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ALCO

Tools for algebraic combinatorics

0.2

4 February 2024

Benjamin Nasmith
Email: bnasmith@proton.me
Homepage: https://github.com/BNasmith/

Abstract

ALCO provides implementations in GAP of octonion algebras, Jordan algebras, and certain important integer subrings of those algebras. It also provides tools to compute the parameters of t-designs in spherical and projective spaces (modeled as manifolds of primitive idempotent elements in a simple Euclidean Jordan algebra). Finally, this package provides tools to explore octonion lattice constructions, including octonion Leech lattices.

Copyright

© 2024 by Benjamin Nasmith

This package may be distributed under the terms and conditions of the GNU Public License Version 3 or (at your option) any later version.

Acknowledgements

This documentation was prepared using the GAPDoc package.

Contents

3 Simple Euclidean Jordan Algebras

3.1 Filters and Basic Attributes

  3.1-1 Jordan Filters
  3.1-2 Jordan Rank
  3.1-3 Jordan Degree
  3.1-4 Trace
  3.1-5 Determinant
  3.1-6 Norm
  3.1-7 GenericMinimalPolynomial

3.2 Jordan Algebra Constructions

  3.2-1 SimpleEuclideanJordanAlgebra
  3.2-2 JordanSpinFactor
  3.2-3 HermitianSimpleJordanAlgebra
  3.2-4 JordanHomotope

3.3 The Albert Algebra

  3.3-1 AlbertAlgebra
  3.3-2 AlbertVectorToHermitianMatrix
  3.3-3 HermitianMatrixToAlbertVector

3.4 The Quadratic Representation

3.4-1 JordanQuadraticOperator
3.4-2 JordanTripleSystem

3.5 Additional Tools and Properties

  3.5-1 HermitianJordanAlgebraBasis
  3.5-2 JordanMatrixBasis
  3.5-3 HermitianMatrixToJordanVector
  3.5-4 JordanAlgebraGramMatrix
  3.5-5 JordanAdjugate
  3.5-6 IsPositiveDefinite

4 Spherical and Projective Designs

4.1 Jacobi Polynomials

4.1-1 JacobiPolynomial
4.1-2 Renormalized Jacobi Polynomials

4.2 Jordan Designs

  4.2-1 Jordan Design Categories
  4.2-2 DesignByJordanParameters
  4.2-3 Jordan Rank and Degree
  4.2-4 DesignQPolynomials
  4.2-5 DesignConnectionCoefficients

4.3 Designs with an Angle Set

  4.3-1 IsDesignWithAngleSet
  4.3-2 Design Angle Sets
  4.3-3 DesignByAngleSet
  4.3-4 DesignNormalizedAnnihilatorPolynomial
  4.3-5 DesignNormalizedIndicatorCoefficients
  4.3-6 IsDesignWithPositiveIndicatorCoefficients
  4.3-7 DesignSpecialBound

4.4 Designs with Angle Set and Cardinality

  4.4-1 Design Cardinality
  4.4-2 Designs at the Special Bound
  4.4-3 DesignAnnihilatorPolynomial
  4.4-4 DesignIndicatorCoefficients
  4.4-5 Design Strength
  4.4-6 Schemes and Tight Designs

4.5 Designs Admitting a Regular Scheme

4.5-1 DesignSubdegrees

4.6 Designs Admitting an Association Scheme

  4.6-1 DesignBoseMesnerAlgebra
  4.6-2 DesignBoseMesnerIdempotentBasis
  4.6-3 DesignIntersectionNumbers
  4.6-4 DesignKreinNumbers
  4.6-5 DesignFirstEigenmatrix
  4.6-6 DesignSecondEigenmatrix
  4.6-7 DesignMultiplicities
  4.6-8 DesignValencies
  4.6-9 DesignReducedAdjacencyMatrices

6 Closure Tools

6.1 Brute Force Method

6.2 Random Choice Methods

6.2-1 RandomClosure
6.2-2 RandomOrbitOnSets

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