# Generalization of the basis theorem for the $D$-type Coxeter groups

Sawsan Khaskeia
Department of Mathematics
Ariel University, Israel
<EMAIL_ADDRESS>Robert Shwartz
Department of Mathematics
Ariel University, Israel
<EMAIL_ADDRESS>
###### Abstract

The $OGS$ for non-abelian groups is an interesting generalization of the basis
of finite abelian groups. The definition of $OGS$ states that every element of
a group has a unique presentation as a product of some powers of specific
generators of the group, in a specific given order. In case of the symmetric
groups $S_{n}$ there is a paper of R. Shwartz, which demonstrates a strong
connection between the $OGS$ and the standard Coxeter presentation of the
symmetric group, which is called the standard $OGS$ of $S_{n}$. In this paper
we generalize the standard $OGS$ of $S_{n}$ to the finite classical Coxeter
group $D_{n}$. We describe the exchange laws for the generalized standard
$OGS$ of $D_{n}$, and we connect it to the Coxeter length of elements of
$D_{n}$.

##  1 Introduction

The fundamental theorem of finitely generated abelian groups states the
following: Let $A$ be a finitely generated abelian group, then there exists
generators $a_{1},a_{2},\ldots a_{n}$, such that every element $a$ in $A$ has
a unique presentation of a form:

$g=a_{1}^{i_{1}}\cdot a_{2}^{i_{2}}\cdots a_{n}^{i_{n}},$

where, $i_{1},i_{2},\ldots,i_{n}$ are $n$ integers such that for $1\leq k\leq
n$, $0\leq i_{k}<|g_{k}|$, where $a_{k}$ has a finite order of $|a_{k}|$ in
$A$, and $i_{k}\in\mathbb{Z}$, where $a_{k}$ has infinite order in $A$. Where,
the meaning of the theorem is that every abelian group $A$ is direct sum of
finitely many cyclic subgroup $A_{i}$ (where $1\leq i\leq k$), for some
$k\in\mathbb{N}$.

######  Definition 1.0.1.

Let $G$ be a non-abelian group. The ordered sequence of $n$ elements $\langle
g_{1},g_{2},\ldots,g_{n}\rangle$ is called an
$Ordered~{}~{}Generating~{}~{}System$ of the group $G$ or by shortened
notation, $OGS(G)$, if every element $g\in G$ has a unique presentation in the
a form

$g=g_{1}^{i_{1}}\cdot g_{2}^{i_{2}}\cdots g_{n}^{i_{n}},$

where, $i_{1},i_{2},\ldots,i_{n}$ are $n$ integers such that for $1\leq k\leq
n$, $0\leq i_{k}<r_{k}$, where $r_{k}||g_{k}|$ in case the order of $g_{k}$ is
finite in $G$, or $i_{k}\in\mathbb{Z}$, in case $g_{k}$ has infinite order in
$G$. The mentioned canonical form is called $OGS$ canonical form. For every
$q>p$, $1\leq x_{q}<r_{q}$, and $1\leq x_{p}<r_{p}$ the relation

$g_{q}^{x_{q}}\cdot g_{p}^{x_{p}}=g_{1}^{i_{1}}\cdot g_{2}^{i_{2}}\cdots
g_{n}^{i_{n}},$

is called exchange law.

In contrast to finitely generated abelian groups, the existence of an $OGS$ is
generally not true for every finitely generated non-abelian group. Even in
case of two-generated infinite non-abelian groups it is not difficult to find
counter examples. For example, the Baumslag-Solitar groups $BS(m,n)$ [4],
where $m\neq\pm 1$ or $n\neq\pm 1$, or most of the cases of the one-relator
free product of a finite cyclic group generated by $a$, with a finite two-
generated group generated by $b,c$ with the relation $a^{2}\cdot b\cdot a\cdot
c=1$ [10], do not have an $OGS$. Even the question of the existence of an
$OGS$ for a general finite non-abelian group is still open. Moreover, contrary
to the abelian case where the exchange law is just $g_{q}\cdot
g_{p}=g_{p}\cdot g_{q}$, in most of the cases of non-abelian groups with the
existence of an $OGS$, the exchange laws are very complicated. Although there
are some specific non-abelian groups where the exchange laws are very
convenient and have very interesting properties. A very good example of it is
the symmetric group $S_{n}$. In 2001, Adin and Roichman [1] introduced a
presentation of an $OGS$ canonical form for the symmetric group $S_{n}$, for
the hyperoctahedral group $B_{n}$, and for the wreath product
$\mathbb{Z}_{m}\wr S_{n}$. Adin and Roichman proved that for every element of
$S_{n}$ presented in the standard $OGS$ canonical form, the sum of the
exponents of the $OGS$ equals the major-index of the permutation. Moreover, by
using an $OGS$ canonical form, Adin and Roichman generalized the theorem of
MacMahon [6] to the $B$-type Coxeter group, and to the wreath product
$\mathbb{Z}_{m}\wr S_{n}$. A few years later, that $OGS$ canonical form was
generalized for complex reflection groups by Shwartz, Adin and Roichman [11].
Recently, Shwartz [9] significantly extended the results of [1], [11], where
the $OGS$ of $S_{n}$ is strongly connected to the Coxeter length and to the
descent set of the elements. Moreover, in [9], there are described the
exchange laws for the $OGS$ canonical forms of the symmetric group $S_{n}$,
which have very interesting and surprising properties. In the paper we try to
generalize the results of [9] to the finite classical Coxeter group $D_{n}$.
Similarly to the symmetric group $S_{n}$, the group $D_{n}$ can be considered
as permutation group as well. Therefore, we recall the notations of
permutations, the $OGS$ of $S_{n}$ and the corresponding exchange laws, from
[9].

######  Definition 1.0.2.

Let $S_{n}$ be the symmetric group on $n$ elements, then :

  * •

The symmetric group $S_{n}$ is an $n-1$ generated simply-laced finite Coxeter
group of order $n!$, which has the presentation of:

$\langle s_{1},s_{2},\ldots,s_{n-1}|s_{i}^{2}=1,~{}~{}(s_{i}\cdot
s_{i+1})^{3}=1,~{}~{}(s_{i}\cdot s_{j})^{2}=1~{}~{}for~{}~{}|i-j|\geq
2\rangle;$

  * •

The group $S_{n}$ can be considered as the permutation group on $n$ elements.
A permutation $\pi\in S_{n}$ is denoted by

$\pi=[\pi(1);~{}\pi(2);\ldots;~{}\pi(n)]$

(i.e., $\pi=[2;~{}4;~{}1;~{}3]$ is a permutation in $S_{4}$ which satisfies
$\pi(1)=2$, $\pi(2)=4$, $\pi(3)=1$, and $\pi(4)=3$);

  * •

Every permutation $\pi\in S_{n}$ can be presented in a cyclic notation, as a
product of disjoint cycles of the form $(i_{1},~{}i_{2},~{}\ldots,~{}i_{m})$,
which means $\pi(i_{k})=i_{k+1}$, for $1\leq k\leq m-1$, and
$\pi(i_{m})=i_{1}$ (i.e., The cyclic notation of $\pi=[3;~{}4;~{}1;~{}5;~{}2]$
in $S_{5}$, is $(1,~{}3)(2,~{}4,~{}5)$);

  * •

The Coxeter generator $s_{i}$ can be considered the permutation which
exchanges the element $i$ with the element $i+1$, i.e., the transposition
$(i,i+1)$;

  * •

We consider multiplication of permutations in left to right order; i.e., for
every $\pi_{1},\pi_{2}\in S_{n}$, $\pi_{1}\cdot\pi_{2}(i)=\pi_{2}(j)$, where,
$\pi_{1}(i)=j$ (in contrary to the notation in [1] where Adin, Roichman, and
other people have considered right to left multiplication of permutations);

  * •

For every permutation $\pi\in S_{n}$, the Coxeter length $\ell(\pi)$ is the
number of inversions in $\pi$, i.e., the number of different pairs $i,j$, s.
t. $i<j$ and $\pi(i)>\pi(j)$;

  * •

For every permutation $\pi\in S_{n}$, the set of the locations of the descents
is defined to be

$des\left(\pi\right)=\\{1\leq i\leq n-1|\pi(i)>\pi(i+1)\\},$

and

$i\in
des\left(\pi\right)~{}~{}if~{}and~{}only~{}if~{}~{}\ell(s_{i}\cdot\pi)<\ell(\pi)$

(i.e., $i$ is a descent of $\pi$ if and only if multiplying $\pi$ by $s_{i}$
in the left side shortens the Coxeter length of the element.);

  * •

For every permutation $\pi\in S_{n}$, the major-index is defined to be

$maj\left(\pi\right)=\sum_{\pi(i)>\pi(i+1)}i$

(i.e., major-index is the sum of the locations of the descents of $\pi$.).

  * •

By [5] Chapter 3.4, every element $\pi$ of $S_{n}$ can be presented uniquely
in the following normal reduced form, which we denote by $norm(\pi)$:

$norm(\pi)=\prod_{u=1}^{n-1}\prod_{r=0}^{y_{u}-1}s_{u-r}.$

such that $y_{u}$ is a non-negative integer where, $0\leq y_{u}\leq u$ for
every $1\leq u\leq n-1$. By our notation of $norm(\pi)$ the Coxeter length of
an element $\pi$ as follow:

$\ell(\pi)=\sum_{u=1}^{n-1}y_{u}.$

For example: Let $m=8$, $y_{2}=2$, $y_{4}=3$, $y_{5}=1$, $y_{8}=4$, and
$y_{1}=y_{3}=y_{6}=y_{7}=0$, then

$norm(\pi)=(s_{2}\cdot s_{1})\cdot(s_{4}\cdot s_{3}\cdot s_{2})\cdot
s_{5}\cdot(s_{8}\cdot s_{7}\cdot s_{6}\cdot s_{5}).$ $\ell(\pi)=2+3+1+4=10.$

######  Theorem 1.0.3.

Let $S_{n}$ be the symmetric group on $n$ elements. For every $2\leq m\leq n$,
define $t_{m}$ to be the product $\prod_{j=1}^{m-1}s_{j}$. The element $t_{m}$
is the permutation

$t_{m}=[m;~{}1;~{}2;\ldots;~{}m-1;~{}m+1;\ldots;~{}n],$

which is the $m$-cycle $(m,~{}m-1,~{}\ldots,~{}1)$ in the cyclic notation of
the permutation. Then, the elements $t_{n},t_{n-1},\ldots,t_{2}$ generates
$S_{n}$, and every element of $S_{n}$ has a unique presentation in the
following $OGS$ canonical form:

$t_{2}^{i_{2}}\cdot t_{3}^{i_{3}}\cdots
t_{n}^{i_{n}},~{}~{}~{}where~{}~{}~{}0\leq i_{k}<k~{}~{}~{}for~{}~{}~{}2\leq
k\leq n$

######  Proposition 1.0.4.

The following holds:

In order to transform the element $t_{q}^{i_{q}}\cdot t_{p}^{i_{p}}$ ($p<q$)
onto the $OGS$ canonical form
$t_{2}^{i_{2}}\cdot t_{3}^{i_{3}}\cdots t_{n}^{i_{n}}$, i.e., according to the
standard $OGS$, one needs to use the following exchange laws:

$t_{q}^{i_{q}}\cdot t_{p}^{i_{p}}=\begin{cases}t_{i_{q}+i_{p}}^{i_{q}}\cdot
t_{p+i_{q}}^{i_{p}}\cdot t_{q}^{i_{q}}&q-i_{q}\geq p\\\ \\\
t_{i_{q}}^{p+i_{q}-q}\cdot t_{i_{q}+i_{p}}^{q-p}\cdot
t_{q}^{i_{q}+i_{p}}&i_{p}\leq q-i_{q}\leq p\\\ \\\
t_{p+i_{q}-q}^{i_{q}+i_{p}-q}\cdot t_{i_{q}}^{p-i_{p}}\cdot
t_{q}^{i_{q}+i_{p}-p}&q-i_{q}\leq i_{p}.\end{cases}$

