Lievis: Lie theory visualisations

An ongoing project to make illuminating visualisations for phenomena in Lie theory. If you find this site interesting, or have suggestions and corrections, do not hesitate to get in touch.

Rank 2 representations

Rank 2 representations

A tour of 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).

The Affine Weyl Group

The 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

Tables Fin and Aff

Rank 2 representations

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 \SL_2

Affine SL2

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

Modular \SL_2 characters

Modular SL2 Characters

Compare four bases of characters for the modular representations of \SL_2: the monomial symmetric functions, Weyl characters, simple characters, and tilting characters.

Simple \SL_2 characters

Simple SL2 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

Simple SL2 Characters

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


The “laboratory” of less polished work.