On the generalised Foulkes conjecture for $\mathrm{SL}_2(\mathbb{C})$ under divisibility conditions

Abstract

The generalised Foulkes conjecture for $\mathrm{SL}_2(\mathbb{C})$ was recently proved under mild divisibility conditions by Raicu, Sam, Weyman, and Yang, who showed surjectivity of the Foulkes--Howe map through a geometric and homological approach. As an immediate corollary, we note partial proofs of related conjectures by Bergeron, Zanello, and Troyka. The main result of this paper is that the dual of the (geometric) Foulkes--Howe map is the (combinatorial) $k$-fold plethystic substitution, which admits an explicit and straightforward definition. We derive several formulas and combinatorial interpretations for its structure constants. We briefly remark on an unexpected corollary that settles a conjecture in condensed matter physics. Finally, we use the combinatorial properties of the $k$-fold map to propose candidate maps towards other variants of Foulkes' conjecture.