A Mixed Robinson--Schensted--Knuth Construction in the Representation Theory of ${\mathrm{GL}}_n$ over Local non-Archimedean Fields

Abstract

We present a simple combinatorial method for computing the socle of representations of ${\mathrm{GL}}_n(F)$, where $F$ is a local non-Archimedean field, parabolically induced from two ladder representations satisfying a permissibility condition. We adapt recent approaches to such problems using the classical Robinson--Schensted--Knuth correspondence (RSK) and introduce a new variant of RSK that we call mixed-RSK. Some combinatorial properties of this map are investigated and then used to resolve a conjecture of Erez Lapid.

32 pages