######  Remark 1.0.5.

The standard $OGS$ canonical form of $t_{q}^{i_{q}}\cdot t_{p}^{i_{p}}$ is a
product of non-zero powers of two different canonical generators if and only
if $q-i_{q}=p$ or $q-i_{q}=i_{p}$, as follow:

  * •

If $q-i_{q}=p$ then by considering $q-i_{q}\geq p$:

$t_{i_{q}+i_{p}}^{i_{q}}\cdot t_{p+i_{q}}^{i_{p}}\cdot
t_{q}^{i_{q}}=t_{i_{q}+i_{p}}^{i_{q}}\cdot t_{q}^{i_{p}}\cdot t_{q}^{i_{q}}$

and by considering $q-i_{q}\leq p$:

$t_{i_{q}}^{p+i_{q}-q}\cdot t_{i_{q}+i_{p}}^{q-p}\cdot
t_{q}^{i_{q}+i_{p}}=t_{i_{q}}^{0}\cdot t_{i_{q}+i_{p}}^{i_{q}}\cdot
t_{q}^{i_{q}+i_{p}};$

  * •

If $q-i_{q}=i_{p}$ then by considering $q-i_{q}\geq i_{p}$:

$t_{i_{q}}^{p+i_{q}-q}\cdot t_{i_{q}+i_{p}}^{q-p}\cdot
t_{q}^{i_{q}+i_{p}}=t_{i_{q}}^{p-i_{p}}\cdot t_{q}^{q-p}\cdot
t_{q}^{q}=t_{i_{q}}^{p-i_{p}}\cdot t_{q}^{q-p}$

and by considering $q-i_{q}\leq i_{p}$:

$t_{p+i_{q}-q}^{i_{q}+i_{p}-q}\cdot t_{i_{q}}^{p-i_{p}}\cdot
t_{q}^{i_{q}+i_{p}-p}=t_{p+i_{q}-q}^{0}\cdot t_{i_{q}}^{p-i_{p}}\cdot
t_{q}^{q-p}.$

Hence we have

$t_{q}^{i_{q}}\cdot t_{p}^{i_{p}}=\begin{cases}t_{i_{q}+i_{p}}^{i_{q}}\cdot
t_{q}^{i_{q}+i_{p}}&q-i_{q}=p\\\ \\\ t_{i_{q}}^{p-i_{p}}\cdot
t_{q}^{q-p}&q-i_{q}=i_{p}.\end{cases}$

Moreover, the most significant achievement of the paper [9] is the definition
of the standard $OGS$ elementary factorization. By using the standard $OGS$
elementary factorization, it is possible to give a very interesting formula
for the Coxeter length and a complete classification of the descent set of any
arbitrary element of $S_{n}$. In the paper we try to generalize the standard
$OGS$ elementary factorization to the $B$-type Coxeter groups, in order to
find similar properties (Coxeter length and the descent set) for the elements
of the group. Hence, we recall the definition of the standard $OGS$ elementary
element and factorization for the symmetric group $S_{n}$ as it is defined in
[9], and theorems concerning the Coxeter length and the descent set of
elements of $S_{n}$ as it is mentioned and proved in [9] .

######  Definition 1.0.6.

Let $\pi\in S_{n}$, where $\pi=\prod_{j=1}^{m}t_{k_{j}}^{i_{k_{j}}}$ is
presented in the standard $OGS$ canonical form, with $i_{k_{j}}>0$ for every
$1\leq j\leq m$. Then, $\pi$ is called standard $OGS$ elementary element of
$S_{n}$, if

$\sum_{j=1}^{m}i_{k_{j}}\leq k_{1}.$

######  Theorem 1.0.7.

[9] Let $\pi=\prod_{j=1}^{m}t_{k_{j}}^{i_{k_{j}}}$ be a standard $OGS$
elementary element of $S_{n}$, presented in the standard $OGS$ canonical form,
with $i_{k_{j}}>0$ for every $1\leq j\leq m$. Then, the following are
satisfied:

  * •

$\ell(\pi)=\sum_{j=1}^{m}k_{j}\cdot
i_{k_{j}}-(i_{k_{1}}+i_{k_{2}}+\cdots+i_{k_{m}})^{2}=\sum_{j=1}^{m}k_{j}\cdot
i_{k_{j}}-\left(maj\left(\pi\right)\right)^{2};$

  * •

Every subword of $\pi$ is a standard $OGS$ elementary element too. In
particular, for every two subwords $\pi_{1}$ and $\pi_{2}$ of $\pi$, such that
$\pi=\pi_{1}\cdot\pi_{2}$, it is satisfied:

$\ell(\pi)=\ell(\pi_{1}\cdot\pi_{2})<\ell(\pi_{1})+\ell(\pi_{2});$

  * •

$\ell(s_{r}\cdot\pi)=\begin{cases}\ell(\pi)-1&r=\sum_{j=1}^{m}i_{k_{j}}\\\
\ell(\pi)+1&r\neq\sum_{j=1}^{m}i_{k_{j}}\end{cases}.$

i.e., $des\left(\pi\right)$ contains just one element, which means
$des\left(\pi\right)=\\{maj\left(\pi\right)\\}$.

######  Definition 1.0.8.

Let $\pi\in S_{n}$. Let $z(\pi)$ be the minimal number, such that $\pi$ can be
presented as a product of standard $OGS$ elementary elements, with the
following conditions:

  * •

$\pi=\prod_{v=1}^{z(\pi)}\pi^{(v)},~{}~{}~{}~{}where~{}~{}~{}~{}\pi^{(v)}=\prod_{j=1}^{m^{(v)}}t_{h^{(v)}_{j}}^{\imath_{j}^{(v)}},$

by the presentation in the standard $OGS$ canonical form for every $1\leq
v\leq z(\pi)$ and $1\leq j\leq m^{(v)}$ such that:

    * –

$\imath_{j}^{(v)}>0;$

    * –

$\sum_{j=1}^{m^{(1)}}\imath_{j}^{(1)}\leq h^{(1)}_{1}$ i.e.,
$maj\left(\pi^{(1)}\right)\leq h^{(1)}_{1}$;

    * –

$h^{(v-1)}_{m^{(v-1)}}\leq\sum_{j=1}^{m^{(v)}}\imath_{j}^{(v)}\leq
h^{(v)}_{1}$ for $2\leq v\leq z$

i.e., $h^{(v-1)}_{m^{(v-1)}}\leq maj\left(\pi^{(v)}\right)\leq
h^{(v)}_{1}~{}~{}for~{}~{}2\leq v\leq z$.

Then, the mentioned presentation is called Standard $OGS$ elementary
factorization of $\pi$. Since the factors $\pi^{(v)}$ are standard $OGS$
elementary elements, they are called standard $OGS$ elementary factors of
$\pi$.

######  Theorem 1.0.9.

[9] Let $\pi=\prod_{j=1}^{m}t_{k_{j}}^{i_{k_{j}}}$ be an element of $S_{n}$
presented in the standard $OGS$ canonical form, with $i_{k_{j}}>0$ for every
$1\leq j\leq m$. Consider the standard $OGS$ elementary factorization of $\pi$
with all the notations used in Definition 1.0.8. Then, the following
properties hold:

  * •

The standard $OGS$ elementary factorization of $\pi$ is unique, i.e., the
parameters $z(\pi)$, $m^{(v)}$ for $1\leq v\leq z(\pi)$, $h^{(v)}_{j}$, and
$\imath_{j}^{(v)}$ for $1\leq j\leq m^{(v)}$, are uniquely determined by the
standard $OGS$ canonical form of $\pi$, such that:

    * –

For every $h^{(v)}_{j}$ there exists exactly one $k_{j^{\prime}}$ (where,
$1\leq j^{\prime}\leq m$), such that $h^{(v)}_{j}=k_{j^{\prime}}$;

    * –

If $h^{(v)}_{j}=k_{j^{\prime}}$, for some $1\leq v\leq z(\pi)$, $1<j<m^{(v)}$,
and $1\leq j^{\prime}\leq m$, then $\imath_{j}^{(v)}=i_{k_{j^{\prime}}}$;

    * –

If $h^{(v_{1})}_{j_{1}}=h^{(v_{2})}_{j_{2}}$, where $1\leq v_{1}<v_{2}\leq
z(\pi)$, $1\leq j_{1}\leq m^{(v_{1})}$, and
$1\leq j_{2}\leq m^{(v_{2})}$, then necessarily $v_{1}=v_{2}-1$,
$j_{1}=m^{(v_{1})}$, $j_{2}=1$, and

$h^{(v_{2}-1)}_{m^{(v_{2}-1)}}=h^{(v_{2})}_{1}=maj\left(\pi_{(v_{2})}\right)=k_{j^{\prime}},$

for some $j^{\prime}$, such that
$\imath_{m^{(v_{2}-1)}}^{(v_{2}-1)}+\imath_{1}^{(v_{2})}=i_{k_{j^{\prime}}}$;

  * •

$\ell(s_{r}\cdot\pi)=\begin{cases}\ell(\pi)-1&r=\sum_{j=1}^{m^{(v)}}\imath_{j}^{(v)}~{}~{}for~{}~{}1\leq
v\leq z(\pi)\\\ \ell(\pi)+1&otherwise\end{cases}.$

i.e.,

$des\left(\pi\right)=\bigcup_{v=1}^{z(\pi)}des\left(\pi^{(v)}\right)=\\{maj\left(\pi^{(v)}\right)~{}|~{}1\leq
v\leq z(\pi)\\};$

  * •

$\displaystyle\ell(\pi)$
$\displaystyle=\sum_{v=1}^{z(\pi)}\ell(\pi^{(v)})=\sum_{v=1}^{z(\pi)}\sum_{j=1}^{m^{(v)}}h^{(v)}_{j}\cdot\imath_{j}^{(v)}-\sum_{v=1}^{z(\pi)}\left(maj\left(\pi^{(v)}\right)\right)^{2}$
$\displaystyle=\sum_{x=1}^{m}k_{x}\cdot
i_{k_{x}}-\sum_{v=1}^{z(\pi)}\left(maj\left(\pi^{(v)}\right)\right)^{2}$
$\displaystyle=\sum_{x=1}^{m}k_{x}\cdot
i_{k_{x}}-\sum_{v=1}^{z(\pi)}{{\left(c^{(v)}\right)}}^{2},~{}~{}where~{}~{}c^{(v)}\in
des\left(\pi\right).$

The paper is organized as follows: In Section 2, we recall some important
definitions and basic properties concerning the group $D_{n}$, Then we
generalize the definition of the standard $OGS$ and we show the arising
exchange laws. In Section 3, we focus on the generalized standard $OGS$
presentation of the elements in two interesting parabolic subgroups of $D_{n}$
which are isomorphic to $S_{n}$. In Section 4, we recall the definition of a
normal form of element’s of $D_{n}$, then we give an algorithm to find the
Coxeter length of elements of $D_{n}$ by using the generalized standard $OGS$.

##  2 Generalization of the standard OGS for the Coxeter group $D_{n}$

First, we recall some important definitions and basic properties concerning
the group $D_{n}$, as it defined in every book about Coxeter groups (for
example [5])

######  Definition 2.0.1.

Let $D_{n}$ be the Coxeter group with $n$ generators, with the following
presentation:

$\left\langle
s_{1^{\prime}},s_{1},\ldots,s_{n-1}\right|s_{i}^{2}=1,\left(s_{i}\cdot
s_{i+1}\right)^{3}=1\text{ for }1\leq i\leq n-1,\left(s_{i}\cdot
s_{j}\right)^{2}=1\text{ for }|i-j|\geq 2,$ $\left(s_{1^{\prime}}\cdot
s_{2}\right)^{3}=1,\quad\left(s_{1^{\prime}}\cdot s_{i})^{2}=1\text{ for
}1\leq i\leq n-1\text{ and }i\neq 2,\right\rangle$

Basic properties of $D_{n}$

  * •

The group $D_{n}$ is a subgroup of $B_{n}$, therefore, it can be presented as
a permutation group of the set $[\pm n]=\\{\pm i\mid 1\leq i\leq n\\}]$ in
such the following holds :

    * –

$\pi(-i)=-\pi(i)\text{ for every }i\in[\pm n].$

    * –

$\\{i|i>0\quad\text{and}\quad\pi(i)<0\\}\quad\text{is even.}$

  * •

The group $D_{n}$, can be considered as even signed permutation group, where
as a subgroup of $B_{n}$, every element $\pi\in D_{n}$ is uniquely determined
by $\pi(i)$ for $1\leq i\leq n$.

