3.1 MiB
push!(LOAD_PATH, "./");
LOAD_PATH3-element Array{String,1}:
"/opt/julia-3c9d75391c/local/share/julia/site/v0.5"
"/opt/julia-3c9d75391c/share/julia/site/v0.5"
"./" workspace()
using GroupAlgebras
include("property(T).jl");WARNING: Method definition redirect_stdout(Function, Any) in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600 overwritten in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600. WARNING: Method definition isnull(Any) in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1678 overwritten in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1678. WARNING: Method definition take!(Main.Base.AbstractIOBuffer) in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1698 overwritten in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1698. WARNING: Method definition redirect_stderr(Function, Any) in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600 overwritten in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600. WARNING: Method definition redirect_stdin(Function, Any) in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600 overwritten in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600.
using JuMP
import SCS: SCSSolver
using LatexPrintBasis of $\mathbb{R}[SL(3,\mathbb{Z})]$
All elements of the group algebra $\mathbb{R}[SL(3,\mathbb{Z})]$ will be written as coefficients vector in the following (partial) basis of the group ring (in total 121 elements)
GROUP = "SL(3,Z)";basis = read_GAP_raw_list("./basis."*GROUP);
convert_to_matrix(x) = hcat(x...);
basis = map(convert_to_matrix, basis);
# lap(basis...);$\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 2 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ -1 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 0 & -1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 2 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 0 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & -1 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & -1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ -1 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ -1 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 2 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & 0 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ -1 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 0 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 1 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 1 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 2 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 1 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -2 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 2 & 0 & -1 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & -1 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 1 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -2 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & -1 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 0 \\ 1 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ 1 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ -1 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 0 \\ -1 & 0 & 2 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 1 & 0 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 1 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & -1 & 2 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & -1 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right] ,\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \\ \end{array} \right]$
The generating set
We set the generating set $S$ of $SL(3,\mathbb{Z})$ to consist of the following $12$ elements:
- 3 elementary matrices
- their transpositions
- inverse elements to the above 6
The basis above consists of all words of length less than or equal to $2$ in $S$ (i.e. $S$ and all double products of elements of $S$, excluding repetitions)
$\left[\left(\begin{array}{rrr} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{array}\right), \\\left(\begin{array}{rrr} 1 & -1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right)\right]$
Elements of $\mathbb{R}[SL(3,\mathbb{Z}]$
Every element $$X = \sum_g a_g g\in \mathbb{R}[SL(3,\mathbb{Z}]$$ of the group ring will be represented as type (container) with two elements inside:
- a vector of coefficients $(a_g)_g$
- a matrix $P=p_{ij}$ which tells us how to compute product of the element with every other element in basis
We define the Laplacian element as $$\Delta_S = \Delta = \frac{1}{2}\sum_{g\in S} (1-g)^_\cdot(1-g),$$ where $\cdot$ denotes the group algebra multiplication and $
$$ \begin{align} g^* &\mapsto g^{-1} \quad \text{for group elements}\\ a_g^* &\mapsto a_g \quad \text{for coefficients.} \end{align} $$ (Note: we work over $\mathbb{R}$; over $\mathbb{C}$ the $*$-involution is given by conjugation.)
Example: $$(1-x+y)^*\cdot(1-x+y) = (1-x^{-1}+y^{-1})(1-x+y) = 1 - x + y - x^{-1} + 1 - x^{-1}y + y^{-1} - y^{-1}x + 1 = 3 - x - x^{-1} +y + y^{-1} - x^{-1}y - yx^{-1}$$
matrix_constraints = read_GAP_raw_list("./constraints."*GROUP);
product_matrix = create_product_matrix(matrix_constraints);
Delta² = read_GAP_raw_list("./delta_sq."*GROUP);
Delta = read_GAP_raw_list("./delta."*GROUP);
Δ = GroupAlgebraElement(Delta, product_matrix);
Δ² = GroupAlgebraElement(Delta², product_matrix);
@assert Δ*Δ == Δ²@show(Δ);
@show(Δ²);Δ = Element of Group Algebra over Int64 [12,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] Δ² = Element of Group Algebra over Int64 [156,-24,-24,-24,-24,-24,-24,-24,-24,-24,-24,-24,-24,1,2,1,1,1,2,2,1,1,1,2,1,2,1,1,1,2,2,1,1,1,1,1,2,1,1,1,2,2,1,1,1,1,1,2,1,1,1,2,2,1,1,1,1,1,2,1,1,1,2,2,1,1,1,1,2,1,1,1,2,1,1,1,1,2,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1]
Eg. the the $1st$ coefficients of those vectors (respecively $12$ i $156$) corresponds to basis vector $\operatorname{Id}$.
