Examples of differential towers of groups For $r \in \mathbb{Z}_{>0}$, we say that a tower $1=G_0 \subseteq G_1 \subseteq \cdots$ of finite groups is an $r$-differential tower if for all $n$ the branching rules for restriction of irreducible (complex) representations from $G_n$ to $G_{n-1}$ are multiplicity free, and we have that
$$
Res^{G_{n+1}}_{G_n} Ind_{G_{n}}^{G_{n+1}} - Ind_{G_{n-1}}^{G_n} Res^{G_{n}}_{G_{n-1}} = r \cdot id
$$
viewed as a linear operator on the representation ring $R(G_n)$.
For example, the tower of symmetric groups $\mathfrak{S}_0 \subseteq \mathfrak{S}_1 \subseteq \mathfrak{S}_2 \subseteq \cdots$, and more generally the tower $(A \text{ wr } \mathfrak{S}_n)_{n \geq 0}$ of wreath products of a fixed abelian group $A$ and the symmetric groups give $r$-differential towers where $r=|A|$.  This is a restatement of the fact that Young's lattice $Y$ and its powers $Y^r$ are $(r$-)differential posets.
Are there any other known examples of differential towers of groups?  I would be interested in even a small modification of these families (involving, say, alternating groups instead of symmetric groups, or allowing the abelian group $A$ to depend on $n$).
 A: After thinking about the problem for a while, I recently posted this paper which answers this question when $r$ is one or prime.  I show:
Theorem: If $r$ is one or prime, then, even without requiring multiplicity-freeness, the towers $(\mathbb{Z}/r \mathbb{Z} \text{ wr } S_n)_{n \geq 0}$ are the only $r$-differential towers of groups.
And I conjecture:
Conjecture: For general $r$:


*

*If we require multiplicity-freeness, then the only examples are $(A \text{ wr } S_n)_{n \geq 0}$ with $A$ an abelian group of order $r$ (this gives the differential poset $Y^r$).

*If we allow multiplicities, then the only examples are $(H \text{ wr } S_n)_{n \geq 0}$ where $H$ is any group of order $r$ (this gives the dual graded graph $(d_1 Y) \times \cdots (d_k Y)$ where $d_1,...,d_k$ are the dimensions of the irreducible $H$-reps, and where $dY$ is a copy of $Y$ with all edges having multiplcity $d$).


I prove the Theorem by inductively showing that the groups must be complex reflection groups, and applying the Shephard-Todd classification.  This method cannot work for general $r$ since $A \text{ wr } S_n$ is not a complex reflection group if $A$ is not cyclic.
