Let $C$ be the field of complex number and $G$ a finite group, then define $C[G]$ be a vector space over $C$, with elemnts of $G$ as the basis. Then any element in $C[G]$ can be written as $\sum_{g \in G} a_g e_g$ where $a_g \in C$ and $g \in G$. We also have a multiplication structure on $C[G]$, carried naturally from the group structure of $G$.
My question here concerns with a hint to Exercise 4.4 in Representation Theory, by William Fulton and Joe Harris, page 518:" More generally, if $A = C[G]$ is a group algebra, call an element $a = \sum a_g e_g$ Hermitian if $a_{g^{-1}} = \overline{a_{g}}$ for all $g$ in the summation. If $a$ and $b$ are idempotent and Hermitian, then $Aab \equiv Aba$." $a$ and $b$ in the original questions are $a_{\lambda}$ and $b_{\lambda}$ defined in Young Symmetrizer. In that case we could use the fact that Young symmetrizer is idempotent.
If Hermitian here means the same thing for matrices, I imagine $a$ and $b$ here as orthogonal projections of $C[G]$, therefore they commute. But here $Aab$ and $Aba$ are really sub-modules, or left ideals of $C[G]$, and an explanation in term of these concepts would be nice.
EDIT: clarify the definition of Hermitian.