Semi-definite programming (optimalisation of a linear functional over the manifold of symmetric matrices)
Our main problem is to find (possibly large) constant $\kappa$ such that $\Delta^2 - \kappa \Delta$ can be decomposed in the group ring as a hermitian sum of squares. That is, there exist a finite set of elements $\xi_1, \xi_2,\ldots, \xi_n \in \mathbb{R}[SL(3,\mathbb{Z})]$ such that $$ \Delta^2 - \kappa\Delta = \sum_{i=1}^n \xi_i^*\xi.$$
Such sum-of-squares decompositions correspond to symmetric, positive semi-definite matrices $A$ such that $$\left(1, x, x^{-1}, \ldots \right)^_A\left(1, x, x^{-1}, \ldots\right)^T = \Delta^2 - \kappa\Delta $$ for some basis $(1, x, x^{-1}, \ldots )$ of $\mathbb{R}[SL(3,\mathbb{Z})]$. Indeed, since $A$ is symmetric and positive semi-definite, there exist a root $\sqrt{A}$, satisfying $\sqrt{A}\sqrt{A}^T = A$. Then we can set $$(\xi_1,\xi_2,\ldots,\xi_{n}) = \left(1,x, x^{-1},\ldots\right)\sqrt{A},$$ and it follows that $$\Delta^2-\kappa\Delta = \left(1,x, x^{-1},\ldots\right)^
In our case we will try to find such $121\times 121$ matrix $A$ (as our basis consists of $121$ elements) and therefore to express $\Delta^2 - \kappa\Delta$ as sum of $121$ hermitian squares.
First we define SDP problem:
SL_3ZZ = create_SDP_problem(matrix_constraints, Δ², Δ);
const TOL=6
setsolver(SL_3ZZ, SCSSolver(eps=10.0^-TOL));
@show(SL_3ZZ);SL_3ZZ = Maximization problem with: * 121 linear constraints * 1 semidefinite constraint * 7382 variables Solver is SCS
SL_3ZZThis includes:
- defining $\kappa$ as a variable
- setting maximizing $\kappa$ as the objective
- defining matrix $A$ of $121\times 121$ variables $a_{i,j}$ and adding constraints:
- $A\succeq 0$, i.e. $A$ is positive semi-definite
- $(1, x, \ldots)^* A (1, x, \ldots)^T = \Delta^2 - \kappa\Delta$ (this is achieved on the matrix: entries which contribute to a given poistion in the resulting vector are constrained by the appropriate coefficient in $\Delta^2 - \kappa\Delta$.
