Reduced decompositions play a role also in the theory of other groups.
The symmetric group 
is in fact a Coxeter group,
which means that
there exists a set of generators 
such that the relations
are of the particular form 
where
In the case of the symmetric group 
the Coxeter system of generators is the
subset
of elementary transpositions 
as we have
seen already.
Furthermore, this leads us to introduce the weak
Bruhat order
on , which is the transitive closure of
The Bruhat order of the symmetric group 
is shown in figure 
.
We note that the number of reduced decompositions (and also the set of reduced decompositions)
can be obtained by going from the identity upwards in all the possible ways until the permutation in
question is reached.
We are now approaching a famous result on reduced sequences for the proof of
which we need a better knowledge of the set of inversions.
In order to describe how 
, arises from
, we use that acts on 
in a canonic way: 
. Keeping this in mind, we easily obtain:
This yields for the corresponding reduced lengths:
in accordance with .
If 
denotes the permutation 
of maximal length,
then
Proof: Clearly 
, and hence
. Moreover
so 
, and
. Finally we note that
which completes the proof.
An expression
of 
in terms of elementary transpositions and minimal
was called a reduced decomposition
of . The set of corresponding
sequences of indices is indicated as follows:
These sequences are called reduced sequences
of .
