PyCox: computing with (finite) Coxeter groups and Iwahori–Hecke algebras

PyCox (opens in a new tab), developed by Meinolf Geck, is a Python package for computations with Coxeter groups and Iwahori–Hecke algebras. Its capabilities include Kazhdan–Lusztig polynomials, cells, and W-graphs, making it a useful tool for computational representation theory.

I am also developing rustcox (opens in a new tab), a Rust rewrite of PyCox’s KL machinery. The project is still under development, with support for equal and unequal parameters and an emphasis on parallel computation. PyCox serves as the reference implementation for checking its outputs.

OpenAI’s recent work on Combinatorial invariance of Kazhdan–Lusztig polynomials (opens in a new tab) raises an interesting algorithmic question. The reported result says that equal-parameter KL polynomials depend only on the abstract Bruhat interval. How can this be turned into practical improvements for software such as PyCox and rustcox?

In particular:

• Does the proof suggest an efficient algorithm based directly on the interval’s order structure?
• Could identifying isomorphic intervals allow computations to be shared across different element pairs or Coxeter groups? When would the savings outweigh the cost of constructing and comparing intervals?
• Could these ideas accelerate cell and W-graph computations, while retaining the generator and descent information those computations require?

I would be interested in concrete algorithms, implementation ideas, and benchmark examples. The initial scope would be the equal-parameter case; extending these ideas to unequal parameters is a separate question.