  * •

An even signed permutation $\pi\in D_{n}$ is denoted by

$[\pi(1);~{}\pi(2);\ldots;~{}\pi(n)]$

where the number of negative elements of the form $\pi(i)$, where $1\leq i\leq
n$ is even. (e.g., $\pi=[2;-4;1;-3]$ is a permutation in $D_{4}$, since for
$i=1,2,3,4$, there are 2 (even number) elements of the form $\pi(i)$ such that
$\pi(i)<0$, namely: $\pi(2)=-4$, $\pi(4)=-3$).

  * •

As in $S_{n}$, for $1\leq i\leq n-1$, the Coxeter generators $s_{i}$ can be
considered the permutation which exchanges the element $i$ with the element
$i+1$ where in addition exchanges $-i$ with $-(i+1)$ as well, and
$s_{{1}^{\prime}}$ is the permutation which exchanges the element $1$ with the
element $-2$ and the element $2$ with the element $-1$.

  * •

$D_{n}\simeq\mathbb{Z}_{2}^{n-1}\rtimes S_{n}$.

  * •

$|D_{n}|=2^{n-1}\cdot n!$.

Now, we recall the definition of a normal form of elements of $D_{n}$ in
Coxeter generators as it defined in [5] Chapter 3.4.

######  Definition 2.0.2.

Every element $\pi\in D_{n}$ can be presented uniquely in the following normal
form :

$\text{norm}(\pi)=\prod_{u=1}^{n-1}\text{norm}_{u}(\pi).$

where,

$\text{norm}_{u}(\pi)\in\prod_{r=0}^{y_{u}-1}s_{u-r}\quad\text{or}\quad\text{norm}_{u}(\pi)\in\prod_{r=0}^{u-2}s_{u-r}\cdot
s_{1^{\prime}}\cdot\prod_{r=1}^{y_{u}^{\prime}}s_{r}.$

where:

$0\leqslant y_{u}\leqslant u,\quad 0\leqslant y_{u}^{\prime}\leqslant u.$

######  Example 2.0.3.

Let

$\pi=[-2;~{}-6;~{}-5;~{}4;~{}-3;~{}-7;~{}8;~{}-1].$

Then,

$\text{norm}(\pi)=s_{2}\cdot(s_{3}\cdot s_{2}\cdot s_{1}\cdot
s_{1^{\prime}})\cdot(s_{4}\cdot s_{3}\cdot s_{2})\cdot(s_{5}\cdot s_{4}\cdot
s_{3}\cdot s_{2}\cdot s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdot s_{3}\cdot
s_{4}\cdot s_{5})\\\ \cdot(s_{7}\cdot s_{6}\cdot s_{5}\cdot s_{4}\cdot
s_{3}\cdot s_{2}\cdot s_{1^{\prime}}).$

Hence,

$\text{norm}_{1}(\pi)=\text{norm}_{6}(\pi)=1\quad\text{norm}_{2}(\pi)=s_{2}\quad\text{norm}_{3}(\pi)=s_{3}\cdot
s_{2}\cdot s_{1}\cdot s_{1^{\prime}}$ $\text{norm}_{4}(\pi)=s_{4}\cdot
s_{3}\cdot s_{2}\quad\text{norm}_{5}(\pi)=s_{5}\cdot s_{4}\cdot s_{3}\cdot
s_{2}\cdot s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdot s_{3}\cdot s_{4}\cdot
s_{5}$ $\text{norm}_{7}(\pi)=s_{7}\cdot s_{6}\cdot s_{5}\cdot s_{4}\cdot
s_{3}\cdot s_{2}\cdot s_{1^{\prime}}.$

###  2.1 The generalized standard OGS for $D_{n}$

In this subsection we define the generalized standard $OGS$ for $D_{n}$.
First, notice, the set of generalized standard $OGS$ contains the standard
$OGS$ for $S_{n}$ which is defined in [9].

Therefore, first we recall the definition of $t_{i}$ as it defined in [9].

######  Definition 2.1.1.

[9] for $i=2,3,\ldots,n$ let $t_{i}$ to be :

$t_{i}=\prod_{j=1}^{n-1}s_{j}.$

######  Remark 2.1.2.

[9] For every $2\leqslant i\leqslant n$, the permutation presentation of
$t_{i}$ satisfies the following properties:

$\begin{array}[]{l}t_{i}(1)=i\\\ t_{i}(j)=j-1,\text{ for }2\leq j\leq i\\\
t_{i}(j)=j,\text{ for }i+1\leq j\leq n\end{array}$

######  Definition 2.1.3.

For $k=1,2,\ldots,n$ let $w_{k}$ to be :

$w_{k}=s_{k}\cdot s_{k-1}\cdots s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdot
s_{3}\cdots s_{k}$

######  Lemma 2.1.4.

For every $1\leqslant k\leqslant n-1$,

$w_{k}^{2}=1.$

###### Proof.

$w_{k}^{2}=$ $=(s_{k}\cdot s_{k-1}\cdots s_{1}\cdot s_{1^{\prime}}\cdot
s_{2}\cdots s_{k-1}\cdot s_{k})(\cdot s_{k}\cdot s_{k-1}\cdots s_{1}\cdot
s_{1^{\prime}}\cdot s_{2}\cdots s_{k-1}\cdot s_{k})=$ $=s_{1}\cdot
s_{1^{\prime}}\cdot s_{1}\cdot s_{1^{\prime}}=s_{1}\cdot s_{1^{\prime}}\cdot
s_{1^{\prime}}\cdot s_{1}=1.$

∎

######  Remark 2.1.5.

For every $1\leqslant k\leqslant n$, the permutation presentation of $w_{k}$
satisfies the following properties:

$\begin{array}[]{l}w_{k}(1)=-1\\\ w_{k}(k+1)=-(k+1)\\\ w_{k}(j)=j,\text{ for
}2\leqslant j\leqslant k\quad\text{and}\quad k+2\leqslant j\leqslant
n.\end{array}$

######  Theorem 2.1.6.

Every $\pi\in D_{n}$ has a unique presentation in the form :

$\pi=w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}\cdot w_{2}^{j_{2}}\cdot
t_{3}^{i_{3}}\cdots w_{n-1}^{j_{n-1}}\cdot t_{n}^{i_{n}}$

Where

$0\leqslant i_{k}\leqslant k-1\quad\text{and}\quad 0\leqslant j_{k}\leqslant
1$

###### Proof.

The proof is by induction on $n$. for $n=2$, it is easy to see that $D_{2}$ is
generated by $w_{1}$ and $t_{2}$. Since $D_{2}$ is a non-cyclic abelian group
of order $4$, every element of $D_{2}$ has a unique presentation in the form
$w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}$, where $j_{1},i_{2}\in\\{0,1\\}$. Now,
assume by induction that theorem holds for every $k$ such that $k\leq n-1$,
i.e., every element of $D_{k}$ has a unique presentation in the form

$w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}\cdot w_{2}^{j_{2}}\cdot t_{3}^{i_{3}}\cdots
w_{k-1}^{j_{k-1}}\cdot t_{k}^{i_{k}}.$

Denote by $\dot{D}_{n-1}$ the parabolic subgroup of $D_{n}$ generated by
$s_{1}^{\prime},s_{1},\ldots s_{n-2}$. Easy to see that $\dot{D}_{n-1}$ is
isomorphic to $D_{n-1}$ (which satisfy the theorem by the induction
hypothesis). Notice, that in the permutation presentation of $D_{n}$, every
element $x\in\dot{D}_{n-1}$ satisfies $x(n)=n$. Now, look at the right cosets
of $\dot{D}_{n-1}$ in $D_{n}$. There are $2n$ different right cosets, where
every two elements $x$ and $y$ in the same right coset of $\dot{D}_{n-1}$ in
$D_{n}$ satisfy $x(n)=y(n)$. Now, notice that for $0\leq i_{n}<n$, the
elements of the form $t_{n}^{i_{n}}$ and $w_{n-1}\cdot t_{n}^{i_{n}}$ satisfy
the following properties:

$t_{n}^{i_{n}}(n)=n-i_{n}\quad\quad w_{n-1}\cdot t_{n}^{i_{n}}(n)=-(n-i_{n}).$

Hence, for $0\leq i_{n}\leq n-1$, $0\leq j_{n-1}\leq 1$, the elements of the
form $w_{n-1}^{j_{n-1}}\cdot t_{n}^{i_{n}}$ gives the $2n$ different images of
$n$ in the permutation presentation. Hence, for $0\leq i_{n}\leq n-1$, $0\leq
j_{n-1}\leq 1$, we have the following $2n$ different right cosets of
$\dot{D}_{n-1}$ in $D_{n}$ :

$\dot{D}_{n-1}w_{n-1}^{j_{n-1}}\cdot t_{n}^{i_{n}}.$

Then the result of the theorem holds for $k=n$. ∎

######  Example 2.1.7.

Consider the element $\pi=[-2;~{}-1;~{}-4;~{}-3]$ of $D_{4}$. Now, we
construct the $OGS$ presentation of the element, as it described in Theorem
2.1.6. First, notice $\pi(4)=-3$. Since, $-3<0$ we have $j_{3}=1$, and hence
$t_{4}^{i_{4}}(-4)=-3$, which applies $i_{4}=4-3=1$. Thus we conclude:

$\pi\in\dot{D}_{3}~{}w_{3}\cdot t_{4}.$

Now, look at $\pi(3)=-4$. Notice, that $[w_{3}\cdot t_{4}](1)=-4$. Hence, we
look for $j_{2}$ and $i_{3}$, such that $[w_{2}^{j_{2}}\cdot
t_{3}^{i_{3}}](3)=1$. Since $1>0$, we have $j_{2}=0$, and hence
$t_{3}^{i_{3}}(3)=1$, which applies $i_{3}=3-1=2$. Thus we conclude:

$\pi\in\dot{D}_{2}~{}t_{3}^{2}\cdot w_{3}\cdot t_{4}.$

Continuing by the same process, looking at $\pi(2)=-1$. Notice, that
$[t_{3}^{2}\cdot w_{3}\cdot t_{4}](-1)=-1$. Hence, we look for $j_{1}$ and
$i_{2}$, such that $[w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}](2)=-1$. Since $-1<0$,
we have $j_{1}=1$, and hence $t_{2}^{i_{2}}(2)=1$, which applies
$i_{2}=2-1=1$. Thus we conclude:

$\pi=w_{1}\cdot t_{2}\cdot t_{3}^{2}\cdot w_{3}\cdot t_{4}.$

######  Remark 2.1.8.

We call the presentation of elements of $D_{n}$ which has been shown in
Theorem 2.1.6 the generalized standard $OGS$ presentation of $D_{n}$.

We show by Theorem 2.1.9 a presentation of a subgroup of $D_{n}$ which is
isomorphic to $B_{m}$ for every $m<n$, by using the generalized standard $OGS$
of $D_{n}$.

######  Theorem 2.1.9.

The elements $w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}\cdots t_{m}^{i_{m}}\cdot
w_{m}^{j_{m}}$, such that $m$ is a positive integer less than $n$, $0\leq
j_{k}\leq 1$ and $0\leq i_{k}\leq k-1$ form a subgroup of $D_{n}$ which is
isomorphic to $B_{m}$.

