Irreducible decomposition of multihypergraph orbit modules

Determine the full irreducible decomposition of the multihypergraph orbit modules arising in symmetric powers of subset permutation representations, including the Foulkes-module case associated with disjoint edges.

Background

For the action of the symmetric group on r-element subsets, the paper identifies a basis of symmetric powers using isomorphism classes of r-uniform multihypergraphs. These orbit spans are explicit permutation modules, but they are generally reducible.

The paper specifically notes that the disjoint-edge summand at n=rd is a Foulkes module and that its general irreducible decomposition is an open problem. More broadly, obtaining irreducible decompositions of the multihypergraph orbit modules would refine the orbit-based polynomial description into representation-theoretic constituents.

References

For example, at $n=rd$ the disjoint-edge summand is the Foulkes module $\Ind_{S_r\wr S_d}{S_{rd}$, whose general irreducible decomposition is an open problem .

— Graded Representation Theory of Equivariant Neural Networks  (2609.25776 - Shayestehfar, 22 Sep 2026) in Section 6, subsection “Symmetric powers are multihypergraph modules”; also Section 7, subsection “Limitations”