This is an interactive browser visualizer and verified prime-field implementation of the Burnside sampling process on the symmetric group, which arises from the finite-field flag variety in Type A as explained in the paper Markov chains on Weyl groups from the geometry of the flag variety.

It's live on the web here, check it out!

The project has two core layers:

• web/: a no-backend browser app that runs the sampler locally through Pyodide
• burnside_sampler/: the pure-Python implementation of the actual sampling algorithm, together with a Sage version used for correctness checks

The mathematical source of truth is still the original Sage notebook, which implements exactly the algorithm described in Section 3 of our paper.

• sampling/sampling_burnside.ipynb

• burnside_sampler/pure_python.py Prime-field Burnside sampler used in the browser app.
• burnside_sampler/oracle_sage.py Sage-backed reference extracted from the notebook for tests.
• tests/ Correctness tests.
• web/ Web app wrapper for the sampler.
• run_tests.sh Runs the Python tests and the Sage-backed verification suite.
• serve_web.sh Serves the repository locally for browser use.

To run it locally:

./serve_web.sh

Then open:

http://localhost:8000/web/index.html

For GitHub Pages, the repository root also includes a small redirecting index.html, so the project site can open directly from the repository URL.

The browser UI:

• simulates the Burnside chain locally
• restricts q to primes
• colors histogram bars by right Steinberg cell via the RSK insertion tableau P
• uses the mathematical kernel in Python for the actual sampling

Run the full verification suite with:

./run_tests.sh

This checks:

• pure-Python finite-field linear algebra and Green-polynomial logic
• quotient and Springer-fiber sampling behavior
• alignment against Sage for the trusted oracle
• RSK / right-cell metadata used by the visualization

• Sage is needed only for oracle verification, not for running the public web app!