Approximate sum of squares decompositions for 36(Adj₅ + k·Op₅) - λΔ₅ ∈ ISAut(F₅)

This is the dataset accompanying On property (T) for Aut(Fₙ) and SLₙ(ℤ) paper (https://arxiv.org/abs/1812.03456). See the appendix thereof and Section 4 of (Aut(F₅) has property (T)) for more details.

Content

1. check_positivity.jl is the script to (re)-produce sum of squares decompositions. In sqadjop.jl the helper functions are defined.
2. Positivity in SL(n,Z).ipynb is the ipython notebook which contains replication for special linear groups.
3. SAutF5_r2/delta.jld contains the Laplacian element (and the multiplication table) of the Group Ring of SAut(F₅).
4. SAutF5_r2/OrbitData.jld contains the data for decomposition and block decomposition.
5. SAutF5_r2/SqAdjOp_coeffs.jld contains the coefficients of elements Sq, Adj, and Op.
6. SAutF5_r2/5.0/ contains the approximate sum of squares decomposition for Adj-2·Op + 5.0Δ.
7. SAutF5_r2/50.0/ contains the approximate sum of squares decomposition for Adj-3·Op + 50.0Δ.

drwxr-xr-x         0 oSAutF5_r2/
-rw-r--r-- 257218404 oSAutF5_r2/OrbitData.jld
-rw-r--r--     86472 oSAutF5_r2/SqAdjOp_coeffs.jld
-rw-r--r-- 172357682 oSAutF5_r2/delta.jld
drwxr-xr-x         0 oSAutF5_r2/50.0/
-rw-r--r--      5166 oSAutF5_r2/50.0/Adj+3Op.log
-rw-r--r-- 344628928 oSAutF5_r2/50.0/solution.jld
drwxr-xr-x         0 oSAutF5_r2/5.0/
-rw-r--r--      5236 oSAutF5_r2/5.0/Adj+2Op.log
-rw-r--r-- 172315832 oSAutF5_r2/5.0/solution.jld

Preparation

The code needs to be run with julia-0.6. To install all of the dependencies run the following code in julias REPL console:

Pkg.add("AbstractAlgebra")
Pkg.add("Nemo") # this may take some time to compile
Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/Groups.jl")
Pkg.checkout("Groups", "AutFn")
Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/GroupRings.jl")
Pkg.checkout("GroupRings", "AutFn")
Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/PropertyT.jl")
Pkg.checkout("PropertyT", "AutFn")
Pkg.resolve()

Replication

To replicate the computations of the approximate sum of squares from the paper you need to unpack the content of oSAutF5_r2.tar.xz in the same folder where check_positivity.jl and sqadjop.jl reside.

To verify that 36(Adj₅ + 3.0·Op₅) - 50.0Δ₅ admits an approximate sum of squares decomposition run

julia check_positivity.jl -k 3 -lambda 50.0

On a modern laptop computer with 8GB of RAM this should finish in less than 3h.

Note: The execution will produce numerous warnings, eg.

WARNING: Scalar and coeffs are in different rings! Promoting result to ...
WARNING: Basis of the GroupRing is not defined.

These are safe to ignore.

At the end of computations you will see lines such as:

INFO: λ is certified to be > 47.40050266779858

INFO: i.e Adj₅ + 3.0·Op₅ - 1.3166806296610716·Δ₅ ∈ Σ²₂ ISAut(F₅)

This means that  Adj₅ + 3.0·Op₅ - λΔ₅ is a sum of Hermitian squares of elements from ISAut(F₅) for every λ < 1.316....

A similar verification for 36(Adj₅ + 2.0·Op₅) - 5.0Δ₅ can be run by executing

julia check_positivity.jl -k 2 -lambda 5.0

Generating the provided files

If You want to generate the multiplication table, orbit decomposition, etc. on Your own delete all *.jld files from the oSAutF5_r2 folder and run check_positivity.jl script with the same parameters again. Note: To do this You need at least 32GB of RAM and spare 24h of Your CPU.

The solutions could be also recreated, but this takes considerably more time (several days, depending on the solver's version and CPU).

This research was supported in part by National Science Center, Poland, grant 2017/26/D/ST1/00103.