Minimal determining sets for certain $W$-graph ideals

TL;DR

This paper investigates specific Kazhdan-Lusztig cells in symmetric groups, extending previous work to identify minimal determining sets for certain W-graph ideals, thereby advancing understanding of their structure.

Contribution

It extends prior research to determine rims of additional Kazhdan-Lusztig cells and identifies minimal determining sets for new classes of W-graph ideals.

Findings

Identified rims of new families of Kazhdan-Lusztig cells.

Determined minimal sets for certain W-graph ideals.

Extended previous methods to broader classes of cells.

Abstract

We consider Kazhdan-Lusztig cells of the symmetric group $S_n$ containing the longest element of a standard parabolic subgroup of $S_n$. Extending some of the ideas in [Beitr{\"a}ge zur Algebra und Geometrie, 59 (2018), no. 3, 523-547] and [Journal of Algebra and Its Applications, 20 (2021), no. 10, 2150181], we determine the rim of some additional families of cells and also of certain induced unions of cells. These rims provide minimal determining sets for certain $W$-graph ideals introduced in [Journal of Algebra, 361 (2012), 188-212].