###### Proof.

The proof is by induction on $m$. For $m=1$, the group generated by $w_{1}$ is
cyclic of order $2$, which is isomorphic to $B_{1}$. Notice, by Theorem 2.1.6,
the subgroup which is generated by $w_{1},t_{2},w_{2},\ldots,w_{m-1},t_{m}$,
denoted by $\dot{D}_{m}$ is isomorphic to $D_{m}$, where the permutation
presentation of every element $x\in\dot{D}_{m}$ satisfies the following
properties:

  * •

$x(k)=k$ for every $m+1\leq k\leq n$;

  * •

$|\\{1\leq k\leq m|x(k)<0\\}|$ is even.

Notice, the permutation presentation $w_{m}^{j_{m}}$ where $j_{m}=0$ or
$j_{m}=1$ satisfies:

  * •

Since $w_{m}^{0}$ is the identity, $w_{m}^{0}(k)=k$ for every $1\leq k\leq n$;

  * •

$w_{m}^{1}(1)=-1\quad w_{m}^{1}(m+1)=-(m+1)\quad w_{m}^{1}(k)=k$ for $2\leq
k\leq n$, where $k\neq m+1$.

Therefore, we have that the permutation presentation of every element of the
form $y=x\cdot w_{m}^{j_{m}}$, where $x\in\dot{D}_{m}$ satisfies $y(k)=k$ for
$m+2\leq k\leq n$, and either $y(m+1)=m+1$ or $y(m+1)=-(m+1)$. Since
$y\in\dot{D}_{m+1}$, $|\\{1\leq k\leq m+1|y(k)<0\\}|$ is even. Hence, we
conclude the following properties:

  * •

If $y(m+1)=m+1$ (i.e., $j_{m}=0$), $|\\{1\leq k\leq m|y(k)<0\\}|$ is even;

  * •

If $y(m+1)=-(m+1)$ (i.e., $j_{m}=1$), $|\\{1\leq k\leq m|y(k)<0\\}|$ is odd.

Hence, the permutation presentation of the elements of the form
$w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}\cdots t_{m}^{i_{m}}\cdot w_{m}^{j_{m}}$
reduced to the set $[\pm m]$, such that $m$ is a positive integer less than
$n$, $0\leq j_{k}\leq 1$ and $0\leq i_{k}\leq k-1$ for every $1\leq k\leq m$,
contains once every sign permutation of $[\pm m]$. Hence, the subgroup is
isomorphic to $B_{m}$. ∎

Now, we define a natural homomorphism from $D_{n}$ to $S_{n}$ by identifying
$s_{1}^{\prime}$ and $s_{1}$, and then we see the connection of the
generalized standard $OGS$ of $D_{n}$ to the standard $OGS$ of the homomorphic
image $S_{n}$.

######  Definition 2.1.10.

Consider the natural homomorphism $\Phi$ from $D_{n}$ to $S_{n}$, such that

  * •

$\Phi(s_{1^{\prime}})=s_{1}$

  * •

$\Phi(s_{i})=s_{i},\quad\text{where}\quad i\neq 1^{\prime}$

Then denote

$\Phi(\pi):=\pi^{\prime}.$

Notice, $\pi^{\prime}$ is the permutation of $S_{n}$ which satisfies:

$\pi^{\prime}(j)=|\pi(j)|,\text{ for }1\leqslant j\leqslant n$

######  Corollary 2.1.11.

Let $\pi\in D_{n}$ be presented by the generalized standard $OGS$ as it is
described in Theorem 2.1.6:

$\pi=w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}\cdot w_{2}^{j_{2}}\cdot
t_{3}^{i_{3}}\cdots w_{n-1}^{j_{n-1}}\cdot t_{n}^{i_{n}}$

Then,

$\pi^{\prime}=t_{2}^{i_{2}}\cdots t_{n}^{i_{n}}.$

###### Proof.

by Definition 2.1.3, $w_{k}=s_{k}\cdot s_{k-1}\cdots s_{2}\cdot s_{1}\cdot
s_{1}^{\prime}\cdot s_{2}\cdots s_{k}$, for every $1\leq k\leq n-1$. Then, by
Definition 2.1.10,

$\Phi(w_{k})=\Phi(s_{k}\cdot s_{k-1}\cdots s_{2}\cdot s_{1}\cdot
s_{1}^{\prime}\cdot s_{2}\cdots s_{k})=s_{k}\cdot s_{k-1}\cdots s_{2}\cdot
s_{1}\cdot s_{1}\cdot s_{2}\cdots s_{k}=1.$

Hence, in case

$\pi=w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}\cdot w_{2}^{j_{2}}\cdot
t_{3}^{i_{3}}\cdots w_{n-1}^{j_{n-1}}\cdot t_{n}^{i_{n}}$

by Definition 2.1.10, we get that

$\pi^{\prime}=\Phi(\pi)=w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}\cdot
w_{2}^{j_{2}}\cdot t_{3}^{i_{3}}\cdots w_{n-1}^{j_{n-1}}\cdot
t_{n}^{i_{n}}=t_{2}^{i_{2}}\cdot t_{3}^{i_{3}}\cdots t_{n}^{i_{n}}.$

∎

Now we show the exchange laws which arise from the generalized standard $OGS$
of $D_{n}$.

######  Proposition 2.1.12.

Consider the elements of the group $D_{n}$ expressed by the generalized
standard $OGS$ presentation. Assume $q>p$, then the following exchange laws
holds:

First, we look at the exchange law of $t_{q}^{i_{q}}\cdot t_{p}^{i_{p}}$ as it
defined in [9] ;

Now, we show the exchange laws for products which involves generator of a form
$w_{k}$.

  * •

$w_{q}\cdot w_{p}=w_{p}\cdot w_{q}.$

  * •

$t_{q}^{i_{q}}\cdot w_{p}=\left\\{\begin{array}[]{ll}w_{i_{q}}\cdot
t_{q}^{i_{q}}&q=p+i_{q}\\\ w_{i_{q}}\cdot w_{p+i_{q}}\cdot
t_{q}^{i_{q}}&q>p+i_{q}\\\ w_{p+i_{q}-q}\cdot w_{i_{q}}\cdot
t_{q}^{i_{q}}&q<p+i_{q}\end{array}\right.$

  * •

$w_{q}\cdot t_{p}^{i_{p}}=w_{i_{p}}\cdot t_{p}^{i_{p}}\cdot w_{q}.$ where
$p<q$

###### Proof.

  * •

First, consider the exchange law for $w_{q}\cdot w_{p}$.

By looking at the permutation presentation of $w_{p}$ and $w_{q}$ we get :

$w_{q}=[-1;~{}2;~{}3;\ldots;~{}p;~{}p+1;\ldots;~{}q;~{}-(q+1);~{}q+2;~{}q+3;\ldots;~{}n].$

$w_{p}=[-1;~{}2;~{}3;\ldots;~{}p;~{}-(p+1);~{}p+2;\ldots;~{}q+1;\ldots;~{}n].$

Hence we get:

$w_{q}\cdot
w_{p}=[-1;~{}2;~{}3;\ldots;~{}p;~{}-(p+1);~{}p+2\ldots;~{}q;~{}-(q+1);~{}q+2;~{}q+3;\ldots;~{}n].$

Then obviously:

$w_{q}\cdot w_{p}=w_{p}\cdot w_{q}.$

  * •

Now, consider the exchange law for $t_{q}^{i_{q}}\cdot w_{p}.$

By looking at the permutation presentation of $w_{p}$:

$1\rightarrow-1,\quad\quad p+1\rightarrow-(p+1)$ $j\rightarrow
j\quad\text{for}\quad j\neq 1\quad\text{or}\quad j\neq p+1.$

and by looking at the permutation presentation of $t_{q}^{i_{q}}$:

$i_{q}+1\rightarrow 1$

Then we get the following three subcases:

    * –

$p+i_{q}<q$;

    * –

$p+i_{q}>q$;

    * –

$p+i_{q}=q$.

The case $p+i_{q}<q$:

$p+i_{q}+1\leqslant q$ $p+i_{q}+1\rightarrow p+1.$

The case $p+i_{q}>q$:

$p+i_{q}-q+1\rightarrow p+1.$

The case $p+i_{q}=q$:

$i_{q}+1\rightarrow 1\rightarrow-1\rightarrow-(i_{q}+1)$ $1\rightarrow
p+1\rightarrow-(p+1)\rightarrow-1.$

  * •

First, consider the permutation presentation of $w_{q}$ and $t_{p}^{i_{p}}$

$w_{q}=[-1;~{}2;~{}3;\ldots;~{}q;~{}-(q+1);~{}q+2;~{}q+3;\ldots;~{}n]$
$t_{p}^{i_{p}}=[p+1-i_{p};~{}p+2-i_{p};\ldots;~{}p;~{}1;~{}2;\ldots;~{}q;~{}q+1;\ldots~{}n].$

$w_{q}\cdot
t_{p}^{i_{p}}=[-(p+1-i_{p});~{}p+2-i_{p};\ldots;~{}p;~{}1;~{}2;\ldots;~{}q;~{}-(q+1);~{}q+2;\ldots~{}n]$
$w_{q}\cdot t_{p}^{i_{p}}=t_{p}^{i_{p}}\cdot w_{p-i_{p}}\cdot w_{q}$

Now, from the previous exchange laws, we have :

$w_{q}\cdot t_{p}^{i_{p}}=w_{i_{p}}\cdot t_{p}^{i_{p}}\cdot w_{q}.$

∎

In Theorem 2.1.6, we introduce a generalized standard $OGS$ for the group
$D_{n}$, which involves two types of generators $t_{k}$ for $2\leq k\leq n$,
and $w_{L}$
for $1\leq L\leq n-1$. In Definition 2.1.10, there was introduced a natural
homomorphism $\Phi$ from $D_{n}$ onto $S_{n}$, where by Corollary 2.1.11, the
presentation of $\pi^{\prime}=\Phi(\pi)$ is derived from the presentation of
$\pi$ by omitting the generators of type $w_{L}$ from the generalized standard
$OGS$ presentation of $\pi$. Hence, it motivates us to find other properties
of the group $D_{n}$ by considering the elements of the group presented in
different forms by using $t_{k}$ and $w_{L}$. We show by the next theorem that
all the elements of $D_{n}$ which can be expressed as product of powers of
$t_{k}$ (i.e., $t_{k_{1}}^{i_{k_{1}}}\cdot t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{m}}^{i_{k_{m}}}$) or as a product of elements of the form $w_{L}$ for
$1\leq L\leq n-1$ (i.e., $w_{L_{1}}\cdot w_{L_{2}}\cdots w_{L_{m}}$) form two
important subgroups of $D_{n}$. We show that the mentioned two subgroups are
strongly connected to the homomorphism $\Phi$ from $D_{n}$ onto $S_{n}$.

######  Theorem 2.1.13.

Consider the group $D_{n}$, with the generalized standard $OGS$ presentation
as it is presented in Theorem 2.1.6.

$\pi=w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}\cdot w_{2}^{j_{2}}\cdot
t_{3}^{i_{3}}\cdots w_{n-1}^{j_{n-1}}\cdot t_{n}^{i_{n}}$

where

$0\leqslant i_{k}\leqslant k-1\quad\text{and}\quad 0\leqslant j_{k}\leqslant
1$

Then the following holds:

  * •

All the elements such that $j_{k}=0$ for every $1\leq k\leq n-1$ form a
subgroup which is isomorphic to $S_{n}$, and contains all elements $\pi\in
D_{n}$ such that the generalized standard $OGS$ presentation of $\pi$, as it
described in Theorem 2.1.6, is the same to the standard $OGS$ presentation of
$\pi^{\prime}=\pi$, as it is described in Theorem 1.0.3. (i.e.,The parabolic
subgroup of $D_{n}$, which is generated by
$\\{s_{1},s_{2},\ldots,s_{n-1}\\}$). We denoted the mentioned subgroup by
$S^{\circ}_{n}$.

  * •

All the elements such that $i_{k}=0$ for every $2\leq k\leq n$ form a subgroup
which is isomorphic to $\mathbb{Z}_{2}^{n-1}$, and contains all elements
$\pi\in D_{n}$ such that $\pi^{\prime}=1$. We denote the mentioned subgroup by
$Id^{\bullet}_{n}$.