Numeric solution
Which takes $< 30s$ (depending on the precision and processor&memory)
solution_status = solve(SL_3ZZ);
κ = SL_3ZZ.objVal;
A = getvalue(getvariable(SL_3ZZ, :A));; # Pobiera wartość numeryczną
@show solution_status;
@show κ;solution_status = :Optimal
κ = 7.100471806882364
----------------------------------------------------------------------------
SCS v1.1.8 - Splitting Conic Solver
(c) Brendan O'Donoghue, Stanford University, 2012-2015
----------------------------------------------------------------------------
Lin-sys: sparse-direct, nnz in A = 17465
eps = 1.00e-06, alpha = 1.80, max_iters = 20000, normalize = 1, scale = 5.00
Variables n = 7382, constraints m = 14884
Cones: primal zero / dual free vars: 121
linear vars: 1
sd vars: 14762, sd blks: 2
Setup time: 1.37e-02s
----------------------------------------------------------------------------
Iter | pri res | dua res | rel gap | pri obj | dua obj | kap/tau | time (s)
----------------------------------------------------------------------------
0| inf inf -nan -inf inf inf 1.90e-02
100| 2.56e-03 1.30e-02 5.25e-03 -7.04e+00 -7.12e+00 2.31e-16 1.01e+00
200| 3.08e-04 1.59e-03 9.57e-05 -7.07e+00 -7.07e+00 2.34e-16 2.11e+00
300| 4.23e-05 1.34e-03 3.46e-05 -7.09e+00 -7.09e+00 2.34e-16 3.16e+00
400| 4.19e-05 1.25e-03 2.11e-05 -7.09e+00 -7.09e+00 2.36e-16 4.27e+00
500| 4.21e-05 1.17e-03 2.61e-05 -7.10e+00 -7.10e+00 2.37e-16 5.33e+00
600| 3.17e-05 1.77e-04 2.36e-05 -7.10e+00 -7.10e+00 2.39e-16 6.48e+00
700| 1.73e-06 2.86e-05 1.06e-06 -7.10e+00 -7.10e+00 2.39e-16 7.65e+00
800| 1.74e-07 5.93e-06 8.33e-08 -7.10e+00 -7.10e+00 2.39e-16 8.76e+00
900| 4.04e-08 1.36e-06 3.18e-09 -7.10e+00 -7.10e+00 2.39e-16 9.80e+00
940| 2.24e-08 7.63e-07 4.00e-09 -7.10e+00 -7.10e+00 2.39e-16 1.02e+01
----------------------------------------------------------------------------
Status: Solved
Timing: Solve time: 1.02e+01s
Lin-sys: nnz in L factor: 39791, avg solve time: 2.20e-04s
Cones: avg projection time: 1.05e-02s
----------------------------------------------------------------------------
Error metrics:
dist(s, K) = 1.7899e-09, dist(y, K*) = 1.0969e-09, s'y/m = 3.7798e-14
|Ax + s - b|_2 / (1 + |b|_2) = 2.2377e-08
|A'y + c|_2 / (1 + |c|_2) = 7.6300e-07
|c'x + b'y| / (1 + |c'x| + |b'y|) = 3.9957e-09
----------------------------------------------------------------------------
c'x = -7.1005, -b'y = -7.1005
============================================================================
using Gadfly
Gadfly.spy(A)
# imshow(A)Computed coordinates of $\Delta^2 - \kappa\Delta$ in the basis (at the beginning)
@show (Δ² - κ*Δ);WARNING: Scalars and coefficients ring are not the same! Trying to promote... WARNING: Adding elements with different base rings!
Δ² - κ * Δ = Element of Group Algebra over Float64 [70.7943,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,1.0,2.0,1.0,1.0,1.0,2.0,2.0,1.0,1.0,1.0,2.0,1.0,2.0,1.0,1.0,1.0,2.0,2.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,2.0,2.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,2.0,2.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,2.0,2.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0]
Checking the correctness
Let us check that the computed (approximated!) floating point matrix $A$ actually is positively semi-definite (up to some floating point error!):
rounded_eigenvalues_A = round(sort(eigvals(A)),TOL)'
@show rounded_eigenvalues_A;rounded_eigenvalues_A = [-2.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 2.0e-9 2.0e-9 2.0e-9 5.85665 5.85665 6.15125 6.15125 6.15125 6.48966 6.48966 6.48966 6.61277 6.61277 7.93277]
Since $A$ is also symmetric we can find its root $\sqrt{A}$, i.e. a matrix satisfying $\sqrt{A}\sqrt{A}^T = A$:
@show A == Symmetric(A)
A_sqrt = real(sqrtm(Symmetric(A)));A == Symmetric(A) = true
Warning
Due to floating-point arithmetic, knowledge of $\sqrt{A}$ only gives an approximate decomposition of $\Delta^2 - \kappa\Delta$ into sum of hermitian squares. Approximate i.e. if we set $X = (1, x, x^{-1}, \ldots)$, then
$$\left| (\Delta^2 - \kappa\Delta) - X^*AX^{T}\right|_\infty < 1.1\cdot 10^{-6}.$$
floating_SOS = resulting_SOS(A_sqrt, Δ)
floating_point_distance = norm(floating_SOS - (Δ²-κ*Δ), Inf)
@show floating_point_distance;floating_point_distance = 2.138915533578256e-8
WARNING: Scalars and coefficients ring are not the same! Trying to promote... WARNING: Adding elements with different base rings!
