Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cell Classification of Gelfand $S_n$-Graphs

Yifeng Zhang

TL;DR

The paper addresses the classification of cells for Gelfand $S_n$-graphs arising from the Iwahori–Hecke algebra by introducing RS-like insertions. It constructs two RS-like frameworks, row Beissinger insertions for $\\Gamma^{\mathsf{row}}$ and column Beissinger insertions for $\\Gamma^{\mathsf{col}}$, and proves that every molecule in these graphs coincides with a cell. This result completes the cell classification for type $A$ Gelfand $S_n$-graphs and solidifies the connection between canonical-basis actions, insertion combinatorics, and the cell/molecule structure of $W$-graphs. The work leverages the quasiparabolic setup, dual module constructions, and canonical bases to derive explicit edge-structures and dominance relations that underpin the cell conjecture proofs. The findings have significance for understanding the representation-theoretic filtration induced by $W$-graphs and for combinatorial models of Hecke-algebra actions in type $A$.

Abstract

Kazhdan and Lusztig introduced the $W$-graphs, which represent the multiplication action of the standard basis on the canonical bais in the Iwahori-Hecke algebra. In the Hecke algebra module, Marberg defined two generalied $W$-graphs, called the Gelfand $W$-graphs. The classification of the molecules of the type $A$ Gelfand $S_n$-graphs are determined by two RSK-like insertion algorithms. We finish the classification of cells by proving that every molecule in the $S_n$-graphs is indeed a cell.

Cell Classification of Gelfand $S_n$-Graphs

TL;DR

The paper addresses the classification of cells for Gelfand -graphs arising from the Iwahori–Hecke algebra by introducing RS-like insertions. It constructs two RS-like frameworks, row Beissinger insertions for and column Beissinger insertions for , and proves that every molecule in these graphs coincides with a cell. This result completes the cell classification for type Gelfand -graphs and solidifies the connection between canonical-basis actions, insertion combinatorics, and the cell/molecule structure of -graphs. The work leverages the quasiparabolic setup, dual module constructions, and canonical bases to derive explicit edge-structures and dominance relations that underpin the cell conjecture proofs. The findings have significance for understanding the representation-theoretic filtration induced by -graphs and for combinatorial models of Hecke-algebra actions in type .

Abstract

Kazhdan and Lusztig introduced the -graphs, which represent the multiplication action of the standard basis on the canonical bais in the Iwahori-Hecke algebra. In the Hecke algebra module, Marberg defined two generalied -graphs, called the Gelfand -graphs. The classification of the molecules of the type Gelfand -graphs are determined by two RSK-like insertion algorithms. We finish the classification of cells by proving that every molecule in the -graphs is indeed a cell.

Paper Structure

This paper contains 16 sections, 35 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasiparabolic sets
  4. Hecke algebra modules
  5. Canonical bases
  6. Cells
  7. Cells
  8. $W$-graphs
  9. Proof of cell conjecture in $\mathbf{m}$ case
  10. Row Beissinger insertion
  11. One-direction edges and dominance order for m
  12. Proof of cell conjecture
  13. Proof of cell conjecture in $\mathbf{n}$ case
  14. Column Beissinger insertion
  15. One-direction edges and dominance order for n
  16. ...and 1 more sections

Key Result

Theorem 1.1

If $y$ and $z$ are in the same cell of $\Gamma^{\mathsf{row}}$ or $\Gamma^{\mathsf{col}}$, then they are in the same molecule. In other words, all molecules of $\Gamma^{\mathsf{row}}$ and $\Gamma^{\mathsf{col}}$ are cells, respectively.

Theorems & Definitions (76)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6: Rains and Vazirani RV
  • Remark 2.7
  • Lemma 2.8: Rains and Vazirani RV
  • Definition 2.9
  • ...and 66 more