Low-rank cells and p-cells

This repository contains code and additional calculations for the paper

The subregular and submaximal p-cells

by Vanessa Miemietz, Marie Roth and Daniel Tubbenhauer.

The paper is available on the arXiv.

The paper determines the subregular and submaximal p-cells in finite Weyl groups. In classical type we give explicit descriptions of the relevant p-canonical basis elements and the resulting p-cells; in exceptional type some of the low-rank calculations are done by computer. This repository contains the corresponding ordinary cell data, p-cell data, raw output and Magma code.

Contact

If you find any errors in the paper please email me:

dtubbenhauer@gmail.com

Same goes for any errors related to this page.

Main files

low-rank-cells.pdf

This PDF records ordinary Kazhdan--Lusztig two-sided cells for finite Coxeter groups in low rank, together with some extra information used in the paper.

For each type considered, the tables include:

• the cell numbering;
• the size of each two-sided cell;
• Lusztig's a-value;
• the number of parabolic longest elements in each cell;
• whether the cell is strongly regular;
• for cells which are not strongly regular, the matrix of intersection sizes between right and left cells;
• in several cases, the corresponding asymptotic Hecke algebra or category.

As a quick check on the parabolic data, the numbers of parabolic longest elements over all two-sided cells add up to 2^rank(W).

low-rank-p-cells.pdf

This PDF records low-rank two-sided p-cells computed from the p-canonical basis. It is the p-analogue of low-rank-cells.pdf.

The note includes:

• characteristic zero comparison data;
• the number and sizes of the p-cells;
• intersection matrices for cells which are not strongly regular;
• positive-characteristic data for types B and C;
• low-rank exceptional data for G2, F4 and E6;
• asymptotic category labels in the cases where we have identified them.

For large intersection matrices we use a compressed notation, explained in the PDF.

pre-computed-data.zip

This archive contains the raw output files used to make low-rank-p-cells.pdf. The file names follow the convention

<type>-<rank>-<characteristic>.txt

for example:

b-6-2.txt
c-6-2.txt
d-6-2.txt
e-6-3.txt
f-4-2.txt
g-2-3.txt

The raw files contain the full cell output behind the tables in the PDF.

GraphCellsH.m

This Magma script computes the p-canonical basis for a finite Coxeter group and then its left, right and two-sided p-cells. For each two-sided p-cell it prints the right-cell/left-cell intersection matrix. It also prints multiplication tables in the p-canonical basis for the diagonal H-cells.

The script computes the p-canonical basis elements themselves as well. We do not list these in low-rank-p-cells.pdf, since there are far too many of them.

For large types the full computation can take a while. The optional save directory allows an interrupted p-canonical basis calculation to be resumed.

Requirements

The Magma script uses:

1. Magma;
2. ASLoc, with ASLoc.spec visible from the working directory;
3. IHecke, available through ASLoc.

A convenient setup is to run the script from a directory where Magma can see ASLoc.

Examples

Create save and output directories once:

mkdir -p saves outputs

Small examples:

magma -b type:=G2 char:=2 saveDir:=saves GraphCellsH.m > outputs/g-2-2.txt
magma -b type:=G2 char:=3 saveDir:=saves GraphCellsH.m > outputs/g-2-3.txt
magma -b type:=F4 char:=2 saveDir:=saves GraphCellsH.m > outputs/f-4-2.txt
magma -b type:=F4 char:=3 saveDir:=saves GraphCellsH.m > outputs/f-4-3.txt

Some larger examples used in the data files are:

magma -b type:=B6 char:=2 saveDir:=saves GraphCellsH.m > outputs/b-6-2.txt
magma -b type:=C6 char:=2 saveDir:=saves GraphCellsH.m > outputs/c-6-2.txt
magma -b type:=D6 char:=2 saveDir:=saves GraphCellsH.m > outputs/d-6-2.txt
magma -b type:=E6 char:=3 saveDir:=saves GraphCellsH.m > outputs/e-6-3.txt

The main arguments are:

• type: Cartan type, for example B6, C6, D6, G2, F4, E6;
• char: characteristic, either 0 or a prime;
• saveDir: optional directory for resumable p-canonical basis data;
• targetLength: optional length cutoff for tests;
• profileName: optional Magma profiling label;
• chatty: optional ASLoc verbosity.

Relation to the paper

The ordinary and p-cell tables collect the low-rank calculations used for comparisons and checks in the paper. The Magma code is also used for the exceptional-type computations where the p-cell structure is determined computationally.

If you only want the results, start with low-rank-cells.pdf and low-rank-p-cells.pdf. If you want to reproduce the calculations, use GraphCellsH.m together with the raw files in pre-computed-data.zip.

The suggested citation is:

@misc{MRTpcells,
  author = {Miemietz, Vanessa and Roth, Marie and Tubbenhauer, Daniel},
  title = {Code, data and more for ``The subregular and submaximal p-cells''},
  year = {2026},
  url = {https://github.com/dtubbenhauer/pcells}
}

Repository contents

README.md
LICENSE
GraphCellsH.m
low-rank-cells.pdf
low-rank-p-cells.pdf
pre-computed-data.zip

Erratum

Empty so far.