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Kostant cuspidal permutations

Samuel Creedon, Volodymyr Mazorchuk

TL;DR

The paper advances Kostant's problem by introducing Kostant cuspidal permutations and proving a robust propagation principle: Kostant negativity propagates along consecutive pattern containment. It then classifies Kostant cuspidal fully commutative involutions and exhibits four infinite families of Kostant cuspidal involutions, establishing that the pool of such elements grows with rank. The results leverage deep KL-cell combinatorics, Serre quotient techniques between categories $\,\mathcal{O}$, and computational tools (GAP/CHEVIE) to connect pattern theory with representation-theoretic properties. These findings offer a structural explanation for the hardness of Kostant's problem by showing that cuspidality can persist and proliferate in larger symmetric groups.

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.

Kostant cuspidal permutations

TL;DR

The paper advances Kostant's problem by introducing Kostant cuspidal permutations and proving a robust propagation principle: Kostant negativity propagates along consecutive pattern containment. It then classifies Kostant cuspidal fully commutative involutions and exhibits four infinite families of Kostant cuspidal involutions, establishing that the pool of such elements grows with rank. The results leverage deep KL-cell combinatorics, Serre quotient techniques between categories , and computational tools (GAP/CHEVIE) to connect pattern theory with representation-theoretic properties. These findings offer a structural explanation for the hardness of Kostant's problem by showing that cuspidality can persist and proliferate in larger symmetric groups.

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.
Paper Structure (27 sections, 33 theorems, 86 equations)

This paper contains 27 sections, 33 theorems, 86 equations.

Table of Contents

  1. Introduction and description of results
  2. Preliminaries
  3. Symmetric group
  4. Hecke algebra
  5. Category $\mathcal{O}$
  6. Kostant's problem
  7. Kåhrström's conjecture
  8. Propagation of Kostant negative consecutive patterns
  9. Main result
  10. Proof of Theorem \ref{['thm-main']}
  11. Kostant cuspidal permutations
  12. Definition and first examples
  13. Left cell invariance
  14. Small ranks
  15. Type $A_6$
  16. ...and 12 more sections

Key Result

Lemma 1

Let $x,y\in S_n$ have disjoint supports, that is, any simple reflection $s$ that appears in a reduced expression of $x$ does not appear in a reduced expression of $y$ and vice versa. Then we have $\underline{{H}}_{xy}=\underline{{H}}_{x}\underline{{H}}_{y}$.

Theorems & Definitions (62)

  • Lemma 1
  • proof
  • Conjecture 2: Kåhrström
  • Theorem 3
  • Corollary 4
  • Definition 5
  • Example 6
  • Proposition 7
  • proof
  • Theorem 8
  • ...and 52 more