The problem [Counting cycles after permuting within rows and columns](https://mathoverflow.net/questions/254756) reminds me of the
following unpublished conjecture of mine. Let $D$ be *any* finite
planar diagram (in the sense of Young diagram, which is a special
case), say with $n$ squares. Put the numbers $1,2,\dots,n$ into the
squares of the diagram. Let $R\_D$ be the subgroup of the symmetric
group $S\_n$ permuting elements within each row, and similarly $C\_D$
for the columns. Let $\chi$ and $\psi$ be any characters of
$S\_n$. Define
 $$ u\_D=\sum\_{\substack{u\in R\_D\\ v\in C\_D}}
 \chi(u)\psi(v)p\_{\rho(uv)}, $$
where $p\_{\rho(uv)}$ is the power sum symmetric function indexed by
the cycle type of $uv$. Then (conjecturally) $u\_D$ is Schur-positive.


This conjecture is open even for diagrams of partitions when $\chi$
and $\psi$ are the trivial character. (In this case, one can show for
hook shapes that $u\_D$ is even $h$-positive, but $h$-positivity fails
in most other cases.) When $D$ is the diagram of a partition
$\lambda$, and where $\chi$ is the trivial character and $\psi$ the
sign character, we have $u\_D= H\_\lambda s\_\lambda$, where $H\_\lambda$
is the product of the hook lengths of $\lambda$. See the slides of
Valentin Féray at <http://fpsac.combinatorics.kr/program>.



A Schur positivity conjecture related to row and column permutations
I heard about this conjecture from Sara Billey at FPSAC, and I think I've got an argument. Let $F : \mathbb{C}[S\_n] \to \mathbb{Z}[x\_1, \ldots, x\_N]^{S\_N}$ be the linear map sending $w \mapsto p\_{\rho(w)}(x\_1, \ldots, x\_N)$, and $V$ a complex vector space with $\dim V = N \geq n$.



> 
> **Lemma**: If $\alpha \in \mathbb{C}[S\_n]$ acts on $V^{\otimes n}$ (on the right) with nonnegative eigenvalues, then $F(\alpha)$ is Schur-positive.
> 
> 
> *Proof*: By a density argument we can assume $\alpha$ acts diagonalizably: say $V^{\otimes n} = \bigoplus\_{\omega} U\_{\omega}$ where $U\_{\omega}$ is the $\omega$-eigenspace of $\alpha$. Each $U\_{\omega}$ is a left $\operatorname{GL}(V)$-module since the left $\operatorname{GL}(V)$-action commutes with $\alpha$. Let $X \in \operatorname{GL}(V)$ have eigenvalues $x\_1, \ldots, x\_n$. The trace of $X \times \alpha$ on $V^{\otimes n}$ (i.e. of the map $z \mapsto Xz\alpha$) is, on the one hand, $\sum\_{\omega} \omega \operatorname{tr}(X|\_{V\_\omega})$.
> Since $V\_\omega$ is a $\operatorname{GL}(V)$-module, $\operatorname{tr}(X|\_{V\_\omega})$ is a Schur-positive polynomial in $x\_1, \ldots, x\_N$. On the other hand, $\operatorname{tr}(X \times \alpha) = F(\alpha)$.
> 
> 
> 


Since $\mathbb{C}[S\_n]$ acts faithfully on $V^{\otimes d}$, the eigenvalues of $\alpha$ acting on $\mathbb{C}[S\_n]$ or on $V^{\otimes d}$ are the same, ignoring multiplicity (maybe the lemma can be modified to work directly on $\mathbb{C}[S\_n]$?). Up to constant factors, $\sum\_{u \in R\_D} \chi(u)u \sum\_{v \in C\_D} \psi(v)v$ is the product of two idempotents, which are both Hermitian with respect to the inner product on $\mathbb{C}[S\_n]$ for which permutations form an orthonormal basis. The product of two positive semidefinite matrices has nonnegative eigenvalues, so $F(\sum\_{u \in R\_D} \chi(u)u \sum\_{v \in C\_D} \psi(v)v)$ is Schur-positive by the lemma.


