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WeylModules

for simple simply-connected algebraic groups

2.1

21 June 2024

Stephen Doty
Email: doty@math.luc.edu
Homepage: https://doty.math.luc.edu
Address:
Department of Mathematics and Statistics
Loyola University Chicago
Chicago, Illinois 60660 USA

Abstract

WeylModules is a GAP Package supporting computer computations with Weyl modules for simple simply-connected algebraic groups.

Copyright

© Copyright 2009--2024 by Stephen R. Doty.
This package is distributed under the terms and conditions of the GNU Public License Version 2 or (at your option) any later version.

Acknowledgements

The development of this software was initiated in June 2003 while the author was visiting the Department of Pure Mathematics and Mathematical Statistics (DPMMS) at the University of Cambridge, and continued during subsequent visits to DPMMS in June 2004 and in May through July of 2007. The author was supported by a Yip Fellowship at Magdalene College, Cambridge in 2007. The final stages of development took place in Chicago and in January 2009 at Universität Bielefeld, where the author was supported by a Mercator grant from the Deutsche Forschungsgemeinschaft (DFG).

The existence of this software owes much to the gentle prodding of Stuart Martin. Thanks are also due to Yutaka Yoshii for testing an earlier version of the software, and Matt Fayers for supplying his GAP code for computing the Mullineux map.

Contents

1.2 Counterexample to Donkin's conjecture

2.1 Constructors

  2.1-1 WeylModule
  2.1-2 QuotientWeylModule
  2.1-3 SubWeylModule
  2.1-4 SubQuotientWeylModule

  2.2-1 IsWeylModule
  2.2-2 IsQuotientWeylModule
  2.2-3 IsSubWeylModule
  2.2-4 IsSubQuotientWeylModule

2.3 Operations common to all four types

  2.3-1 BasisVecs
  2.3-2 Character
  2.3-3 DecompositionNumbers
  2.3-4 Dim
  2.3-5 DominantWeights
  2.3-6 DominantWeightSpaces
  2.3-7 Weights
  2.3-8 WeightSpaces
  2.3-9 WeightSpace
  2.3-10 TheCharacteristic
  2.3-11 TheLieAlgebra

2.4 Operations common to Weyl modules, submodules, and quotients

  2.4-1 MaximalVectors
  2.4-2 IsAmbiguous
  2.4-3 AmbiguousMaxVecs

2.5 Operations on Weyl modules and their quotients

  2.5-1 ActOn
  2.5-2 Generator
  2.5-3 GensSocleLayers
  2.5-4 PrintSocleLayers
  2.5-5 SocleSeries
  2.5-6 SocleWeyl

2.6 Operations on Weyl modules

2.6-1 MaximalSubmodule
2.6-2 SimpleQuotient

2.7 Operations on quotients

2.7-1 AmbientWeylModule
2.7-2 DefiningKernel

2.8 Operations on submodules

  2.8-1 AmbientWeylModule
  2.8-2 Generators
  2.8-3 IsWithin
  2.8-4 NextSocle
  2.8-5 GensNextSocle
  2.8-6 SocleLengthTwoQuotient
  2.8-7 TwoFactorQuotientsContaining

2.9 Operations on subquotients

  2.9-1 AmbientQuotient
  2.9-2 Generators
  2.9-3 IsWithin
  2.9-4 NextSocle

3 Weights and Characters

  3.2-1 DecomposeCharacter
  3.2-2 DifferenceCharacter
  3.2-3 ProductCharacter
  3.2-4 SimpleCharacter

4 Schur Algebras

4.1 Constructor and filter

4.1-1 SchurAlgebraWeylModule
4.1-2 IsSchurAlgebraWeylModule

4.2 Decomposition matrices

  4.2-1 SchurAlgebraDecompositionMatrix
  4.2-2 SymmetricGroupDecompositionNumbers
  4.2-3 SymmetricGroupDecompositionMatrix

  4.3-1 CompositionToWeight
  4.3-2 WeightToComposition
  4.3-3 AllPartitions
  4.3-4 BoundedPartitions
  4.3-5 Conjugate
  4.3-6 pRegular
  4.3-7 pRegularPartitions
  4.3-8 pRestricted
  4.3-9 pRestrictedPartitions
  4.3-10 Mullineux

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