  We prove some Schur positivity results for the chromatic symmetric function
$X_G$ of a (hyper)graph $G$, using connections to the group algebra of the
symmetric group. The first such connection works for (hyper)forests $F$: we
describe the Schur coefficients of $X_F$ in terms of eigenvalues of a product
of Hermitian idempotents in the group algebra, one factor for each edge (a more
general formula of similar shape holds for all chordal graphs). Our main
application of this technique is to prove a conjecture of Taylor on the Schur
positivity of certain $X_F$, which implies Schur positivity of the formal group
laws associated to various combinatorial generating functions. We also
introduce the pointed chromatic symmetric function $X_{G,v}$ associated to a
rooted graph $(G,v)$. We prove that if $X_{G,v}$ and $X_{H,w}$ are positive in
the generalized Schur basis of Strahov, then the chromatic symmetric function
of the wedge sum of $(G,v)$ and $(H,w)$ is Schur positive.
