In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these group ...
, a length function is a
function that assigns a number to each element of a
group.
Definition
A length function ''L'' : ''G'' → R
+ on a
group ''G'' is a function satisfying:
:
Compare with the
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...
s for a
metric and a
filtered algebra.
Word metric
An important example of a length is the
word metric: given a
presentation of a group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—an ...
by generators and relations, the length of an element is the length of the shortest word expressing it.
Coxeter group
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 ref ...
s (including the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
) have combinatorial important length functions, using the simple reflections as generators (thus each simple reflection has length 1). See also:
length of a Weyl group element.
A
longest element of a Coxeter group is both important and unique up to conjugation (up to different choice of simple reflections).
Properties
A group with a length function does ''not'' form a
filtered group, meaning that the
sublevel set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is cal ...
s
do not form
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
s in general.
However, the
group algebra of a group with a length functions forms a
filtered algebra: the axiom
corresponds to the filtration axiom.
{{PlanetMath attribution, id=4365, title=Length function
Group theory
Geometric group theory