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

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



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

Requirements: Python 2.7, NZMATH.


python 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 D 8 4 -v
type: D
*** 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]


This project is licensed under the GNU General Public License v3.0.

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