# Lie theory visualisations

An ongoing project to make illuminating visualisations for phenomena in Lie theory. This site is currently a draft and is changing often. If you have suggestions, get in touch.

## Rank 2 representations

A journey through the representations of rank 2 reductive groups, starting from Weyl characters and ending at simple characters in positive characteristic. Simple characters are calculated on-the-fly for almost all rank 2 groups (some are missing for $G_2$ in positive characteristic).

## Tables Fin and Aff

Table Fin contains data for the crystallographic root systems, including number of roots, order of the Weyl group, coefficients of the highest root and its dual, and the exponents of the Weyl group. Tables Aff contain further data about the affine Dynkin diagrams.

## Affine Weyl Group

An interactive visualisation of the affine Weyl group associated to a rank 2 Weyl group, showing various data about each group element (length, number of reduced expressions, etc). It can also calculate the cells and the cell ordering (by calculating the KL basis) - the figure to the right shows the right cells in Ã₂

## Affine $\SL_2$

A picture of the root system of Ã₂, showing the affine root system in the geometric representation, and the chamber geometry in its dual.

## Simple $\SL_2$ characters

The simple characters of $\SL_2$ in positive characteristic are given essentially by binomial coefficients modulo a prime. This visualisation shows the binomial coefficients modulo a prime, for any choice of prime.

## Euclidean root systems

Euclidean root systems are the starting point for the combinatorics of Lie groups and Lie algebras. This page is currently being revised.