The $\ell$-modular local theta correspondence in type II and partial permutations

In this paper we compute the multiplicities appearing in the ${\overline{\mathbb{F}}_\ell}$-modular theta correspondence in type II over a non-archimedean field $\mathrm{F}$, where $\ell$ is a prime not dividing the residue cardinality of $\mathrm{F}$. Unlike for representations with complex coefficients, highly non-trivial multiplicities can emerge. We show that these multiplicities are precisely governed by the action of symmetric groups on the set of partial permutations, and the ${\overline{\mathbb{F}}_\ell}$-representation of symmetric groups these give rise to. The problem is thus reduced to certain branching problems in the modular representation theory of symmetric groups. In particular, if $d$ is the order of the residue cardinality of $\mathrm{F}$ in ${\overline{\mathbb{F}}_\ell}$, and the rank of the involved general linear groups is bounded above by $ d\ell$, the behavior of the theta correspondence can be predicted via explicit algorithms coming from Pieri's Formula.