alexbecker / exact-bgc

Compute the stable representation decomposition of the cohomologies of the pure braid groups.

exact-bgc

Copyright 2014 Alex Becker

Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at

http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.

This license applies to all code in this repository.

This library is designed for computing the stable representation decompositions of the cohomology groups H^i(PConf_n(C)) of the pure braid groups. Calculations I have performed so far are located in results/.

GMP is required to use this library. It can be installed using your package manager or from http://gmplib.org. To compile the library, run "make MAX_N=n" where n is the largest value of n you want to work with, usually 3i+1 if you want to compute H^i(PConf_n(C)).

To calculate the representation decomposition, run:

./evaluate n max_i threads prime1 prime2 ... primek

where threads is the number of threads to use, prime1 through primek are prime numbers or prime powers. This will produce the decompositions for i up to max_i. Choose prime1 through primek to be larger than you expect any of the multiplicities of any irrep in H^i(PConf_n(C)) will be, and such that their product is larger than the dimension of any H^i(PConf_n(C)). If full_decomp.out.primej match for all j, this certifies the results.

The files in cluster/ are intended for use on a distributed cluster where sharing memory is slow or impossible. These are intended for use only by experienced Linux users; the following procedure should work for Ubuntu machines, but has only been tested on Ubuntu Trusty on AWS c3 instances and your machine may vary in ways I cannot predict. Compile the cluster versions with "make cluster MAX_N=n". If you have compiled anything for a different value of n, run "make clean" first. On a machine with network permissions appropriate for an NFS filesystem, run aws_master.sh to set it up as a master. Replace each of the values in aws_slave.sh enclosed in percent signs with the appropriate value. Run aws_slave.sh on each slave to connect it to the master and automatically start computing the character tables. Once every slave has finished, run

python combine_partial_tables.py n max_i

on a single machine to combine the tables. Then run

./evaluate_from_file n max_i filename threads prime1 prime2 ... primek

and verify the result as in the non-cluster case.

A note on performance: The parallelization of character_table works very well on machines with up to 16 cores, but on 32 cores the overhead becomes significant. Therefore it is ideal to run on a cluster of 16 core machines. The parallelization of evaluate_from_file is less good, and it gains little with more than 4 cores, but it is very fast relative to character_table so this is not a major issue.

RESULTS:

Results computed so far are located in results/. Run

python visualizations.py max_i

to see visualizations of the results for i up to max_i.

CONJECTURES:

It is conjectured that the multiplicity of V(lambda) in H^i(PConf_n(C)) stabilizes at n=k+i+1, where k is the sum of lambda. This can be tested for i up to max_i by running

./evaluate_multiple_n max_n max_i threads prime

where n = 3*max_i + 1 and prime is sufficiently, then running

python test_stabilization_conjecture.py max_i

and checking that it returns true. Testing on a cluster has yet to be automated.

KNOWN BUGS:

• evaluate_from_file has a memory leak when using multiple primes.