Welcome to the website for Fully Commutative Kazhdan–Lusztig cells in Coxeter Groups, a Summer Research in Mathematics project at the University of Colorado Boulder. Our team includes high school student Rachel Castro, undergradaute students Joel Courtney, Thomas Magnuson, Natalie Schoenhals, graduate student Chase Meadors, and postdoctoral fellow Tianyuan Xu. The goal of our project is to implement certain useful objects associated to Coxeter groups in the mathematical software SageMath (Sage), with a view towards potential applications in combinatorial representation theory. These objects include fully commutative elements, heaps, certain Kazhdan–Lusztig cells, and certain generalized Temperley–Lieb algebras and diagrams. A more detailed overview of the project is available here.

The main references for the project are:

• Combinatorics of Coxeter Groups by Anders Bjorner and Francesco Brenti.
• On the fully commutative elements of Coxeter groups by John Stembridge.
• Kazhdan–Lusztig Cells with Unequal Parameters by Cédric Bonnafé.
• The Sage documentation, especially the parts on Coxeter groups and word combinatorics.

This project is a continuation of the Experimental Mathematics Lab project Implementing Fully Commutative Elements in SageMath. We use the Sage server at the CU math department to share our code. Informal notes and Sage worksheets from our meetings can be found here.