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Construction of precise matchings for the local homology of finite and affine Artin groups

# giove91/precise-matchings

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# Precise matchings

Construction of precise matchings for the local homology of finite and affine Artin groups (see https://arxiv.org/abs/1709.01358).

## Installation

It is enough to clone this repository. No installation is needed. After cloning, you can run the test suite via `python test.py` or `pypy test.py`.

Requirements: Python 2.7, NZMATH.

## Usage

`python check_matching.py A|B|D|E|F|H|tA|tB|tC|tD|tE|tF|tG|tI n [d] [-v|-vv] [-l]`

The first argument is the Coxeter type, where `t` stands for "tilde" and denotes affine types.

The second argument `n` (integer >= 1) is the size of the Coxeter system.

The third optional argument `d` (integer >= 2) indicates which local component to check. If `d` is not specified, all relevant local components are checked.

Optional arguments `-v` and `-vv` ask for more output, and `-l` asks for a LaTeX-friendly description of the torsion part of the local homology (one row per homology group, starting from the 0-th).

By default, the program constructs a matching and checks that it is precise. It also computes the ranks of the boundary matrices of the Morse complex (they coincide with the ranks of the d-localized homology groups). With the `-v` option, critical simplices (with their d-weights) are also printed. With the `-vv` option the matching itself is also printed, together with the non-zero incidence numbers between critical simplices in the Morse complex.

### Example: D_8, d=4

`python check_matching.py D 8 4 -v`
``````type: D
n=8
*** d=4 ***
Critical simplices:
(1, 2, 3, 6, 7) 	w=1
(2, 3, 6, 7) 	w=0
(2, 3, 5, 6, 7) 	w=1
(1, 3, 6, 7) 	w=0
(1, 2, 3, 4, 5, 6, 7) 	w=3
(1, 2, 3, 4, 6, 7) 	w=2
(1, 3, 4, 6, 7) 	w=1
(1, 2, 3, 4, 6, 7, 8) 	w=3
(1, 2, 3, 5, 6, 7) 	w=2
(1, 2, 3, 4, 5, 6, 7, 8) 	w=4
The matching is precise.
Ranks (from 1-dim to 8-dim): [0, 0, 0, 0, 2, 1, 1, 1]
``````

## Licence

Construction of precise matchings for the local homology of finite and affine Artin groups

## Releases 1

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Sep 28, 2017

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