For $n$ a positive integer, let $[n]=\{1,2,\ldots,n\}$. Consider two set partitions $\mathcal{A}=\{A_1,\ldots,A_p\}$ and $\mathcal{B}=\{B_1,\ldots,B_q\}$ of the set $[n]$. We will denote by $G(\mathcal{A})$ the Young subgroup of permutations $\sigma$ of $[n]$ which preserve the blocks of $\mathcal{A}$ (i.e., satisfy $\sigma(A_i)\subset A_i$ for all $i$), and similarly for $\mathcal{B}$.
We will say that $\mathcal{A}$ and $\mathcal{B}$ are transverse iff for all $i,j$, the set $A_i\cap B_j$ has at most one element. Now define the group algebra element $$ Y_{\mathcal{A},\mathcal{B}}=\left(\sum_{\sigma\in G_{\mathcal{A}}}\sigma\right)\times \left(\sum_{\rho\in G_{\mathcal{B}}}{\rm sgn}(\rho)\ \rho\right)\ . $$
These elements generalize the Young symmetrizers, the latter corresponding to the greedy case where, given $\mathcal{A}$, the blocks of $\mathcal{B}$ are as big as allowed by transversality?
Q1: Has there been a systematic study of these more general group algebra elements?
Q2: If not, why not?
Of course, with hindsight, one could say the interesting pairs of transverse partitions are the greedy ones (i.e., corresponding to rows and columns of a Young tableau), because they successfully explain the representation theory of the symmetric group. I am looking more for a reason to a priori predict this success, like: "among all pairs of transvers set partitions, the greedy ones exactly are the ones such that $Y_{\mathcal{A},\mathcal{B}}$ satisfies magic property $X$".
I was trying to read Young's QSA 1 article, and his definition comes out of the blue in Section 15.
Q3: Does anyone have any clue as to how Young got there?