The irreducible representations of the symmetric group algebras $A_n=KS_n$ over a the complex numbers (or any field of characteristic 0) $K$ satisfy the following properties:

• The irreducible representations of $A_n$ are in natural bijection to paritions of $n$.
• We have natural subalgebra inclusions $A_k \subseteq A_{k+1}$ for all $k$ and an irreducible $A_n$-representation $V$ restricts to a direct sum of distinct irreducible $A_{n-1}$-representations giving a poset structure (Hasse diagram has arrows from those restricted irreducible representations to $V$) that is isomorphic to the Young lattice for partitions (paritions ordered by "inclusion" via their Young diagrams).

Question: Is there a sequence of algebras $B_n$ that satisfy the same properties when we replace "partitions" (which are 2-dimensional) by "plane paritions" (which are 3-dimensional)?

So the following properties should be satisfied for the irreducible representations of those algebras $B_n$:

• The irreducible representations of $B_n$ are in natural bijection to plane partitions with $n$ blocks.
• We have natural subalgebra inclusions $B_k \subseteq B_{k+1}$ for all $k$ and an irreducible $B_n$-representation $V$ restricts to a direct sum of distinct irreducible $B_{n-1}$-representations giving a poset structure that is isomorphic to the lattice of plane partitions (plane paritions ordered by "inclusion").

It would be especially interesting whether this is possible when choosing $B_n$ to be a semigroup algebra. Since I have never seen such a thing for group algebras, it is probably impossible to realise plane partitions via group representations, but Im not sure.

Question 2: Is there a class of groups $G_n$ such that the group algebras $B_n=KG_n$ over the complex numbers satisfy those properties to realise plane partitions via irreducible representations?

Most likely the answer is no, but I do not see an easy argument.

If it is not possible with irreducible representations alone for algebras, maybe it is possible with indecomposable representations instead for algebras $B_n$ that might be not semisimple.