How to generalise $(\wedge^2 \chi)(g) = \frac{1}{2}(\chi(g)^2-\chi(g^2))$?

Question

One can decompose $\bigotimes^2 V = \bigvee^2 V \oplus \bigwedge^2 V$, getting a corresponding decomposition for representations, say when $V$ is a module for some finite group $G$. One then has the relation between the characters $\chi$, $\vee^2 \chi$ and $\wedge^2 \chi$ afforded by the $G$-modules $V$, $\bigvee^2 V$ and $\bigwedge^2 V$, given by $(\wedge^2 \chi)(g) = \frac{1}{2} (\chi(g)^2-\chi(g^2))$ and $(\vee^2 \chi)(g) = \frac{1}{2} (\chi(g)^2 + \chi(g^2))$.

More generally, one can decompose $\bigotimes^r V$ by Schur-Weyl duality in terms of the irreducible representations of the symmetric group $S_r$. Can we use this to give similar formulas for the representations coming up in this decomposition?

Answer 1

From the GAP manual:

The symmetrization $\chi^{[\lambda]}$ of the character $\chi$ with the character $\lambda$ of the symmetric group $S_n$ of degree $n$ is defined by $$ \chi^{[\lambda]}(g) = \frac{1}{n!}\left( \sum_{{\rho \in S_n}} \lambda(\rho) \prod_{{k=1}}^n \chi(g^k)^{{a_k(\rho)}} \right)$$
where $a_k(\rho)$ is the number of cycles of length $k$ in $\rho$.

You might also be interested in Murnaghan's refinement of these ideas which often can give smaller constituents. All of these ideas are important as they only rely on the power map of the character table, and so allow one to pretty quickly find a lot of nearly irreducible characters from very little information (which is then followed by LLL to magically make them irreducible).

Answer 2

Well, yes, but I'm not sure what Schur-Weyl duality has to do with it. The eigenvalues of $g \in G$ acting on $\Lambda^n V$ are the elementary symmetric polynomials of the eigenvalues of $g$ acting on $V$, so the character value can be read off the characteristic polynomial of $g$ as a linear endomorphism of $V$. Similarly, the eigenvalues of $g$ acting on $\text{Sym}^n V$ are the complete homogeneous symmetric polynomials of the eigenvalues of $g$, and can be related to the character of $\Lambda^n V$. With some work one can rewrite these as linear combinations of $\chi(g^m)$; for example $\displaystyle \Lambda^3 \chi(g) = \frac{1}{6} \left( \chi(g)^3 - 3 \chi(g^2) \chi(g) + 2 \chi(g^3) \right)$.