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

charactersin 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 $\ell$.