apply such a property to the regular representation. Then $\chi_\ell$ would be $\ell$ times the sum of irreducible characters of degree $\ell$.

Answer by Jim Humphreys for Multiple factors of the character of a representation

2010-11-18T14:46:18Z

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.

Multiple factors of the character of a representation - MathOverflow

Multiple factors of the character of a representation

Answer by Jim Humphreys for Multiple factors of the character of a representation