###### Proof.

Consider the generalized standard $OGS$ presentation of $D_{n}$, where every
element $\pi\in D_{n}$ is presented in the following form:

$\pi=w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}\cdot w_{2}^{j_{2}}\cdot
t_{3}^{i_{3}}\cdots w_{n-1}^{j_{n-1}}\cdot t_{n}^{i_{n}}$

where

$0\leqslant i_{k}\leqslant k-1\quad\text{and}\quad 0\leqslant j_{k}\leqslant
1.$

The elements of $D_{n}$ which we get by considering $j_{k}=0$ for $1\leq k\leq
n-1$, are of the form

$t_{2}^{i_{2}}\cdot t_{3}^{i_{3}}\cdots t_{n}^{i_{n}}$ (1)

where,

$0\leqslant i_{k}\leqslant k-1\quad\text{for}\quad 2\leq k\leq n.$

Hence, by Theorem 1.0.3, the set of all elements of $D_{n}$ which generalized
standard $OGS$ presentation can be expressed by Equation 1 form a subgroup of
$D_{n}$ which is isomorphic to $S_{n}$. Now, by Theorem 1.0.3,
$t_{k}=\prod_{r=1}^{k-1}s_{r}$, for $2\leq k\leq n$. Hence, there is no
occurrence of the Coxeter generator $s_{1}^{\prime}$ in the normal form of
$t_{k}$ for $2\leq k\leq n$. Thus by Corollary 2.1.11, the generalized
standard $OGS$ presentation of $\pi$ is the same to the standard $OGS$
presentation of $\pi^{\prime}$, and by considering $\pi$ as an element of
$S_{n}$, $\pi$ presents the same permutation as $\pi^{\prime}$.

Moreover, If there is $1\leq k\leq n-1$ such that $j_{k}\neq 0$ in the
generalized standard $OGS$ presentation of an element $\pi\in D_{n}$, then by
Corollary 2.1.11, the standard $OGS$ presentation of $\pi^{\prime}$ (as it is
described in Theorem 1.0.3) is different from the generalized standard $OGS$
presentation of $\pi$ (as it is described in Theorem 2.1.6), since there is an
appearance of $w_{k}$ in the generalized standard $OGS$ presentation of an
element $\pi\in D_{n}$, which does not occur in the standard $OGS$
presentation of $\pi^{\prime}$.

Now, consider the elements of $D_{n}$ such that $i_{k}=0$ for $2\leq n$ in
generalized standard $OGS$ presentation. Then, we get the elements of the
form:

$w_{1}^{j_{1}}\cdot w_{2}^{j_{2}}\cdots w_{n-1}^{j_{n-1}}$ (2)

where,

$0\leqslant j_{k}\leqslant 1\quad\text{for}\quad 1\leq k\leq n-1.$

Notice, by Proposition 2.1.12, $w_{q}\cdot w_{p}=w_{p}\cdot w_{q}$ for every
$1\leq p,q\leq n-1$, and by Lemma 2.1.4, $w_{k}^{2}=1$ for every $1\leq k\leq
n-1$. Therefore, the elements of $D_{n}$ which generalized standard $OGS$
presentation can be expressed by Equation 2 form an abelian group which is
isomorphic to $\mathbb{Z}_{2}^{n-1}.$ Now, by Corollary 2.1.11, every element
$\pi\in D_{n}$ such that the generalized standard $OGS$ presentation of it can
be expressed by Equation 2 satisfies $\pi^{\prime}=1$. Now, we prove that the
generalized standard $OGS$ presentation of an element $\pi$ such that
$\pi^{\prime}=1$ can be expressed by Equation 2. We prove it by the
permutation presentation of $\pi$. By Definition 2.1.10, if $\pi^{\prime}=1$,
then $|\pi(k)|=k$ for every $1\leq k\leq n$. Hence, either $\pi(k)=k$ or
$\pi(k)=-k$ for every $1\leq k\leq n$. Let $k_{1},k_{2},\ldots k_{q}$ be $q$
integers such that

  * •

$2\leq k_{1}<k_{2}<\ldots<k_{q}\leq n$.

  * •

$\pi(k_{1})=-k_{1},\quad\pi(k_{2})=-k_{2},\quad\ldots,\quad\pi(k_{q})=-k_{q}$.

  * •

If $k\neq k_{p}$ for any $1\leq p\leq q$ and $2\leq k\leq n$, then $\pi(k)=k$.

Now, since by Definition 2.0.1, every $\pi$ such that $\pi^{\prime}=1$,
satisfies that the number of $k$ such that $\pi(k)=-k$ for $1\leq k\leq n$ is
even, $\pi(1)$ is determined by the parity of $q$ (i.e, if $q$ is even then
$\pi(1)=1$ and if $q$ is odd then $\pi(1)=-1$. By Remark 2.1.5, for every
$1\leq k\leq n-1$, $w_{k}$ satisfies the following

$\begin{array}[]{l}w_{k}(1)=-1\\\ w_{k}(k+1)=-(k+1)\\\ w_{k}(j)=j,\text{ for
}2\leqslant j\leqslant k\quad\text{and}\quad k+2\leqslant j\leqslant
n.\end{array}.$

Thus we get that

$\pi=w_{k_{1}-1}\cdot w_{k_{2}-1}\cdots w_{k_{q}-1}.$

Hence, every $\pi\in D_{n}$ such that $\pi^{\prime}=1$, the generalized
standard $OGS$ presentation of $\pi$ can be expressed by Equation 2.

∎

######  Corollary 2.1.14.

Let $\pi\in D_{n}$, then $\pi$ can be presented in the form:

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\mu-1}^{\circ}\cdot\pi_{\mu-1}^{\bullet}\cdot\pi_{\mu}^{\circ}.$

such that the following holds:

  * •

For $1\leq j\leq\mu$, $\pi_{j}^{\circ}\in S^{\circ}_{n}$ (as it is defined in
Theorem 2.1.13) such that :

    * –

For $j=1$ or $j=\mu$, either $\pi_{j}^{\circ}=1$ or the generalized standard
$OGS$ presentation of $\pi_{j}^{\circ}$ has the following form

$\pi_{j}^{\circ}=t_{k_{r_{j-1}+1}}^{i_{k_{r_{j-1}+1}}}\cdots
t_{k_{r_{j}}}^{i_{k_{r_{j}}}}$

for some integer $r_{j}$;

    * –

For $2\leq j\leq\mu-1$, the generalized standard $OGS$ presentation of
$\pi_{j}^{\circ}$ has the following form:

$\pi_{j}^{\circ}=t_{k_{r_{j-1}+1}}^{i_{k_{r_{j-1}+1}}}\cdots
t_{k_{r_{j}}}^{i_{k_{r_{j}}}}$

for some integer $r_{j}$,

  * •

For $1\leq j\leq\mu-1$, $\pi_{j}^{\bullet}\in Id^{\bullet}_{n}$ (as it is
defined in Theorem 2.1.13) such that:

    * –

The generalized standard $OGS$ presentation of $\pi_{j}^{\bullet}$ has the
following form:

$\pi_{j}^{\bullet}=w_{L_{j_{1}}}\cdot w_{L_{j_{2}}}\cdots
w_{L_{j_{\nu_{j}}}},$

where $\nu_{j}$ denoted the number of non-zero powers of elements of the form
$w_{L_{j}}$ in the generalized standard $OGS$ presentation (as it is described
in Theorem 2.1.6) of $\pi_{j}^{\bullet}$, $L_{j_{1}}\geq k_{r_{j}}$,
$L_{j_{u}}>L_{j_{u-1}}$ for every $2\leq u\leq\nu_{j}$, and
$L_{j_{\nu_{j}}}<t_{k_{r_{j}}+1}$.

Then, the generalized standard $OGS$ presentation of $\pi$ is presented as
follow:

  * •

In case $\pi_{1}^{\circ}\neq 1$ and $\pi_{\mu}^{\circ}\neq 1$:

$\pi=t_{k_{1}}^{i_{k_{1}}}\cdots t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot
t_{k_{r_{\mu-1}+1}}^{i_{k_{r_{\mu-1}+1}}}\cdots
t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}.$

and then,

$\pi^{\prime}=\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\mu}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\cdot t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots
t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdots t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}.$

  * •

In case $\pi_{1}^{\circ}\neq 1$ and $\pi_{\mu}^{\circ}=1$:

$\pi=t_{k_{1}}^{i_{k_{1}}}\cdots t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}.$

and then,

$\pi^{\prime}=\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\mu}^{\circ}=\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\mu-1}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\cdot t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots
t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdots t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}.$

  * •

In case $\pi_{1}^{\circ}=1$ and $\pi_{\mu}^{\circ}\neq 1$ (Notice, in this
case $r_{1}=0$):

$\pi=w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot
t_{k_{r_{\mu-1}+1}}^{i_{k_{r_{\mu-1}+1}}}\cdots
t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}.$

and then,

$\pi^{\prime}=\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\mu}^{\circ}=\pi_{2}^{\circ}\cdot\pi_{3}^{\circ}\cdots\pi_{\mu}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot t_{k_{r_{2}+1}}^{i_{k_{r_{2}+1}}}\cdots
t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdots t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}.$

  * •

In case $\pi_{1}^{\circ}=1$ and $\pi_{\mu}^{\circ}=1$ (Notice, in this case
$r_{1}=0$):

$\pi=w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}.$

and then,

$\pi^{\prime}=\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\mu}^{\circ}=\pi_{2}^{\circ}\cdot\pi_{3}^{\circ}\cdots\pi_{\mu-1}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot t_{k_{r_{2}+1}}^{i_{k_{r_{2}+1}}}\cdots
t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdots t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}.$

###### Proof.

The generalized standard $OGS$ of $\pi$ as follow,

$\pi=w_{1}^{j_{1}}\cdot t_{2}^{i_{2}}\cdot w_{2}^{j_{2}}\cdots t_{n}^{i_{n}},$

If $i_{k}=0$ or $j_{k}=0$, we may omit $t_{k}^{i_{k}}$ $w_{k}^{j_{k}}$ since
$t_{k}^{0}=w_{k}^{0}=1$. Then by considering generators $t_{k}$ and $w_{k}$ in
the generalized standard $OGS$ with non-zero powers only, we get the result of
the proposition. ∎

In Theorem 2.1.6 we introduced the generalized standard $OGS$ presentation of
the elements in the group $D_{n}$, by using two types of generators, $t_{k}$
for
$2\leq k\leq n$ and $w_{L}$ for $1\leq L\leq n-1$. In Theorem 2.1.13, we show
that the elements of $D_{n}$ which can be presented by using just one of the
types of the generators ($t_{k}$ or $w_{L}$) in the generalized standard $OGS$
presentation, form a subgroup of $D_{n}$, where the subgroup which contains
only products of elements of the form $w_{L}$ (for $1\leq L\leq n-1$) is
denoted by $Id_{n}^{\bullet}$ and the subgroup which contains only products of
elements of the form $t_{k}$ (for $2\leq k\leq n$) is denoted by
$S_{n}^{\circ}$. Now, we show an algorithm to transform an element $\pi\in
D_{n}$ presented by the generalized standard $OGS$ presentation, to a
presentation of $\pi$ as a product of an element in $Id_{n}^{\bullet}$ and an
element in $S_{n}^{\circ}$, where by Theorem 2.1.17 we give an explicit
formula for it in case where $\pi^{\prime}$ is a standard $OGS$ elementary
element (as it is defined in Definition 1.0.6). We start with the following
lemma, which we use in the proof of Theorem 2.1.17.

######  Lemma 2.1.15.

Let $\pi=\pi_{1}\cdot w_{q}$ such that the following holds:

  * •

$\pi_{1}\in S^{\circ}_{n}$;

  * •

$\pi_{1}=t_{k_{1}}^{i_{k_{1}}}\cdot t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{m}}^{i_{k_{m}}}$ is a standard $OGS$ elementary element (as it is
defined in Definition 1.0.6) by considering $\pi_{1}$ as an element of
$S_{n}$;

  * •

$q\geq k_{m}$. i.e., the presentation of

$\pi=t_{k_{1}}^{i_{k_{1}}}\cdot t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{m}}^{i_{k_{m}}}\cdot w_{q}$

is a generalized standard $OGS$ presentation of $\pi\in D_{n}$ as it is
presented in Theorem 2.1.6.

