Coxeter groups form a large family of groups with natural connections to algebra, combinatorics and geometry. In particular, the partitioning of a Coxeter group into certain subsets called Kazhdan–Lusztig cells is very interesting to representation theory, a branch of mathematics which studies symmetry in linear spaces. Elements of a Coxeter group can be studied via words on Coxeter generators, and in this project we will learn about the so-called fully commutative or FC elements. These elements enjoy especially nice combinatorial properties: the set of words representing a fixed FC element will always satisfy a certain nice condition, and to each FC element we may often associate two combinatorial gadgets—a partially ordered set called a heap, and a diagram called a Temperley–Lieb diagram.
The goal of the project is implement FC elements and related notions in the mathematical software SageMath (Sage), including heaps, Temperley–Lieb diagrams, Temperley–Lieb algebras, and Kazhdan–Lusztig cells consisting entirely of FC elements or FC cells. We will emphasize developing code that have potential applications to combinatorial representation theory. For example, FC cells in Coxeter groups of type A (which are precisely symmetric groups) can be characterized in terms of Temperley–Lieb diagrams, a fact which can help us count each cell. We will use our code to test if FC cells can be characterized and counted in similar ways for other types of Coxeter groups.
This project will build upon a similar Math
Lab project on FC elements of Coxeter groups. However, previous
experience with Coxeter groups or Sage is not necessary.