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SingerAlg — A GAP 4 Package

(Version 1.0.1)

Thomas Breuer
Email: sam@Math.RWTH-Aachen.De
Homepage: http://www.math.rwth-aachen.de/~Thomas.Breuer

Copyright

© 2019–2021

This package may be distributed under the terms and conditions of the GNU Public License Version 3 or later, see http://www.gnu.org/licenses.

Contents

1 Introduction to the SingerAlg Package

1.1 Aims of the SingerAlg Package

1.2 Theoretical Background: Singer algebras

1.3 GAP-Julia Integration in the SingerAlg Package

1.4 Installation of the SingerAlg Package

1.5 Acknowledgements

2 Tutorial for the SingerAlg package

2.1 How to Study Singer Algebras

2.2 Number Theoretic Caveats

2.2-1 Large values of e
2.2-2 Primality tests

2.3 Some Examples from the Papers

2.4 Example: The case n = 4

3 Functions for Singer algebras

3.1 Singer Algebras as Algebraic Structures

  3.1-1 SingerAlgebra
  3.1-2 IsSingerAlgebra
  3.1-3 ParametersOfSingerAlgebra
  3.1-4 DimensionsLoewyFactors
  3.1-5 LoewyVectorAbbreviated and LoewyVectorExpanded
  3.1-6 LoewyLength
  3.1-7 SufficientCriterionForLoewyBoundAttained
  3.1-8 MinimalDegreeOfSingerAlgebra
  3.1-9 RadicalSeriesOfAlgebra
  3.1-10 SocleSeriesOfAlgebra
  3.1-11 LoewyStructureInfo
  3.1-12 CoefficientsQadicReversed

3.2 Visualization of Singer Algebras

  3.2-1 DisplaySingerMonomials
  3.2-2 BrowseSingerMonomials
  3.2-3 Changing the display format: User preference DisplayFunction

3.3 Singer Algebras as Combinatorial Structures

  3.3-1 GeneratingSubsetOfCanonicalBasisOfSingerAlgebra
  3.3-2 SingerAlg.MultTable
  3.3-3 SingerAlg.BasisOfSum and SingerAlg.BasisOfIntersection
  3.3-4 SingerAlg.BasisOfProductSpace
  3.3-5 SingerAlg.BasisOfIdeal
  3.3-6 SingerAlg.BasisOfAnnihilator
  3.3-7 SingerAlg.BasesOfRadicalSeries
  3.3-8 SingerAlg.BasesOfSocleSeries
  3.3-9 SingerAlg.BasisOfPowers
  3.3-10 SingerAlg.BasisOfPMRoots
  3.3-11 SingerAlg.BasisOfPC

3.4 Permutation Isomorphism of Singer Algebras

3.4-1 SingerAlg.IsInducedAlgebraIsomorphism
3.4-2 SingerAlg.ProposedPermutationIsomorphism

3.5 Invariants of Singer Algebras

  3.5-1 ConsiderInvariantsByParameters
  3.5-2 SingerAlg.InfoFromInvariantString
  3.5-3 SingerAlg.NumberOfProductsInSubspace
  3.5-4 SingerAlg.LL4QuoDerDim
  3.5-5 SingerAlg.SubquoDerDim

3.6 Miscellaneous variables related to Singer Algebras

  3.6-1 SingerAlg
  3.6-2 OrderModExt
  3.6-3 SingerAlg.ContentsOfDataFile
  3.6-4 GrayCodeSwitchIndexIterator
  3.6-5 SingerAlg.RightDerivationsDimension

4 The Database of Low Dimensional Singer Algebras

4.2 Access to the Database of Singer Algebras

  4.2-1 OneSingerAlgebraInfo and AllSingerAlgebraInfos
  4.2-2 DisplaySingerAlgebras
  4.2-3 BrowseSingerAlgebras
  4.2-4 IdSingerAlgebra

4.3 On the Classification of Singer Algebras by Isomorphism Type

  4.3-1 The Datastructure that Describes our Knowledge about the Distribution to Isomorphism Types
  4.3-2 Isomorphism of Algebras with Loewy Vector (1, k, 1, ..., 1)
  4.3-3 Canonical Isomorphisms of Singer Algebras
  4.3-4 Permutation Isomorphisms of Singer Algebras
  4.3-5 Combinatorial Invariants Distinguishing Singer Algebras
  4.3-6 Other Invariants Distinguishing Singer Algebras
  4.3-7 Inspect some of the Canonical Isomorphisms

4.4 Files of the Database of Singer Algebras

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