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on Tableaux problem 46

The statement

46.[Gaetz2018a]*Conj. 1.6 Let $\mathfrak{G}:\{e\}=G_{0}\subset G_{1}\subset G_{2}\subset\cdots$ be a tower of groups and let $r\in\mathbb{Z}_{>0}$ . Define $P$ to be the splitting diagram of the representations of the tower of groups, with weights that are the multiplicities of the splittings. The tower is $r$ -dual if $P$ is a self-dual graded graph with differential degree $r$ . The tower is $r$ -differential if the tower is $r$ -dual and the weights of $P$ are all 1, it is a differential poset. Show that: (a) If $\mathfrak{G}$ is $r$ -differential then $P(\mathfrak{G})\cong\mathbb{Y}^{\otimes r}$ and there exists an abelian group $A$ of order $r$ , not depending on $n$ , such that $G_{n}\cong A\wr S_{n}$ for all $n\geq 0$ . (b) If $\mathfrak{G}$ is $r$ -dual then $P(\mathfrak{G})\cong(d_{1}\mathbb{Y})\times\cdots\times(d_{k}\mathbb{Y})$ for some $d_{1},\ldots,d_{k}$ and there exists a group $H$ of order $r$ , not depending on $n$ , such that $G_{n}\cong H\wr S_{n}$ for all $n\geq 0$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.