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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?

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    $\begingroup$ Q1: Ricky Ini Liu's thesis dspace.mit.edu/handle/1721.1/60196 comes to mind. $\endgroup$ Aug 19, 2020 at 20:47
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    $\begingroup$ Possibly related: mathoverflow.net/questions/254782/… $\endgroup$ Aug 19, 2020 at 21:40
  • $\begingroup$ Thanks to both. I did not know about the terminology of general "diagram" and hence did not know what to google. Indeed, this is exactly the situation I am describing with general transverse pairs of set partitions. So Q1's answer is yes, although perhaps the relevant literature might be a bit small. My question about property $X$ still stands as well as Q3. $\endgroup$ Aug 19, 2020 at 23:15
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    $\begingroup$ An interesting case is when $D$ is the Rothe diagram of a permutation: then you get the corresponding Stanley symmetric function out: see hal.inria.fr/hal-01229705/document. $\endgroup$ Aug 19, 2020 at 23:52
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    $\begingroup$ I should have added that Ricky Liu's thesis is interesting for giving some results on the general case, but much more has been done in some more restrictive situations, such as the "%-avoiding diagrams" of Shimozono and Reiner. I would love to see an actual text (as opposed to a bunch of papers) exposing this material. $\endgroup$ Aug 20, 2020 at 3:56

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