Kostant cuspidal permutations

TL;DR

This paper investigates Kostant cuspidal permutations within the context of Kostant's problem, establishing invariance properties, classifying certain involutions, and exploring the complexity of Kostant cuspidal elements as the rank increases.

Contribution

It introduces the notion of Kostant cuspidal permutations, proves their invariance under Kazhdan-Lusztig left cells, and classifies infinite families of Kostant cuspidal involutions.

Findings

Kostant cuspidality is an invariant of Kazhdan-Lusztig left cells.

Four infinite families of Kostant cuspidal involutions are described.

The number of Kostant cuspidal elements can grow arbitrarily large with rank.

Abstract

In relation to Kostant's problem for simple highest weight modules over the general linear Lie algebra, we prove a persistence result for Kostant negative consecutive patterns. Inspired by it, we introduce the notion of a Kostant cuspidal permutation as a minimal Kostant negative consecutive pattern. It is shown that Kostant cuspidality is an invariant of a Kazhdan-Lusztig left cell. We describe four infinite families of Kostant cuspidal involutions, including a complete classification of Kostant cuspidal fully commutative involutions. In particular, we show that the number of new Kostant cuspidal elements can be arbitrarily large, when the rank grows. This provides some potential explanation why Kostant's problem is hard.