Then,

$\pi=\pi_{1}\cdot w_{q}=\begin{cases}w_{q}\cdot\pi_{1}&\text{if}\quad
maj(\pi_{1})=k_{1}\\\ w_{maj(\pi_{1})}\cdot w_{q}\cdot\pi_{1}&\text{if}\quad
maj(\pi_{1})<k_{1}.\end{cases}$

###### Proof.

The proof is by induction on $m$. We start with m=1: By Proposition 2.1.12:

$w_{q}\cdot t_{k_{1}}^{i_{k_{1}}}=w_{i_{k_{1}}}\cdot
t_{k_{1}}^{i_{k_{1}}}\cdot w_{q},$

where, $k_{1}\leq q$.

By Lemma 2.1.4, $w_{i_{k_{1}}}^{2}=1$. Therefore,

$t_{k_{1}}^{i_{k_{1}}}\cdot w_{q}=w_{i_{k_{1}}}\cdot w_{q}\cdot
t_{k_{1}}^{i_{k_{1}}}.$

Now, assume by induction the lemma holds for $m=j$,
i.e., for $\pi_{1}=t_{k_{1}}^{i_{k_{1}}}\cdot t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{j}}^{i_{k_{j}}}$ a standard $OGS$ elementary element by considering
$\pi_{1}$ as an element of $S_{n}$ and $q\geq k_{j}$,

$\pi_{1}\cdot w_{q}=\begin{cases}w_{q}\cdot\pi_{1}&\text{if}\quad
maj(\pi_{1})=k_{1}\\\ w_{maj(\pi_{1})}\cdot w_{q}\cdot\pi_{1}&\text{if}\quad
maj(\pi_{1})<k_{1}.\end{cases}.$

Now, consider $\pi_{1}=t_{k_{1}}^{i_{k_{1}}}\cdot t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{j}}^{i_{k_{j}}}\cdot t_{k_{j+1}}^{i_{k_{j+1}}}$ a standard $OGS$
elementary element by considering $\pi_{1}$ as an element of $S_{n}$. Then, by
[9], Theorem 28, the subword $\pi_{2}=t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{j}}^{i_{k_{j}}}\cdot t_{k_{j+1}}^{i_{k_{j+1}}}$ of $\pi_{1}$ is a
standard $OGS$ elementary element as well by considering $\pi_{2}$ as an
element of $S_{n}$. Since, $\pi_{2}$ is a proper subword of $\pi_{1}$ (i.e.,
$\pi_{2}\neq\pi_{1}$), by [9], Theorem 28, $maj(\pi_{2})<k_{2}$. Then, by the
induction assumption:

$\pi_{1}\cdot w_{q}=t_{k_{1}}^{i_{k_{1}}}\cdot t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{j+1}}^{i_{k_{j+1}}}\cdot w_{q}=t_{k_{1}}^{i_{k_{1}}}\cdot\pi_{2}\cdot
w_{q}=t_{k_{1}}^{i_{k_{1}}}\cdot w_{maj(\pi_{2})}\cdot w_{q}\cdot\pi_{2}$
$=(t_{k_{1}}^{i_{k_{1}}}\cdot w_{q})\cdot
w_{maj(\pi_{2})}\cdot\pi_{2}=w_{i_{k_{1}}}\cdot w_{q}\cdot
t_{k_{1}}^{i_{k_{1}}}\cdot w_{maj(\pi_{2})}\cdot\pi_{2}.$

Notice,

$maj(\pi_{2})=\sum_{x=2}^{j+1}i_{k_{x}}<\sum_{x=1}^{j+1}i_{k_{x}}=maj(\pi_{1}).$

Since, $\pi_{1}$ is a standard $OGS$ elementary element, by Definition 1.0.6,
$maj(\pi_{1})\leq k_{1}=i_{k_{1}}+maj(\pi_{2})\leq k_{1}$. Hence,
$maj(\pi_{2})<maj(\pi_{1})\leq k_{1}$. Then, by applying Proposition 2.1.12,

$t_{k_{1}}^{i_{k_{1}}}\cdot w_{maj(\pi_{2})}=\begin{cases}w_{i_{k_{1}}}\cdot
t_{k_{1}}^{i_{k_{1}}}&\text{if}\quad
i_{k_{1}}+maj(\pi_{2})=maj(\pi_{1})=k_{1}\\\ w_{i_{k_{1}}}\cdot
w_{maj(\pi_{1})}\cdot t_{k_{1}}^{i_{k_{1}}}&\text{if}\quad
i_{k_{1}}+maj(\pi_{2})=maj(\pi_{1})<k_{1}.\end{cases}.$

Hence we get

$\pi_{1}\cdot w_{q}=w_{i_{k_{1}}}\cdot w_{q}\cdot(t_{k_{1}}^{i_{k_{1}}}\cdot
w_{maj(\pi_{2})})\cdot\pi_{2}=$ $=\begin{cases}w_{i_{k_{1}}}\cdot w_{q}\cdot
w_{i_{k_{1}}}\cdot t_{k_{1}}^{i_{k_{1}}}\cdot\pi_{2}&\text{if}\quad
maj(\pi_{1})=k_{1}\\\ w_{i_{k_{1}}}\cdot w_{q}\cdot w_{i_{k_{1}}}\cdot
w_{maj(\pi_{1})}\cdot t_{k_{1}}^{i_{k_{1}}}\cdot\pi_{2}&\text{if}\quad
maj(\pi_{1})<k_{1}.\end{cases}.$

Since by Lemma 2.1.4, $w_{i_{k_{1}}}^{2}=1$ and by the definition of $\pi_{2}$
in the lemma we have $t_{k_{1}}^{i_{k_{1}}}\cdot\pi_{2}=\pi_{1}$. Hence,

$w_{i_{k_{1}}}\cdot w_{q}\cdot
w_{i_{k_{1}}}\cdot(t_{k_{1}}^{i_{k_{1}}}\cdot\pi_{2})=w_{q}\cdot\pi_{1}\quad
w_{i_{k_{1}}}\cdot w_{q}\cdot w_{i_{k_{1}}}\cdot
w_{maj(\pi_{1})}\cdot(t_{k_{1}}^{i_{k_{1}}}\cdot\pi_{2})=w_{maj(\pi_{1})}\cdot
w_{q}\cdot\pi_{1}.$

Hence, we get for $\pi_{1}=t_{k_{1}}^{i_{k_{1}}}\cdot
t_{k_{2}}^{i_{k_{2}}}\cdots t_{k_{j}}^{i_{k_{j}}}\cdot
t_{k_{j+1}}^{i_{k_{j+1}}}:$

$\pi_{1}\cdot w_{q}=\begin{cases}w_{q}\cdot\pi_{1}&\text{if}\quad
maj(\pi_{1})=k_{1}\\\ w_{maj(\pi_{1})}\cdot w_{q}\cdot\pi_{1}&\text{if}\quad
maj(\pi_{1})<k_{1}.\end{cases}.$

Hence, the theorem holds for every $\pi=\pi_{1}\cdot w_{q}$ such that
$\pi_{1}$ is a standard $OGS$ elementary element by considering $\pi_{1}$ as
an element of $S_{n}$.

∎

######  Definition 2.1.16.

Let $\pi\in D_{n}$. Consider the presentation of $\pi$ as it is presented in
Corollary 2.1.14, with all the notations of the corollary. i.e.,

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\mu-1}^{\circ}\cdot\pi_{\mu-1}^{\bullet}\cdot\pi_{\mu}^{\circ}$
$=\pi_{1}^{\circ}\cdot w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot\pi_{\mu}^{\circ},$

where,

$\text{either}\quad\pi_{1}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\quad\text{or}\quad\pi_{1}^{\circ}=1,$

and,

$\text{either}\quad\pi_{\mu}^{\circ}=t_{k_{r_{\mu-1}+1}}^{i_{k_{r_{\mu-1}+1}}}\cdots
t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}\quad\text{or}\quad\pi_{\mu}^{\circ}=1.$

Then for every $1\leq\alpha\leq\mu$, we define $maj_{\alpha}(\pi)$ and
$\rho_{\alpha}(\pi)$ to be

$maj_{\alpha}(\pi)=maj(\prod_{j=1}^{\alpha}\pi_{j}^{\circ})=\sum_{j=1}^{r_{\alpha}}i_{k_{j}},$
$\rho_{\alpha}(\pi)=maj(\pi^{\prime})-maj_{\alpha}(\pi)=maj(\prod_{j=\alpha+1}^{\mu}\pi_{j}^{\circ})=\sum_{j=r_{\alpha}+1}^{r_{\mu}}i_{k_{j}}.$

For every $L_{\alpha_{j}}$, where $1\leq\alpha\leq\mu$,$1\leq
j\leq\nu_{\alpha}$ we define $\varrho_{L_{\alpha_{j}}}(\pi)$ to be

$\varrho_{L_{\alpha_{j}}}(\pi)=\begin{cases}L_{\alpha_{j}}-\rho_{\alpha}(\pi)&\text{if}\quad
L_{\alpha_{j}}\geq maj(\pi^{\prime})\\\ L_{\alpha_{j}}&\text{if}\quad
L_{\alpha_{j}}<maj(\pi^{\prime})\end{cases}$

######  Theorem 2.1.17.

Let $\pi\in D_{n}$ such that $\pi^{\prime}$ is a standard $OGS$ elementary
element of $S_{n}$. Consider the presentation of $\pi$ as it presented in
Corollary 2.1.14, with all the notations. i.e.,

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\mu-1}^{\circ}\cdot\pi_{\mu-1}^{\bullet}\cdot\pi_{\mu}^{\circ}$
$=\pi_{1}^{\circ}\cdot w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot\pi_{\mu}^{\circ},$

where,

$\text{either}\quad\pi_{1}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\quad\text{or}\quad\pi_{1}^{\circ}=1,$

and,

$\text{either}\quad\pi_{\mu}^{\circ}=t_{k_{r_{\mu-1}+1}}^{i_{k_{r_{\mu-1}+1}}}\cdots
t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}\quad\text{or}\quad\pi_{\mu}^{\circ}=1.$

Then, $\pi$ can be presented uniquely in the form:

$\pi=\pi^{\bullet}\cdot\pi^{\circ}\quad\text{such that}\quad\pi^{\bullet}\in
Id^{\bullet}_{n},\quad\pi^{\circ}\in S^{\circ}_{n}$

($Id^{\bullet}_{n}$ and $S^{\circ}_{n}$ are defined in Theorem 2.1.13)

where,

$\pi^{\bullet}=\prod_{\alpha=1}^{\mu-1}w_{maj_{\alpha~{}|~{}maj_{\alpha}(\pi)<k_{1}}(\pi)}^{-0.5\cdot(-1)^{\nu_{\alpha}}+0.5}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot w_{L_{2_{1}}}\cdots
w_{L_{2_{\nu_{2}}}}\cdots w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}},$

where, for every $\pi\in D_{n}$, $maj_{\alpha}(\pi)$ for
$1\leq\alpha\leq\mu-1$ as it is defined in Definition 2.1.16.

$\pi^{\circ}=\prod_{j=1}^{r_{\mu}}t_{k_{j}}^{i_{k_{j}}}.$

Notice, considering $\pi^{\circ}$ as an element of $S_{n}$, the standard $OGS$
presentation (as it is presented in Theorem 1.0.3) of $\pi^{\circ}$ is the
same to the standard $OGS$ presentation of $\pi^{\prime}$ (i.e., $\pi^{\circ}$
and $\pi^{\prime}$ present the same element of $S_{n}$.).

