hecke

Calculating decomposition matrices of Hecke algebras

1.5.4

27 August 2024

This package may be distributed under the terms and conditions of the GNU Public License Version 2 or higher.

Acknowledgements

Hecke is a port of the GAP 3 package Specht 2.4 to GAP 4. Specht 2.4 was written by Andrew Mathas, who allowed Dmitriy Traytel to use his source code as the basis for hecke.

1 Decomposition numbers of Hecke algebras of type A

1.1 Description

1.2 The modular representation theory of Hecke algebras

1.3 Two small examples

1.4 Overview over this manual

1.5 Credits

2 Installation of the hecke-Package

3 Specht functionality

3.1 Porting notes

3.1-1 Structure of Hecke
3.1-2 Renamings

3.2 Specht functions

  3.2-1 Specht
  3.2-2 Simple information access
  3.2-3 The functions MakeSpecht, MakePIM and MakeSimple
  3.2-4 Decomposition numbers of the symmetric groups
  3.2-5 Hecke algebras over fields of positive characteristic
  3.2-6 The Fock space and Hecke algebras over fields of characteristic zero
  3.2-7 Schur
  3.2-8 DecompositionMatrix
  3.2-9 CrystalDecompositionMatrix
  3.2-10 DecompositionNumber

3.3 Partitions in Hecke

3.4 Inducing and restricting modules

  3.4-1 RInducedModule
  3.4-2 SInducedModule
  3.4-3 RRestrictedModule
  3.4-4 SRestrictedModule

3.5 Operations on decomposition matrices

  3.5-1 InducedDecompositionMatrix
  3.5-2 IsNewIndecomposable
  3.5-3 InvertDecompositionMatrix
  3.5-4 AdjustmentMatrix
  3.5-5 SaveDecompositionMatrix
  3.5-6 CalculateDecompositionMatrix
  3.5-7 MatrixDecompositionMatrix
  3.5-8 DecompositionMatrixMatrix
  3.5-9 AddIndecomposable
  3.5-10 RemoveIndecomposable
  3.5-11 MissingIndecomposables

3.6 Calculating dimensions

3.6-1 SimpleDimension
3.6-2 SpechtDimension

3.7 Combinatorics on Young diagrams

  3.7-1 Schaper
  3.7-2 IsSimpleModule
  3.7-3 MullineuxMap
  3.7-4 MullineuxSymbol
  3.7-5 PartitionMullineuxSymbol
  3.7-6 GoodNodes
  3.7-7 NormalNodes
  3.7-8 GoodNodeSequence
  3.7-9 PartitionGoodNodeSequence
  3.7-10 GoodNodeLatticePath
  3.7-11 LittlewoodRichardsonRule
  3.7-12 InverseLittlewoodRichardsonRule
  3.7-13 EResidueDiagram
  3.7-14 HookLengthDiagram
  3.7-15 RemoveRimHook
  3.7-16 AddRimHook

3.8 Operations on partitions

  3.8-1 ECore
  3.8-2 IsECore
  3.8-3 EQuotient
  3.8-4 CombineEQuotientECore
  3.8-5 EWeight
  3.8-6 ERegularPartitions
  3.8-7 IsERegular
  3.8-8 ConjugatePartition
  3.8-9 PartitionBetaSet
  3.8-10 ETopLadder
  3.8-11 Dominates
  3.8-12 LengthLexicographic
  3.8-13 Lexicographic
  3.8-14 ReverseDominance

3.9 Miscellaneous functions on modules

  3.9-1 Specialized
  3.9-2 ERegulars
  3.9-3 SplitECores
  3.9-4 Coefficient
  3.9-5 InnerProduct

3.10 Semi-standard and standard tableaux

  3.10-1 Tableau
  3.10-2 SemiStandardTableaux
  3.10-3 StandardTableaux
  3.10-4 ConjugateTableau
  3.10-5 ShapeTableau
  3.10-6 TypeTableau

References

Index

hecke

Calculating decomposition matrices of Hecke algebras

Copyright

Acknowledgements

Contents