Math Software

Sheafhom: Sparse Linear Algebra and Topology

Sheafhom is a free package for large sparse linear algebra computations over the integers and other exact number types. It includes graphs and windows to show the progress of the computation in real time--see the demo. The package is motivated by algebraic topology and number theory.

Papers

Sheafhom has been used for work on the cohomology of congruence subgroups of arithmetic groups, specifically Γ₀(N) in SL(4, Z).

• Resolutions of the Steinberg Module for GL(n), with Avner Ash and Paul Gunnells. Is paper V in the series below. J. Algebra, 2012.
• Torsion in the cohomology of congruence subgroups of SL(4, Z) and Galois representations, with Avner Ash and Paul Gunnells. Is paper IV in the series below. J. Algebra, 2011.
• Lecture notes on Hecke Operators for Arithmetic Groups via Cell Complexes. April 14, 2009.
• Cohomology of congruence subgroups of SL(4, Z) III, with Avner Ash and Paul Gunnells. Math. Comp., 2010.
• Lecture notes on Hecke Operators for Arithmetic Groups via Cell Complexes. June 5, 2008. Has more about the number theory than the 4/14/2009 notes.
• Cohomology of congruence subgroups of SL(4, Z) II, with Avner Ash and Paul Gunnells. J. Number Theory, 2008.
• Sheafhom: software for sparse integer matrices. A survey article in the 2007 volume of Pure and Applied Math Quarterly honoring the sixtieth birthday of Bob MacPherson.
• Cohomology of congruence subgroups of SL(4, Z), with Avner Ash and Paul Gunnells. J. Number Theory, 2002.

Downloads

Here is a prerelease copy of Sheafhom 2.2 as of March 4, 2008, with notes on installation and the tutorial. The code is stable and has good documentation. What's missing from an official 2.2 release is an updated tutorial and a description of how the code has improved since 2.1:

• better support for the fields Z/(p), not just Z;
• a better implementation of Z/(p) on 64-bit machines;
• routines that handle large matrices by swapping intermediate data between RAM and the disk.

Here is Sheafhom 2.1. The latest release version is dated March 17, 2005. The zip file contains

• an introduction, in the form of a talk proposal in abstract.txt
• installation notes in README.txt
• the source code
• reference documentation

Click here for Sheafhom 2.0, which is in Java.

Here is the source code for Sheafhom 1.x from 1999. It is in Common Lisp, specially optimized for CMU Common Lisp. Compared to Sheafhom 2.x, it has much more support for algebraic topology: vector spaces and maps over Q, (co)chain complexes, direct sums, tensor products, wedge products, topological cell complexes, sheaves on cell complexes, injective resolutions of complexes of sheaves, and toric varieties. Sheafhom 2.x is essentially the linear-algebra back end of the 1999 Sheafhom, but over principal ideal domains like Z instead of Q, and with better performance in speed and memory use.

Group Theory

Repthy is a freeware Java package for finite groups and the characters of their representations over the complex numbers. It has a demo.