Coxeter group | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. Standard references include and . Formally, a Coxeter group can be defined as a group with the presentation where and for . The condition means no relation of the form should be imposed. The pair where is a Coxeter group with generators is called a Coxeter system. Note that in general is not uniquely determined by . For example, the Coxeter groups of type and are isomorphic but the Coxeter systems are not equivalent (see below for an explanation of this notation). A number of conclusions can be drawn immediately from the above definition. The relation means that for all ; as such the generators are involutions. If , then the generators and commute. This follows by observing that together with implies that Alternatively, since the generators are involutions, , so , and thus is equal to the commutator. In order to avoid redundancy among the relations, it is necessary to assume that . This follows by observing that together with implies that Alternatively, and are conjugate elements, as . The Coxeter matrix is the symmetric matrix with entries .

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (14)

Coxeter Combinatorics For Sum Formulas In The Representation Theory Of Algebraic Groups

Jonathan Gruber

Let G be a simple algebraic group over an algebraically closed field F of characteristic p >= h, the Coxeter number of G. We observe an easy 'recursion formula' for computing the Jantzen sum formula o

AMER MATHEMATICAL SOC2022

Irreducible components of exotic Springer fibres

Neil John Saunders

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (t

2018

Equivalences between blocks of p-local Mackey algebras

Baptiste Thierry Pierre Rognerud

Let G be a finite group and (K, O, k) be a p-modular system. Let R = O or k. There is a bijection between the blocks of the group algebra and the blocks of the so-called p-local Mackey algebra mu(1)(R

Elsevier2015

Related people (1)

Jacques Thévenaz

Related concepts (164)

Related courses (8)

MATH-335: Coxeter groups

Study groups generated by reflections

MATH-334: Representation theory

Study the basics of representation theory of groups and associative algebras.

AR-302(n): Studio BA6 (Lapierre)

L'atelier traite de la forme architecturale du point de vue de du logement comme système de stockage. Une occasion de questionner de manière critique la place du logement dans le monde capitaliste. Ma

Related lectures (56)

Coxeter Groups: Simple Reflections and Conjugacy

Explores the theorem that an element sending all simple roots to simple roots is the identity in Coxeter groups.

McKay Graphs of Finite Subgroups of SU(2)

Explores McKay graphs for finite subgroups of SU(2) and the corresponding Coxeter graphs.

Coxeter groups: reflections, rotations

Reviews Coxeter groups, reflections, rotations, and fundamental regions in finite orthogonal transformations.

Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts.

Generalized permutation matrix

In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is An invertible matrix A is a generalized permutation matrix if and only if it can be written as a product of an invertible diagonal matrix D and an (implicitly invertible) permutation matrix P: i.

Dynkin diagram

In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra.