Submit progress

on Tableaux problem 29

The statement

29.[Stan1999a]*Ch. 7 Exer. 71(e) Given a finite group $G$ , define $\psi_{G}$ as the character of the action of $G$ on itself by conjugation. That is, the character of permutation representation $\rho:G\rightarrow\mathfrak{S}_{G}$ defined by $\rho(x)(y)=xyx^{-1}$ , where $\mathfrak{S}_{G}$ is the group of permutations of $G$ . Define $\kappa_{n\lambda}=\left<\psi_{\mathfrak{S}_{n}},\chi^{\lambda}\right>$ , where $\chi^{\lambda}$ is the irreducible representation of $\mathfrak{S}_{n}$ indexed by the partition $\lambda$ . $\kappa_{n\lambda}>0$ with the sole exception of $n=2,\lambda=(1,1)$ . Is there a “nice” combinatorial interpretation of the numbers $\kappa_{n\lambda}$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.