Erez Lapid’s conjecture on socles of induced representations

The conjecture predicts that a representation built from ladder representations has an irreducible socle, namely $\operatorname{soc}(\Lambda(\mathfrak m)) \cong Z(\mathfrak m)$, with multiplicity one. The 2019 work established occurrence through an iterated-socle construction but explicitly left irreducibility open.

Known results

• For every multisegment $\mathfrak m$, $Z(\mathfrak m)$ occurs in the RSK-standard module $\Lambda(\mathfrak m)$ through an iterated-socle construction (2019).

August 27, 2026 mixed-RSK preprint

A new preprint introduces mixed-RSK and applies it to representations induced from two ladder representations, claiming to resolve the corresponding socle calculation. This is a substantive advance for that representation class, but it does not establish the full conjecture in the supplied evidence and remains unverified.

Current status (as of August 2026): The conjecture is proved only in the established partial settings; mixed-RSK claims progress for induction from two ladder representations, while the general statement remains open and unverified.