There is a well-known construction of minimal idempotents in the group algebra of the symmetric group $\mathbb C[S\_n]$ using row symmetrizers and column antisymmetrizers. But these idempotents are not \*-idempotents with respect to the \*-structure on $\mathbb C[S\_n]$ which sends $g$ to $g^{-1}$. (In other words, they are not orthogonal projections with respect to the usual inner product.)


Is there in the literature somewhere a construction of minimal \*-idempotents for $\mathbb C[S\_n]$? My goal is to do some explicit calculations, so simple and explicit constructions are preferred.



Minimal *-idempotents for the group algebra of the symmetric group
There is a well-known construction of the primitive idempotents for the symmetric group over the rationals that is due to Murphy (and possibly Jucys?). It can be found in his paper "A new construction of Young's seminormal representation of the symmetric group", J. Algebra, 69 (1981), 287-297. 


To describe this for $1\le k\le n$ let $L\_k=(1,k)+\dots+(k,k-1)$ be the $k$th Jucys-Murphy element. Notice that $L\_k^\*=L\_k$ for all $k$. You can check that $L\_kL\_m=L\_mL\_k$, for $1\le k,m\le n$. Now for each standard tableau $t$ define
$$ E\_t = \prod\_{k=1}^n\prod\_{c\ne c\_k(t)}\frac{L\_k-c}{c\_k(t)-c}, $$
where $c\_k(t)$ is the content of $k$ in $t$ --- that is, $c\_k(t)=c-r$ if $k$ appears in row $r$ and column $c$ of $t$. In the second product $c$ runs over all possible possible contents in all standard tableaux of size $n$.


Murphy shows that $\{E\_t\mid t \text{ standard}\}$ is a complete set of pairwise orthogonal idempotents in $\mathbb{Q}S\_n$. In fact, this is almost immediate if you just consider how these elements act on the seminormal representations of the symmetric group. For a more modern account see Okounkov and Vershik's paper "A new approach to representation theory of symmetric groups", Selecta Math. (N.S.), 2 (1996), 581–605.


By construction, $E\_t^\*=E\_t$ so these idempotents have the property that you want.


For calculations these idempotents can be constructed recursively. They become a mess, however, if you expand them in the permutation basis of the group algebra, so you should try and avoid this if possible.


