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 $G$ ?
> 
> 
> 



Hadamard-like inequalites for positive definite symmetric matrices
If I interpret correctly the terms and notations, this seems to be a result of Schur. See p.4 [here](http://math.ecnu.edu.cn/~zhan/papers/ZhanICCM.pdf).


