.. _SectCartanIntro:

.. _introduction:

Introduction
============

The functions in this chapter handle basic descriptions of Coxeter systems. A
*Coxeter system* is a group :math:`G` with finite generating set
:math:`S=\{s_1,\dots,s_n\}`, defined by relations :math:`s_i^2=1` for
:math:`i=1,\dots,n` and

.. math:: s_is_js_i\cdots = s_js_is_j\cdots

for :math:`i,j=1,\dots,n` with :math:`i<j`, where each side of this relation has
length :math:`m_{ij}\ge 2`. Traditionally, :math:`m_{ij}=\infty` signifies that
the corresponding relation is omitted but for technical reasons :math:`m_{ij}=0`
is used in Magma instead. The group :math:`G` is called a *Coxeter group* and
:math:`S` is called the set of *Coxeter generators*. Since every group in Magma
has a preferred generating set, no distinction is made between a Coxeter system
and its Coxeter group. See :cite:`BourbakiLie` for more details on
the theory of Coxeter groups.

The *rank* of the Coxeter system is :math:`n=|S|`. A Coxeter system is said to
be *reducible* if there is a proper subset :math:`I` of :math:`\{1,\dots,n\}`
such that :math:`m_{ij}=2` or :math:`m_{ji}=2` whenever :math:`i\in I` and
:math:`j\notin I`. In this case, :math:`G` is an (internal) direct product of
the Coxeter subgroups :math:`W_I=\langle s_i \mid i \in I \rangle` and
:math:`W_{I^c}=\langle s_i \mid i \notin I \rangle`. Note that an *irreducible*
Coxeter group may still be a nontrivial direct product of abstract subgroups
(for example, :math:`W(G_2)\cong S_2\times S_3`). Two Coxeter systems are
*Coxeter isomorphic* (or *graph isomorphic*) if there is a group isomorphism
between them which takes Coxeter generators to Coxeter generators. In other
words, the two groups are the same modulo renumbering of the generators.

Coxeter groups and their representations as reflection groups have a number of
useful descriptions. In this chapter, Coxeter matrices, Coxeter graphs, Cartan
matrices, and Dynkin digraphs will be discussed. The classification of finite
and affine Coxeter groups provides a naming system for these groups. In
Chapters :ref:`ChapRootSys` and :ref:`ChapRootDtm`,
finite root systems and root data, which provide a more detailed description of
finite Coxeter groups, are discussed. Coxeter groups themselves are discussed in
Chapter :ref:`ChapGrpCox`; reflection representations of Coxeter
groups are discussed in Chapter :ref:`ChapGrpRfl`.