Some thoughts and experiments on Bergman’s compact amalgamation problem

This page contains the source code and explanations related to the computational results presented in the paper:
Michael Joswig, Mario Kummer, Andreas Thom, and Claudia He Yun : Some thoughts and experiments on Bergman’s compact amalgamation problem
In: Beiträge zur Algebra und Geometrie, Vol. not yet known, pp. not yet known
DOI: 10.1007/s13366-023-00709-8 ARXIV: https://arxiv.org/abs/2304.08365 CODE: https://mathrepo.mis.mpg.de/CompactAmalgamation/index.html

Abstract

We study the question whether copies of \(S^1\) in \(\mathrm{SU}(3)\) can be amalgamated in a compact group. This is the simplest instance of a fundamental open problem in the theory of compact groups raised by George Bergman in 1987. Considerable computational experiments suggest that the answer is positive in this case. We obtain a positive answer for a relaxed problem using theoretical considerations.

Amalgamating representations of the circle group

Representations of the circle group \(S^1 = \{z \in \mathbb{C} : \|z\|=1\}\) in \(\mathrm{SU}(3)\) are given by \((a,b,c)\in \mathbb{Z}^3\) with \(a+b+c=0\) and \(\mathrm{gcd}(a,b,c)=1\), up to conjugation. Given two such representations \(\psi_{a,b,c}\) and \(\psi_{a',b',c'}\), we look for injective representations \(f,g: \mathrm{SU}(3) \to \mathrm{U}(n)\) such that \(f \circ \psi_{a,b,c} = g \circ \psi_{a',b',c'}\). Unitary representations of \(\mathrm{SU}(3)\) are semi-simple. The irreducible representations have characters Schur polynomials \(s_\lambda(x_1,x_2,x_3)\), where \(\lambda\) is either the partition \((1,1,1)\) or a partition with at most two parts. Therefore, our goal is to find positive linear combinations \(P,Q\) of suitable Schur polynomials such that \(P(z^a,z^b,z^c) = Q(z^{a'},z^{b'},z^{c'})\). By Proposition 3.1, we can accomplish this by solving a particular integer linear program \((\mathrm{ILP}_k)\) on page 5. We set up our linear program using the computer algebra system OSCAR. We solve this linear program in OSCAR and SCIP.

In Table 2 Column 4, we only record the dimension of the amalgamated representations. For explicit descriptions of these representations, see this page.

Setting up computations

See the following for a walkthrough of Section 3. We give a detailed explanation of Example 3.3 and show how we produced Tables 1 and 2.

To reprodice these computations please do the following.

Configure SCIP

Install SCIP and record the path of the binary file. In the file amalgamation.jl, find the following function and replace /Users/yun/Documents/scipoptsuite-8.0.2/scip/bin/scip by your path.

function scip(filename,outputname)
    run(`/Users/yun/Documents/scipoptsuite-8.0.2/scip/bin/scip -f $filename -l $outputname -q`);
end

Checking Conjecture 4.9

Let

\[F_{a,b} = (1+z+\dots+z^{b-1})^a\cdot (1+z^{-1}+\dots+z^{-(b-1)})^a.\]

Let \(v=(v_1,v_2,v_3) \in \mathbb{Z}^3\) be a vector such that \(v_1+v_2+v_3=0\) and \(\mathrm{gcd}(v_1,v_2,v_3)=1\).

Conjecture 4.9. There are natural numbers \(a_0,b_0 > 0\) such that for all \(a \geq a_0\), all \(b\) divisible by \(b_0\) there is \(N\in \mathbb{N}\) and a Schur positive symmetric polynomial \(P\) in three variables such that

\[N\cdot F_{a,b}=P_v.\]

To verify this conjecture, we pick values for \(a,b\) and \(N\) and search for appropriate \(P\). We can set up a similar integer linear program by writing

\[P=\lambda_1S_1+\dots+\lambda_kS_k,\]

where \(\{S_1,\dots,S_k\}\) are Schur polynomials in some ordering. Then the linear program is given by matching coefficients of \(P_v\) with \(F_{a,b}\) while requiring \(\lambda_i \geq 0\).

Note that once \(a\) and \(b\) are chosen, they define a finite set of Schur polynomials that can appear in \(P\) in the following sense. Let \((r,t)\) be a partition. Let \(v=(v_1,v_2,v_3)\) as before, assuming \(v_1\geq v_2 > 0\). Then \((s_{r,t})_v(z)\) has leading deg \(rv_1+tv_2\) and trailing deg \(-rv_1+(t-r)v_2\). In the meantime, the polynomial \(F_{a,b}\) has leading degree \((b-1)a\) and trailing degree \((1-b)a\). Therefore, all Schur polynomials \(s_{r,t}\) that appear in \(P\) must satisfy

\[rv_1+tv_2 \leq (b-1)a,\]

\[-rv_1+(t-r)v_2 \geq (1-b)a.\]

There are only finitely many such partitions. In our program, we use this finite set of partitions and order them lexicographically.

All computations are done in Julia and OSCAR. In Table 3 Column 4, only dimensions of the representations given by \(P\) are recorded. For explicit descriptions, see this page.

See the following page for more details.

This notebook is also available for download conjecture_section4.ipynb and source code.

Project page created: 19/05/2023

Project contributors: Michael Joswig, Mario Kummer, Andreas Thom, Claudia He Yun

Corresponding author of this page: Claudia Yun, clyun@mis.mpg.de

Software used: Julia (version 1.8.5), OSCAR (version 0.12.0), SCIP (version 8.0.2)

System setup used:

  • Hydra at the MPI MiS: 4x16-core Intel Xeon E7-8867 v3 CPU (3300 MHz) on Debian GNU/Linux 5.10.149-2 (2022-10-21) x86_64

  • MacBook Pro: 8-core Intel i7-6700HQ CPU (2600 MHz) on Darwin Kernel 21.6.0 (2022-09-29) x86_64

License for code of this project page: MIT License (https://spdx.org/licenses/MIT.html)

License for all other content of this project page (text, images, …): CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

Last updated 21/08/2023.