Checking Rationally
A_sqrt_rational = Array{Rational{BigInt},2}(round((10^TOL)*A_sqrt,0))
Δ_rational = convert(Rational{BigInt}, Δ )
Δ²_rational = convert(Rational{BigInt}, Δ²);
@assert Δ_rational*Δ_rational == Δ²_rational
κ_rational = rationalize(BigInt, κ)
# @show(norm(Δ-Δ_rational, Inf))
# @show(norm(Δ²-Δ²_rational, Inf))
# @show(norm(κ-κ_rational))
# @show(maxabs((10^TOL)*A_sqrt-A_sqrt_rational)/10^TOL);
const EOI = Δ²_rational - κ_rational*Δ_rational
rational_SOS = resulting_SOS(A_sqrt_rational, Δ)//10^(2*TOL)
rational_distance = norm(rational_SOS - EOI, Inf)
@show rational_distance;rational_distance = 2.478455540080956980681875675141175225375506936376126899286240020874535316025564e-08
WARNING: redefining constant EOI
A_sqrt_corrected = correct_to_augmentation_ideal(A_sqrt_rational);corrected_rational_SOS = resulting_SOS(A_sqrt_corrected, Δ_rational)//10^(2*TOL)
corrected_rational_distance = maxabs((corrected_rational_SOS - EOI).coordinates)
@show corrected_rational_distance;
@show float(corrected_rational_distance)
@show corrected_rational_SOS;corrected_rational_distance = 6890048458160433089//278140993900000000000000000
float(corrected_rational_distance) = 2.477178340937981774070305427207290927854845779351333510856487954758832836686713e-08
corrected_rational_SOS = Element of Group Algebra over Rational{BigInt}
Rational{BigInt}[856611546105105027971//12100000000000000000,-61856499365614410153867//3660250000000000000000,-61856499401009337536803//3660250000000000000000,-15464124843152043280303//915062500000000000000,-123712998732695761484897//7320500000000000000000,-12371299880650121655707//732050000000000000000,-30928249687378680613633//1830125000000000000000,-61856499365614410153867//3660250000000000000000,-61856499401009337536803//3660250000000000000000,-15464124843152043280303//915062500000000000000,-123712998732695761484897//7320500000000000000000,-12371299880650121655707//732050000000000000000,-30928249687378680613633//1830125000000000000000,14641000007137914480999//14641000000000000000000,7320499996092440137641//3660250000000000000000,732049999783051160223//732050000000000000000,14641000022446941509181//14641000000000000000000,7320499967674868142629//7320500000000000000000,7320499989759088262769//3660250000000000000000,7320500001150290737663//3660250000000000000000,3660249999405107033689//3660250000000000000000,7320499992069458508027//7320500000000000000000,14641000016359338409379//14641000000000000000000,14640999979442740804507//7320500000000000000000,3660250003390840556239//3660250000000000000000,29281999950059726074187//14641000000000000000000,14640999940234784904093//14641000000000000000000,14641000016525125604429//14641000000000000000000,14641000036047887346733//14641000000000000000000,7320500001150290737663//3660250000000000000000,183012500224866605213//91506250000000000000,14641000006632970905911//14641000000000000000000,7320499996626971377563//7320500000000000000000,14641000066442331685761//14641000000000000000000,3660249996903194902729//3660250000000000000000,14640999995395380819701//14641000000000000000000,14640999980078524199127//7320500000000000000000,1830125004333791082409//1830125000000000000000,14640999999566255079679//14641000000000000000000,3660249999405107033689//3660250000000000000000,183012500224866605213//91506250000000000000,29281999944617089370199//14641000000000000000000,14640999984195447441533//14641000000000000000000,3660249993641583625871//3660250000000000000000,1464099999045648424271//1464100000000000000000,14641000047353912781063//14641000000000000000000,1464100000684876251263//1464100000000000000000,29281999994407596424839//14641000000000000000000,14640999975064632