# On a certain expansion in term of Schur functions

This question is related to this other one

A Schur positivity conjecture related to row and column permutations

by Richard Stanley (thanks to Sam Hopkins for letting me know about it).

Consider a Young subgroup $$S_{\lambda}$$ of the symmetric group $$S_n$$, corresponding to some integer partition $$\lambda$$ of $$n$$. Let $$\tau$$ be some permutation and define the symmetric function

$$F(\tau)=\sum_{\sigma\in S_{\lambda}}p_{c(\tau\sigma)}$$ where $$p_{\mu}$$ is the usual power sum symmetric function and $$c(\rho)$$ denotes the integer partition given by the cycle-type of the permutation $$\rho$$.

Q: What is known about the Schur function expansion of $$F(\tau)$$, given the double coset class of $$\tau$$ for the Young subgroup?

## 1 Answer

One fact is that $$F(\tau)$$ is Schur positive if and only if $$\tau\in S_\lambda$$. More generally, if $$K$$ is any coset (left or right) of any subgroup $$G$$ of $$S_n$$, then $$\sum_{\sigma\in K}p_{c(\sigma)}$$ is Schur positive if and only if $$K=G$$. The only known proof for the "if" part requires representation theory; see Enumerative Combinatorics, vol. 2, page 396. For the "only if" part, it is easy to see that if a nonzero linear combination $$\sum_{\lambda\vdash n} a_\lambda p_\lambda$$ of power sums is Schur positive, then $$a_{1^n}>0$$.

• Thank you for letting me know about this iff characterization. Do you know any articles where the $F(\tau)$ have been discussed? – Abdelmalek Abdesselam Aug 20 '20 at 20:33
• @AbdelmalekAbdesselam: I don't know of any such articles. – Richard Stanley Aug 22 '20 at 1:34