The ALCO Package for GAP

The ALCO package provides tools for algebraic combinatorics, most of which was written for GAP during the author’s Ph.D. program. This package 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.

Legal

The ALCO package provides tools for algebraic combinatorics in GAP.

Copyright (C) 2024 Benjamin Nasmith

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/.

Setup

1. Install GAP. The ALCO package was prepared using version 4.12.

2. Clone this repository in your GAP installation as c:/gap-4.XX.Y/pkg/alco.

3. Open a GAP session and use the command LoadPackage(“alco”); to import all the commands from this package.

Example Session

The ALCO package allows users to construct the octonion arithmetic (integer ring). In the example below, we construct the octonion arithmetic and verify that the basis vectors define an $E_8$ lattice relative to the inner product shown:

gap> O := OctavianIntegers;
OctavianIntegers
gap> g := List(Basis(O), x -> List(Basis(O), y -> Norm(x+y) - Norm(x) - Norm(y)));;
gap> Display(g);
[ [   2,   0,  -1,   0,   0,   0,   0,   0 ],
  [   0,   2,   0,  -1,   0,   0,   0,   0 ],
  [  -1,   0,   2,  -1,   0,   0,   0,   0 ],
  [   0,  -1,  -1,   2,  -1,   0,   0,   0 ],
  [   0,   0,   0,  -1,   2,  -1,   0,   0 ],
  [   0,   0,   0,   0,  -1,   2,  -1,   0 ],
  [   0,   0,   0,   0,   0,  -1,   2,  -1 ],
  [   0,   0,   0,   0,   0,   0,  -1,   2 ] ]
gap> IsGossetLatticeGramMatrix(g);
true

We can also construct simple Euclidean Jordan algebras, including the Albert algebra:

gap> J := AlbertAlgebra(Rationals);
<algebra-with-one of dimension 27 over Rationals>
gap> SemiSimpleType(Derivations(Basis(J)));
"F4"
gap> i := Basis(J){[1..8]};
[ i1, i2, i3, i4, i5, i6, i7, i8 ]
gap> j := Basis(J){[9..16]};
[ j1, j2, j3, j4, j5, j6, j7, j8 ]
gap> k := Basis(J){[17..24]};
[ k1, k2, k3, k4, k5, k6, k7, k8 ]
gap> e := Basis(J){[25..27]};
[ ei, ej, ek ]
gap> List(e, IsIdempotent);
[ true, true, true ]
gap> Set(i, x -> x^2);
[ ej+ek ]
gap> Set(j, x -> x^2);
[ ei+ek ]
gap> One(J);
ei+ej+ek
gap> Determinant(One(J));
1
gap> Trace(One(J));
3  

The ALCO package also provides tools to construct octonion lattices, including octonion Leech lattices.

gap> short := Set(ShortestVectors(g,4).vectors, y -> LinearCombination(Basis(OctavianIntegers), y));;
gap> s := Filtered(short, x -> x^2 + x + 2*One(x) = Zero(x))[1];
(-1)*e1+(-1/2)*e2+(-1/2)*e3+(-1/2)*e4+(-1/2)*e8
gap> gens := List(Basis(OctavianIntegers), x -> x*[[s,s,0],[0,s,s],ComplexConjugate([s,s,s])]);;
gap> gens := Concatenation(gens);; 
gap> L := OctonionLatticeByGenerators(gens, One(O)*IdentityMat(3)/2);
<free left module over Integers, with 24 generators>
gap> IsLeechLatticeGramMatrix(GramMatrix(L));
true