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Kostant's problem for permutations of shape $(n-2,1,1)$ and $(n-3,2,1)$

Samuel Creedon, Volodymyr Mazorchuk

TL;DR

The paper delivers a complete combinatorial treatment of Kostant's problem for simple modules in the sl$_n$ principal block corresponding to permutations with RS-shapes $(n-2,1,1)$ and $(n-3,2,1)$. By relating Kostant positivity to left KL preorders, consecutive patterns, and Hecke-algebra data, it derives explicit criteria and reductions via Bruhat-walks and shift maps, including a full characterization for shape $(n-2,1,1)$: a permutation is Kostant negative precisely when it consecutively contains the pattern $14325$. For shape $(n-3,2,1)$, the involutions split into 13 types; indecomposability holds for all, and Kostant negativity is governed by the presence of four patterns: $2143$, $14325$, $1536247$, or $1462537$, with asymptotic shape results established for the associated two-row shape $(2,1)$. The Indecomposability Conjecture is verified in both families, while Kåhrström’s conjecture is resolved in the $(n-2,1,1)$ case and remains conditional (via CM25-2) in the $(n-3,2,1)$ case. Together, these results provide precise, shape-specific combinatorial criteria for Kostant positivity and deepen understanding of the interaction between Hecke-algebra combinatorics and category $ ilde{O}$ representations in type $A$.

Abstract

For a permutation $z$ in the symmetric group $\mathrm{S}_{n}$, denote by $L_{z}$ the corresponding simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. In this paper, we provide a combinatorial answer to Kostant's problem for the modules $L_{z}$ when $z$ has shape (associated Young diagram/integer partition via Robinson-Schensted correspondence) equal to $(n-2,1,1)$ or $(n-3,2,1)$. Moreover, we verify that certain closely related conjectures hold for such permutations, including the Indecomposability Conjecture, which states that applying any indecomposable projective functor to the corresponding simple highest weight module outputs either an indecomposable module or zero.

Kostant's problem for permutations of shape $(n-2,1,1)$ and $(n-3,2,1)$

TL;DR

The paper delivers a complete combinatorial treatment of Kostant's problem for simple modules in the sl principal block corresponding to permutations with RS-shapes and . By relating Kostant positivity to left KL preorders, consecutive patterns, and Hecke-algebra data, it derives explicit criteria and reductions via Bruhat-walks and shift maps, including a full characterization for shape : a permutation is Kostant negative precisely when it consecutively contains the pattern . For shape , the involutions split into 13 types; indecomposability holds for all, and Kostant negativity is governed by the presence of four patterns: , , , or , with asymptotic shape results established for the associated two-row shape . The Indecomposability Conjecture is verified in both families, while Kåhrström’s conjecture is resolved in the case and remains conditional (via CM25-2) in the case. Together, these results provide precise, shape-specific combinatorial criteria for Kostant positivity and deepen understanding of the interaction between Hecke-algebra combinatorics and category representations in type .

Abstract

For a permutation in the symmetric group , denote by the corresponding simple highest weight module in the principal block of the BGG category for the Lie algebra . In this paper, we provide a combinatorial answer to Kostant's problem for the modules when has shape (associated Young diagram/integer partition via Robinson-Schensted correspondence) equal to or . Moreover, we verify that certain closely related conjectures hold for such permutations, including the Indecomposability Conjecture, which states that applying any indecomposable projective functor to the corresponding simple highest weight module outputs either an indecomposable module or zero.
Paper Structure (27 sections, 48 theorems, 394 equations, 1 figure)

This paper contains 27 sections, 48 theorems, 394 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric group
  4. Consecutive patterns
  5. Compatible Bruhat walks
  6. Robinson-Schensted correspondence
  7. Hecke algebra
  8. Kazhdan-Lusztig bases
  9. Lusztig's $\mathbf{a}$-function
  10. Dual Kazhdan-Lusztig bases
  11. Kazhdan-Lusztig preorders
  12. Categories $^{\mathbb{Z}}\mathcal{O}_{0}$ and $^{\mathbb{Z}}\mathcal{P}_{0}$
  13. Decategorification
  14. Indecomposability conjecture
  15. Kostant's problem
  16. ...and 12 more sections

Key Result

Theorem A

An involution with shape $\lambda=(n-2,1,1)$, where $n\geq 3$, is Kostant negative if and only if it consecutive contains the pattern $14325$.

Figures (1)

  • Figure 1: Vertices are all the left cells of $\mathrm{S}_{4}$, with permutations written in one-line notation. An arrow $P\rightarrow Q$ implies $p<_{L}^{(4)}q$ for all $p\in P$ and $q\in Q$. The involution $x=s_{2}s_{1}s_{3}s_{2}$ is underlined. A pattern $p$ is teal if $x\leq_{L}^{(4)}p$, and red if $x\not\leq_{L}^{(4)}p$.

Theorems & Definitions (112)

  • Theorem A
  • Theorem B
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • Example 2.4
  • Lemma 2.5
  • Example 2.6
  • Lemma 2.7
  • proof
  • ...and 102 more