This GAP Package supports digital computer computations with Weyl modules for a given simple simply-connected algebraic group \(G\) in positive characteristic \(p\). Actually the group \(G\) itself never appears in any of the computations, which take place instead using the algebra of distributions (also known as the hyperalgebra) of \(G\), taken over the prime field. One should refer to [Jan03] for the definition of the algebra of distributions, and other basic definitions and properties related to Weyl modules.
The algorithms are based on the method of [Irv86] (see also [Xi99]) and build on the existing Lie algebra functionality in GAP. In principle, one can work with arbitrary weights for an arbitrary (simple) root system; in practice, the functionality is limited by the size of the objects being computed. If your Weyl module has dimension in the thousands, you may have to wait a very long time for certain computations to finish.
The package is possibly most useful for doing computations in characteristic \(p\), where \(p\) is relatively small relative to the Coxeter number. The general theory of Weyl modules [Jan03] includes a number of basic properties that break down (or are not known to hold) if the characteristic is too small. In such cases, explicit computations are often useful.
Recall that a maximal vector is a weight vector which is killed by the positive unipotent radical; equivalently, it is killed by the positive part of the algebra of distributions.
The main technical idea underlying this package is the following fact: computing all the maximal vectors in a given Weyl module \(V\) classifies the nonzero Weyl modules \(W\) for which a nonzero homomorphism from \(W\) into \(V\) exists. Such homological information is a powerful aid to understanding structural properties of the Weyl module \(V\). The implementation of this idea involves a brute force search through each dominant weight space, examing all linear combinations (over the prime field) and compiling a list of the ones which are maximal. This exploits the pleasant fact that for Weyl modules of small dimension, the weight spaces tend to be small enough to be manageable.
Although most of the functions deal with the simple simply-connected case, there are a few functions for dealing with Schur algebras and symmetric groups. Those commands are limited in scope, and provided mainly as a convenience.
In 2019, Chris Bendel, Dan Nakano, Cornelius Pillen, and Paul Sobaje [BNPS20] found the first counterexample to Donkin's tilting module conjecture using this package. This important advance led to further development of the package.
The initial release was Version 1.0 in 2009.
1. Versions 1.0 and 1.1 were released in 2009. The initial development was spurred by work on the paper [BDM11].
2. Version 2.0 was released on 29 February 2024. The SubmoduleStructure command was eliminated. Support was added for subquotients.
3. Version 2.1 (this version) was released on 21 June 2024. The documentation was completely rewritten. There is more support for submodules and subquotients, including the new functions: SocleLengthTwoQuotient, TwoFactorQuotientsContaining. These can be used to obtain Ext information about a Weyl module.
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