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A characterization of Kazhdan-Lusztig right cells containing smooth elements

Zhanqiang Bai, Zheng-an Chen

TL;DR

The paper provides a complete classification of Kazhdan–Lusztig right cells for $W=S_n$ in type $A$ whose elements are all smooth Schubert; it translates smoothness into pattern-avoidance and tableaux data via the Robinson–Schensted correspondence and establishes precise tableau-shape criteria. It also explores families of right cells with only non-smooth elements using invariant subsequences that persist under Knuth moves, and introduces the SEF algorithm to identify smooth elements within a given right cell, leveraging hook-length and determinantal formulas to count cells and classify elements. The findings illuminate the relationship between KL right cells, associated varieties, and smoothness of Schubert varieties, and reveal that smooth right cells become increasingly rare as $n$ grows, with practical implications for understanding irreducible associated varieties and smooth loci in type $A$ geometry.

Abstract

Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$. Its Weyl group is the symmetric group $S_n$. In this paper, we want to describe some Kazhdan-Lusztig right cells containing smooth elements which parameterize the smooth Schubert varieties. These elements are closely related to the study of associated varieties of highest weight modules of $\mathfrak{sl}(n,\mathbb{C})$. Firstly, we give a complete classification of the KL right cells containing only smooth elements. Then we give a sufficient condition for a KL right cell to contain only non-smooth elements by using invariant subsequences and a sufficient condition for a KL right cell to contain some smooth elements. Finally, we give an efficient algorithm to find out all the smooth elements in a given KL right cell.

A characterization of Kazhdan-Lusztig right cells containing smooth elements

TL;DR

The paper provides a complete classification of Kazhdan–Lusztig right cells for in type whose elements are all smooth Schubert; it translates smoothness into pattern-avoidance and tableaux data via the Robinson–Schensted correspondence and establishes precise tableau-shape criteria. It also explores families of right cells with only non-smooth elements using invariant subsequences that persist under Knuth moves, and introduces the SEF algorithm to identify smooth elements within a given right cell, leveraging hook-length and determinantal formulas to count cells and classify elements. The findings illuminate the relationship between KL right cells, associated varieties, and smoothness of Schubert varieties, and reveal that smooth right cells become increasingly rare as grows, with practical implications for understanding irreducible associated varieties and smooth loci in type geometry.

Abstract

Let be the Lie algebra . Its Weyl group is the symmetric group . In this paper, we want to describe some Kazhdan-Lusztig right cells containing smooth elements which parameterize the smooth Schubert varieties. These elements are closely related to the study of associated varieties of highest weight modules of . Firstly, we give a complete classification of the KL right cells containing only smooth elements. Then we give a sufficient condition for a KL right cell to contain only non-smooth elements by using invariant subsequences and a sufficient condition for a KL right cell to contain some smooth elements. Finally, we give an efficient algorithm to find out all the smooth elements in a given KL right cell.
Paper Structure (12 sections, 23 theorems, 33 equations, 1 figure, 1 table)

This paper contains 12 sections, 23 theorems, 33 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Associated variety
  4. Pattern avoidance
  5. Hecke algebra and cells
  6. Robinson--Schensted insertion algorithm
  7. Smooth KL right cells
  8. Examples of KL right cells with only non-smooth elements
  9. Invariant subsequences with the pattern $3412$
  10. Invariant subsequences with the pattern $4231$
  11. Young tableaux with two columns
  12. Right cells containing some smooth elements

Key Result

Proposition 2.3

Let $\mathfrak{g}$ be a reductive Lie algebra and $I$ be a primitive ideal in $U(\mathfrak{g})$.Then $V(I)$ is the closure of a single nilpotent coadjoint orbit $\mathcal{O}_I$ in $\mathfrak{g}^*$. In particular, for a highest weight module $L(\lambda)$, we have $V(\mathrm{Ann} (L(\lambda)))=\overli

Figures (1)

  • Figure 1: Cells containing smooth elements, total cells and their ratio

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: Jo85
  • Proposition 2.4: Jo84
  • Proposition 2.5: BoB3
  • Definition 2.6
  • Proposition 2.7: LS
  • Proposition 2.8: Spa
  • Proposition 2.9: Jo84; BoB3
  • Definition 2.10: Robinson--Schensted insertion algorithm
  • ...and 36 more