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

Summary · qwen2.5:32b

The paper calculates multiplicities in the $\ell$-modular local theta correspondence for type II over a non-archimedean field, showing these multiplicities are governed by symmetric group actions on partial permutations. Unlike complex coefficients, significant multiplicities can occur here; if $d$ is the order of the residue cardinality and the rank of involved general linear groups is bounded above by $d\ell$, Pieri's Formula provides explicit algorithms to predict correspondence behavior.

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This paper discusses the l-modular local theta correspondence in type II and partial permutations. It's a complex mathematical topic that involves non-trivial multiplicities, symmetric groups, and branching problems in the modular representation theory of symmetric groups. If you're into advanced mathematics, this could be an interesting read!

Excerpt

arXiv:2601.12497v2 Announce Type: replace Abstract: 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.