306707//14641000000000000000000,7320499992069458508027//7320500000000000000000,14641000006632970905911//14641000000000000000000,29281999944617089370199//14641000000000000000000,7320500000333219948029//3660250000000000000000,7320499992019080827157//7320500000000000000000,14641000054090253575191//14641000000000000000000,1830124997852019617543//1830125000000000000000,14640999972662172154527//14641000000000000000000,14641000018226868015163//14641000000000000000000,14640999974689174763789//7320500000000000000000,14641000016359338409379//14641000000000000000000,7320499996626971377563//7320500000000000000000,14640999984195447441533//14641000000000000000000,7320500000333219948029//3660250000000000000000,1464100001823840820327//732050000000000000000,14640999974031184330179//14641000000000000000000,14640999992877629828481//14641000000000000000000,14640999993881730546339//14641000000000000000000,7320499998227593232481//7320500000000000000000,14640999979442740804507//7320500000000000000000,14641000066442331685761//14641000000000000000000,3660249993641583625871//3660250000000000000000,7320499992019080827157//7320500000000000000000,1464100001823840820327//732050000000000000000,7320499989042658737571//7320500000000000000000,14640999976090361196409//14641000000000000000000,7320500000676886976897//7320500000000000000000,14641000007137914480999//14641000000000000000000,7320499996092440137641//3660250000000000000000,3660249996903194902729//3660250000000000000000,1464099999045648424271//1464100000000000000000,14641000054090253575191//14641000000000000000000,7320499989759088262769//3660250000000000000000,3660250003216240179597//3660250000000000000000,585640001766286492171//585640000000000000000,14640999979386235471421//14641000000000000000000,3660250003390840556239//3660250000000000000000,29281999950059726074187//14641000000000000000000,14641000047353912781063//14641000000000000000000,1830124997852019617543//1830125000000000000000,14640999974031184330179//14641000000000000000000,7320499989042658737571//7320500000000000000000,3660250015723313277677//3660250000000000000000,1464099997582115792329//1464100000000000000000,732049999783051160223//732050000000000000000,14640999995395380819701//14641000000000000000000,14640999980078524199127//7320500000000000000000,14640999972662172154527//14641000000000000000000,14640999992877629828481//14641000000000000000000,14640999976090361196409//14641000000000000000000,3660250003216240179597//3660250000000000000000,29281999996316763843//29282000000000000000,14641000022446941509181//14641000000000000000000,14640999940234784904093//14641000000000000000000,1464100000684876251263//1464100000000000000000,292819999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@show norm(floating_SOS - (Δ²-κ*Δ), Inf)
@show norm(rational_SOS - (Δ²-κ*Δ), Inf)
@show norm(corrected_rational_SOS - (Δ²-κ*Δ), Inf);norm(floating_SOS - (Δ² - κ * Δ),Inf) = 1.2806626159544976e-7 norm(rational_SOS - (Δ² - κ * Δ),Inf) = 1.166245585359110984718427062034606933593749999999999999999999999999999995038896e-06 norm(corrected_rational_SOS - (Δ² - κ * Δ),Inf) = 1.166384273185079156291407049740364737022477545932654873301004029779386648940086e-06
WARNING: Scalars and coefficients ring are not the same! Trying to promote... WARNING: Adding elements with different base rings! WARNING: Scalars and coefficients ring are not the same! Trying to promote... WARNING: Adding elements with different base rings! WARNING: Adding elements with different base rings! WARNING: Scalars and coefficients ring are not the same! Trying to promote... WARNING: Adding elements with different base rings! WARNING: Adding elements with different base rings!