###### Proof.

The proof is by induction on $\mu$. If $\mu=1$, then $\pi=\pi_{1}^{\circ}$,
and then the theorem holds trivially. Hence, we start with $\mu=2$. If $\mu=2$
and $\pi_{1}^{\circ}=1$, then $\pi^{\bullet}=\pi_{1}^{\bullet}$ and
$\pi^{\circ}=\pi_{2}^{\circ}$. Hence, the theorem holds trivially. Therefore,
let assume $\mu=2$ and $\pi_{1}^{\circ}\neq 1$. Then,

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}=\begin{cases}t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\cdot w_{L_{1_{1}}}\cdots
w_{L_{1_{\nu_{1}}}}\cdot t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots
t_{k_{r_{2}}}^{i_{k_{r_{2}}}}&\text{if}\quad\pi_{2}^{\circ}\neq 1\\\
t_{k_{1}}^{i_{k_{1}}}\cdots t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\cdot
w_{L_{1_{1}}}\cdots
w_{L_{1_{\nu_{1}}}}&\text{if}\quad\pi_{2}^{\circ}=1,\end{cases}$

where,

  * •

$k_{r_{1}}\leq L_{1_{1}}$;

  * •

$L_{1_{u}}<L_{1_{u+1}}$ for $1\leq u\leq\nu_{1}-1$;

  * •

$L_{1_{\nu_{1}}}<k_{r_{1}+1}$.

Since by Corollary 2.1.11, the standard $OGS$ presentation (as it is described
in Theorem 1.0.3) of $\pi^{\prime}$ is the same to the generalized standard
$OGS$ presentation of $\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}$ (as it is
described in Theorem 2.1.6), we get that by considering
$\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}$ as an element of $S_{n}$, it presents
the same element to $\pi^{\prime}$. Therefore,
$\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}$ is a standard $OGS$ elementary element
by considering it as an element of $S_{n}$, and then by [9], Theorem 28,
$\pi_{1}^{\circ}$ and $\pi_{2}^{\circ}$ are standard $OGS$ elementary elements
as well, by considering $\pi_{1}^{\circ}$ and $\pi_{2}^{\circ}$ as elements of
$S_{n}$. Then, by applying Lemma 2.1.15 on $\pi^{\circ}\cdot w_{L_{1_{u}}}$
for $1\leq u\leq\nu_{1}$, we have:

$\pi_{1}^{\circ}\cdot w_{L_{1_{u}}}=\begin{cases}w_{maj(\pi_{1}^{\circ})}\cdot
w_{L_{1_{u}}}\cdot\pi_{1}^{\circ}&\text{if}\quad maj(\pi_{1}^{\circ})<k_{1}\\\
w_{L_{1_{u}}}\cdot\pi_{1}^{\circ}&\text{if}\quad
maj(\pi_{1}^{\circ})=k_{1}\end{cases}$

Since by Definition 2.1.16, $maj_{1}(\pi)=maj(\pi_{1}^{\circ})$, we get

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}=\begin{cases}t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\cdot w_{L_{1_{1}}}\cdots
w_{L_{1_{\nu_{1}}}}\cdot t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots
t_{k_{r_{2}}}^{i_{k_{r_{2}}}}&\text{if}\quad\pi_{2}^{\circ}\neq 1\\\
t_{k_{1}}^{i_{k_{1}}}\cdots t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\cdot
w_{L_{1_{1}}}\cdots
w_{L_{1_{\nu_{1}}}}&\text{if}\quad\pi_{2}^{\circ}=1,\end{cases}$
$=\begin{cases}(w_{maj_{1}}(\pi))^{\nu_{1}}\cdot w_{L_{1_{1}}}\cdots
w_{L_{1_{\nu_{1}}}}\cdot\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}&\text{if}\quad\pi_{2}^{\circ}\neq
1\\\ (w_{maj_{1}}(\pi))^{\nu_{1}}\cdot w_{L_{1_{1}}}\cdots
w_{L_{1_{\nu_{1}}}}\cdot\pi_{1}^{\circ}&\text{if}\quad\pi_{2}^{\circ}=1\quad\text{and}\quad
maj_{1}(\pi)<k_{1}\\\ w_{L_{1_{1}}}\cdots
w_{L_{1_{\nu_{1}}}}\cdot\pi_{1}^{\circ}&\text{if}\quad\pi_{2}^{\circ}=1\quad\text{and}\quad
maj_{1}(\pi)=k_{1}.\end{cases}$

Hence, the theorem holds for $\mu=2$.

Now, assume by induction the theorem holds for $\mu=\alpha$, and we prove it
for $\mu=\alpha+1$.

Hence, we assume by induction:

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\alpha-1}^{\circ}\cdot\pi_{\alpha-1}^{\bullet}\cdot\pi_{\alpha}^{\circ}$
$=\pi_{1}^{\circ}\cdot w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\alpha-1}}}^{i_{k_{r_{\alpha-1}}}}\cdot w_{L_{{\alpha-1}_{1}}}\cdots
w_{L_{{\alpha-1}_{\nu_{\alpha-1}}}}\cdot\pi_{\alpha}^{\circ}$
$=\prod_{u=1}^{\alpha-1}\big{(}w_{maj_{u~{}|~{}maj_{u}(\pi)<k_{1}}(\pi)}\big{)}^{\nu_{u}}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot w_{L_{2_{1}}}\cdots
w_{L_{2_{\nu_{2}}}}\cdots$ $\cdots w_{L_{{\alpha-1}_{1}}}\cdots
w_{L_{{\alpha-1}_{\nu_{\alpha-1}}}}\cdot\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\alpha}^{\circ}.$

where,

$\text{either}\quad\pi_{1}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\quad\text{or}\quad\pi_{1}^{\circ}=1,$

and

$\text{either}\quad\pi_{\alpha}^{\circ}=t_{k_{r_{\alpha-1}+1}}^{i_{k_{r_{\alpha-1}+1}}}\cdots
t_{k_{r_{\alpha}}}^{i_{k_{r_{\alpha}}}}\quad\text{or}\quad\pi_{\alpha}^{\circ}=1.$

and

  * •

$k_{r_{u}}\leq L_{u_{1}}$ for $1\leq u\leq\alpha-1$;

  * •

$L_{u_{v}}<L_{u_{v+1}}$ for $1\leq u\leq\alpha-1$, and $1\leq v\leq\nu_{u}-1$;

  * •

$L_{u_{\nu_{u}}}<k_{r_{u}+1}$ $1\leq u\leq\alpha-1$ in case
$\pi_{\alpha}^{\circ}\neq 1$;

  * •

$L_{u_{\nu_{u}}}<k_{r_{u}+1}$ $1\leq u\leq\alpha-2$ in case
$\pi_{\alpha}^{\circ}=1$ (Since, there is no $k_{r_{\alpha-1}+1}$ in case
$\pi_{\alpha}^{\circ}=1$).

Now, consider $\mu=\alpha+1$. Then,

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\alpha-1}^{\circ}\cdot\pi_{\alpha-1}^{\bullet}\cdot\pi_{\alpha}^{\circ}\cdot\pi_{\alpha}^{\bullet}\cdot\pi_{\alpha+1}^{\circ}$
$=\pi_{1}^{\circ}\cdot w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\alpha}}}^{i_{k_{r_{\alpha}}}}\cdot w_{L_{{\alpha}_{1}}}\cdots
w_{L_{{\alpha}_{\nu_{\alpha}}}}\cdot\pi_{\alpha+1}^{\circ},$

where,

$\text{either}\quad\pi_{1}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\quad\text{or}\quad\pi_{1}^{\circ}=1,$

and

$\text{either}\quad\pi_{\alpha+1}^{\circ}=t_{k_{r_{\alpha}+1}}^{i_{k_{r_{\alpha}+1}}}\cdots
t_{k_{r_{\alpha+1}}}^{i_{k_{r_{\alpha+1}}}}\quad\text{or}\quad\pi_{\alpha+1}^{\circ}=1.$

and

  * •

$k_{r_{u}}\leq L_{u_{1}}$ for $1\leq u\leq\alpha$;

  * •

$L_{u_{v}}<L_{u_{v+1}}$ for $1\leq u\leq\alpha$, and $1\leq v\leq\nu_{u}-1$;

  * •

$L_{u_{\nu_{u}}}<k_{r_{u}+1}$ $1\leq u\leq\alpha$ in case
$\pi_{\alpha+1}^{\circ}\neq 1$;

  * •

$L_{u_{\nu_{u}}}<k_{r_{u}+1}$ $1\leq u\leq\alpha-1$ in case
$\pi_{\alpha+1}^{\circ}=1$ (Since, there is no $k_{r_{\alpha}+1}$ in case
$\pi_{\alpha+1}^{\circ}=1$).

In case $\pi_{1}^{\circ}=1$,

$\pi=\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\alpha-1}^{\circ}\cdot\pi_{\alpha-1}^{\bullet}\cdot\pi_{\alpha}^{\circ}\cdot\pi_{\alpha}^{\bullet}\cdot\pi_{\alpha+1}^{\circ}$
$=w_{L_{1_{1}}}\cdots
w_{L_{1_{\nu_{1}}}}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\alpha-1}^{\circ}\cdot\pi_{\alpha-1}^{\bullet}\cdot\pi_{\alpha}^{\circ}\cdot\pi_{\alpha}^{\bullet}\cdot\pi_{\alpha+1}^{\circ}.$

Notice,

$\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\alpha-1}^{\circ}\cdot\pi_{\alpha-1}^{\bullet}\cdot\pi_{\alpha}^{\circ}\cdot\pi_{\alpha}^{\bullet}\cdot\pi_{\alpha+1}^{\circ}=$
$=\prod_{u=2}^{\alpha}\big{(}w_{maj_{u~{}|~{}maj_{u}(\pi)<k_{1}}(\pi)}\big{)}^{\nu_{u}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot w_{L_{3_{1}}}\cdots
w_{L_{3_{\nu_{3}}}}\cdots$ $\cdots w_{L_{{\alpha}_{1}}}\cdots
w_{L_{{\alpha}_{\nu_{\alpha}}}}\cdot\pi_{2}^{\circ}\cdots\pi_{\alpha+1}^{\circ}.$

by the induction hypothesis.

Hence, in case $\pi_{1}^{\circ}=1$ we have:

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\alpha-1}^{\circ}\cdot\pi_{\alpha-1}^{\bullet}\cdot\pi_{\alpha}^{\circ}\cdot\pi_{\alpha}^{\bullet}\cdot\pi_{\alpha+1}^{\circ}=$
$=\prod_{u=1}^{\alpha}\big{(}w_{maj_{u~{}|~{}maj_{u}(\pi)<k_{1}}(\pi)}\big{)}^{\nu_{u}}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot w_{L_{2_{1}}}\cdots
w_{L_{2_{\nu_{2}}}}\cdots$ $\cdots w_{L_{{\alpha}_{1}}}\cdots
w_{L_{{\alpha}_{\nu_{\alpha}}}}\cdot\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\alpha+1}^{\circ}.$

Hence, the theorem holds for $\mu=\alpha+1$ in case $\pi_{1}^{\circ}=1$.
Now, we consider

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\alpha-1}^{\circ}\cdot\pi_{\alpha-1}^{\bullet}\cdot\pi_{\alpha}^{\circ}\cdot\pi_{\alpha}^{\bullet}\cdot\pi_{\alpha+1}^{\circ}$

where

$\pi_{1}^{\circ}\neq 1.$

Then,

$\pi=t_{k_{1}}^{i_{k_{1}}}\cdots t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\alpha}}}^{i_{k_{r_{\alpha}}}}\cdot w_{L_{{\alpha}_{1}}}\cdots
w_{L_{{\alpha}_{\nu_{\alpha}}}}\cdot\pi_{\alpha+1}^{\circ}$

By the induction hypothesis:

