Plane partitions as irreducible representations

Question

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 over a finite semigroup or at least finite dimensional algebras.

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.

Answer 1

The "path model" described in Section 2.3.11 of Goodman, de la Harpe, and Jones' book "Coxeter graphs and towers of algebras" gives a construction for a sequence of finite-dimensional semisimple algebras whose branching rules give any graded poset with finite ranks that you like (or more generally, weighted versions too). Furthermore, this sequence is essentially uniquely determined.

It is an interesting question whether the algebras one gets from this construction have other natural descriptions, for example as semigroup algebras as you suggest, but they definitely exist.