Welcome to the website for Implementing Fully Commutative Elements in SageMath, an Experimental Mathematics Lab project at the University of Colorado Boulder. Our team includes mentors Sarah Salmon, Tianyuan Xu and undergraduate students Natalie Schoenhals, Saurabh Totey and Wei Qu. The goal of our project is to implement fully commutative (FC) elements of Coxeter groups and the heaps of FC elements in the software SageMath, with a view towards applications to combinatorial representation theory. SageMath, or Sage, is a computer algebra system based on Python, and a main component of the project is writing Sage code to create, identify, and count FC elements for any given Coxeter group. 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
• The enumeration of fully commutative elements of Coxeter groups by John Stembridge
• Fully commutative elements in finite and affine Coxeter groups by Riccardo Biagioli, Frédéric Jouhet and Philippe Nadeau
• Length enumeration of fully commutative elements in finite and affine Coxeter groups by Riccardo Biagioli, Frédéric Jouhet and Philippe Nadeau
• The Sage documentation, especially the parts on Coxeter groups and word combinatorics.