Let $S$ be any positive semi-definite symmetric matrix (Hermitian psd matrices work as well). The **Hadamard inequality** is that
$$\det S\le\prod_{i=1}^n s_{ii}.$$
My question is whether there are some other upper bounds of $\det S$ in terms of a partial sum
$$\sum_{\sigma\in F}\epsilon(\sigma)\prod_{i=1}^n s_{i\sigma(i)},$$
where $F$ is some subset of ${\frak S}_n$. The Hadamard inequality corresponds to the case $F=({\rm id})$.

More precisely, I am interested in the validity of $$\det S\le\sum_{\epsilon(\sigma)=1}\prod_{i=1}^n s_{i\sigma(i)}.\qquad\qquad(1)$$

This latter inequality is true at least when $n\le4$, essentially because the difference $$\sum_{\epsilon(\sigma)=-1}\prod_{i=1}^n s_{i\sigma(i)}$$ contains enough many non-negative terms (the half if $n=4$, all of them if $n=2$ or $3$). This reason (which can be combined to the Cauchy-Schwarz inequality) fails if $n\ge5$, and therefore I suspect that the inequality (1) could be false if $n\ge5$.

**Edit**. Felix Goldberg's answer tells us that inequality (1) is true. More generally, Schur proved that if $G$ is a subgroup of ${\frak S}$, and $\chi$ is a character of $G$, then
$$\chi(e)\det S\le\sum_{\sigma\in G}\chi(\sigma)\prod_{i=1}^n s_{i\sigma(i)}=:d_\chi(S).$$
This gives (1) when $G={\frak A}_n$ and $\chi={\bf 1}$.

This leads me raising a second question:

If $G$ is a subgroup of ${\frak S}_n$, let $K_G$ be the set of class functions $f$ over $G$ with the property that, for every positive definite symmetric matrix $S$, one has $$f(e)\det S\le\sum_{\sigma\in G}\chi(\sigma)\prod_{i=1}^n s_{i\sigma(i)}=:d_\chi(S).$$ Clearly, $K_G$ is a convex cone, which contains all the characters. Is it equal to the convex cone spanned by the ireeducible characters of $G$ ?