$\pi=\prod_{u=1}^{\alpha-1}\big{(}w_{maj_{u~{}|~{}maj_{u}(\pi)<k_{1}}(\pi)}\big{)}^{\nu_{u}}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot w_{L_{2_{1}}}\cdots
w_{L_{2_{\nu_{2}}}}\cdots$ $\cdots w_{L_{{\alpha-1}_{1}}}\cdots
w_{L_{{\alpha-1}_{\nu_{\alpha-1}}}}\cdot\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\alpha}^{\circ}\cdot
w_{L_{{\alpha}_{1}}}\cdots
w_{L_{{\alpha}_{\nu_{\alpha}}}}\cdot\pi_{\alpha+1}^{\circ}.$

Now, consider

$(\prod_{j=1}^{\alpha}\pi_{j}^{\circ})\cdot w_{L_{\alpha_{k}}}$

for every $1\leq k\leq\nu_{\alpha}$. Then by Lemma 2.1.15 we conclude:

$(\prod_{j=1}^{\alpha}\pi_{j}^{\circ})\cdot
w_{L_{\alpha_{k}}}=\begin{cases}w_{maj_{\alpha}(\pi)}\cdot
w_{L_{\alpha_{k}}}\cdot\prod_{j=1}^{\alpha}\pi_{j}^{\circ}&\text{if}\quad
maj_{\alpha}(\pi)<k_{1}\\\
w_{L_{\alpha_{k}}}\cdot\prod_{j=1}^{\alpha}\pi_{j}^{\circ}&\text{if}\quad
maj_{\alpha}(\pi)=k_{1}\end{cases}$

Hence, we get:

$\pi=\prod_{u=1}^{\alpha}w_{maj_{u~{}|~{}maj_{u}(\pi)<k_{1}}(\pi)}^{\nu_{u}}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot w_{L_{2_{1}}}\cdots
w_{L_{2_{\nu_{2}}}}\cdots w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot$
$\cdot\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\alpha}^{\circ}\cdot\pi_{\alpha+1}^{\circ}.$

Now, since by Lemma 2.1.4, $w_{maj_{\alpha}(\pi)}^{2}=1$ for all
$1\leq\alpha\leq\mu-1$, and since the following holds:

$-0.5\cdot(-1)^{\nu_{\alpha}}+0.5=\left\\{\begin{array}[]{ll}0&\quad\text{if}\quad\nu_{\alpha}\mod
2=0\\\ \\\ 1&\quad\text{if}\quad\nu_{\alpha}\mod 2=1\par\end{array}\right.$

we get the result of the theorem.

∎

Now, we give the combinatorial meaning of Theorem 2.1.17 by considering
$\pi\in D_{n}$ as a sign-permutation.

######  Corollary 2.1.18.

Let $\pi\in D_{n}$ such that $\pi^{\prime}$ is a standard $OGS$ elementary
element of $S_{n}$. Consider the presentation of $\pi$ as it presented in
Corollary 2.1.14, with all the notations. i.e.,

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\mu-1}^{\circ}\cdot\pi_{\mu-1}^{\bullet}\cdot\pi_{\mu}^{\circ}$
$=\pi_{1}^{\circ}\cdot w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot\pi_{\mu}^{\circ},$

where,

$\text{either}\quad\pi_{1}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\quad\text{or}\quad\pi_{1}^{\circ}=1,$

and,

$\text{either}\quad\pi_{\mu}^{\circ}=t_{k_{r_{\mu-1}+1}}^{i_{k_{r_{\mu-1}+1}}}\cdots
t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}\quad\text{or}\quad\pi_{\mu}^{\circ}=1.$

For $1\leq\alpha\leq\mu-1$, let $maj_{\alpha}(\pi)$ be as it is defined in
Definition $maj_{\alpha}(\pi)$. The subset $N(\pi)$ of the
$\\{1,2,\ldots,n\\}$ is defined as follow:

  * •

For $1\leq\alpha\leq\mu-1$, $maj_{\alpha}(\pi)+1\in N(\pi)$ if and only if
$\nu_{\alpha}$ (as it is defined in Corollary 2.1.14) is odd and
$maj_{\alpha}(\pi)<k_{1}$.

  * •

For $1\leq\alpha\leq\mu-1$ and $1\leq j\leq\nu_{\alpha}$, $L_{\alpha_{j}}+1\in
N(\pi)$.

  * •

If $2\leq x\leq n$ and $x\neq maj_{\alpha}(\pi)+1$ and $x\neq
L_{\alpha_{j}}+1$ for any $1\leq\alpha\leq\mu-1$ and any $1\leq
j\leq\nu_{\alpha}$, then $x\notin N(\pi)$.

  * •

$1\in N(\pi)$ if and only if the number of elements $x$ such that $2\leq x\leq
n$ and $x\in N(\pi)$ is odd.

Then, $\pi$ is the following sign-permutation of $D_{n}$:

$\begin{cases}\pi(x)=\pi^{\prime}(x)&\text{if}\quad x\notin N(\pi)\\\
\pi(x)=-\pi^{\prime}(x)&\text{if}\quad x\in N(\pi).\end{cases}$

###### Proof.

Let

$\pi=\pi_{1}^{\circ}\cdot w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot\pi_{\mu}^{\circ},$

where,

$\text{either}\quad\pi_{1}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\quad\text{or}\quad\pi_{1}^{\circ}=1,$

and,

$\text{either}\quad\pi_{\mu}^{\circ}=t_{k_{r_{\mu-1}+1}}^{i_{k_{r_{\mu-1}+1}}}\cdots
t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}\quad\text{or}\quad\pi_{\mu}^{\circ}=1.$

Assume, $\pi^{\prime}$ is a standard $OGS$ elementary element as it is defined
in Definition 1.0.6. Then, by Theorem 2.1.17,

$\pi=\pi^{\bullet}\cdot\pi^{\circ},$

such that

$\pi^{\bullet}=\prod_{u=1}^{\alpha}w_{maj_{u~{}|~{}maj_{u}(\pi)<k_{1}}(\pi)}^{\nu_{u}}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot w_{L_{2_{1}}}\cdots
w_{L_{2_{\nu_{2}}}}\cdots w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}$

and

$\pi^{\circ}=\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\mu}^{\circ}$

where, by Theorem 2.1.17, the presentation of $\pi^{\circ}$ in terms of
generalized standard $OGS$ (as it is presented in Theorem 2.1.6) is same to
the standard $OGS$ presentation of $\pi^{\prime}$ (as it is presented in
Theorem 1.0.3). Hence, for $1\leq x\leq n$ (i.e., $x>0$)
$\pi^{\circ}(x)=\pi^{\prime}(x)$. Since, by Theorem 2.1.17, $\pi^{\bullet}\in
Id_{n}^{\bullet}$, and by Theorem 2.1.13,
$\big{(}Id_{n}^{\bullet}\big{)}^{\prime}=1$, we get by Definition 2.1.10,
$|\pi^{\bullet}(x)|=x$ for $1\leq x\leq n$ (i.e., either $\pi^{\bullet}(x)=x$
or $\pi^{\bullet}(x)=-x$ for $1\leq x\leq n$. By Remark 2.1.5, for every
$1\leq x\leq n-1$, $w_{x}$ satisfies the following properties

$\begin{array}[]{l}w_{x}(1)=-1\\\ w_{x}(x+1)=-(x+1)\\\ w_{x}(j)=j,\text{ for
}2\leqslant j\leqslant x\quad\text{and}\quad x+2\leqslant j\leqslant
n.\end{array}.$

Hence, for every $1\leq\alpha\leq\mu-1$, $w_{maj_{\alpha}(\pi)}$ satisfies the
following properties

$\begin{array}[]{l}w_{maj_{\alpha}(\pi)}(1)=-1\\\
w_{maj_{\alpha}(\pi)}(maj_{\alpha}(\pi)+1)=-(maj_{\alpha}(\pi)+1)\\\
w_{maj_{\alpha}(\pi)}(j)=j,\text{ for }2\leqslant j\leqslant
maj_{\alpha}(\pi)\quad\text{and}\quad maj_{\alpha}+2\leqslant j\leqslant
n.\end{array}.$

and for every $1\leq\alpha\leq\mu-1$, and $1\leq q\leq\nu_{\alpha}$,
$w_{L_{{\alpha}_{q}}}(\pi)$ satisfies the following

$\begin{array}[]{l}w_{L_{{\alpha}_{q}}}(1)=-1\\\
w_{L_{{\alpha}_{q}}}(L_{{\alpha}_{q}}+1)=-(L_{{\alpha}_{q}}+1)\\\
w_{L_{{\alpha}_{q}}}(j)=j,\text{ for }2\leqslant j\leqslant
L_{{\alpha}_{q}}\quad\text{and}\quad L_{{\alpha}_{q}}+2\leqslant j\leqslant
n.\end{array}.$

Hence, for $2\leq x\leq n$,

  * •

$\pi^{\bullet}(x)=$
$\prod_{u=1}^{\alpha}w_{maj_{u~{}|~{}maj_{u}(\pi)<k_{1}}(\pi)}^{\nu_{u}}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot w_{L_{2_{1}}}\cdots
w_{L_{2_{\nu_{2}}}}\cdots w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}(x)=-x$

if one of the following holds:

    * –

For $1\leq\alpha\leq\mu-1$, $x=maj_{\alpha}(\pi)+1$ and $\nu_{\alpha}$ is odd
and $maj_{\alpha}(\pi)<k_{1}$.

    * –

For $1\leq\alpha\leq\mu-1$ and $1\leq j\leq\nu_{\alpha}$,
$x=L_{\alpha_{j}}+1$.

  * •

$\pi^{\bullet}(x)=$
$\prod_{u=1}^{\alpha}w_{maj_{u~{}|~{}maj_{u}(\pi)<k_{1}}(\pi)}^{\nu_{u}}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot w_{L_{2_{1}}}\cdots
w_{L_{2_{\nu_{2}}}}\cdots w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}(x)=x$

for all the rest of values of $2\leq x\leq n$.

Notice, $\pi^{\bullet}(1)=-1$ if and only if the number of elements $x$ such
that
$2\leq x\leq n$ and $\pi^{\bullet}(x)=-x$ is odd.

Notice, by the definition of $N(\pi)$,

$\pi^{\bullet}(x)=-x\quad\text{if and only if}\quad x\in N(\pi).$

Now, since for every $1\leq x\leq n$,

$\pi^{\circ}(x)=\pi^{\prime}(x),\quad\pi^{\circ}(-x)=-\pi^{\prime}(x),$

we get

$\pi(x)=\pi^{\bullet}\cdot\pi^{\circ}(x)=\pi^{\prime}(x)\quad\text{if and only
if}\quad\pi^{\bullet}(x)=x$

and

$\pi(x)=\pi^{\bullet}\cdot\pi^{\circ}(x)=-\pi^{\prime}(x)\quad\text{if and
only if}\quad\pi^{\bullet}(x)=-x$

∎

######  Remark 2.1.19.

In case of $\pi\in D_{n}$ where $\pi^{\prime}$ is not a standard $OGS$
elementary element, there is possible to decompose $\pi$ in the form

$\pi=\pi^{\bullet}\cdot\pi^{\circ}$

such that

$\pi^{\bullet}\in Id_{n}^{\bullet}\quad\quad\pi^{\circ}\in S_{n}^{\circ},$

where $Id_{n}^{\bullet}$ and $S_{n}^{\circ}$ are subgroups of $D_{n}$ as are
defined in Theorem 2.1.13. Since the formula for the description
$\pi^{\bullet}$ is very complicated in case of general $\pi\in D_{n}$, we just
describe briefly the algorithm for calculating $\pi^{\bullet}$. The standard
$OGS$ presentation of $\pi^{\circ}$ is the same to the standard $OGS$
presentation of $\pi^{\prime}$ (i.e., we get the standard $OGS$ presentation
of $\pi^{(\circ)}$ by just omitting the elements of the form $w_{L}$ from the
generalized standard $OGS$ presentation of $\pi$). Let $\pi\in D_{n}$ such
that $\pi^{\prime}$ is not a standard $OGS$ elementary element of $S_{n}$.
First, consider the standard $OGS$ elementary factorization of $\pi^{\prime}$
as it is defined in Definition 1.0.8:

$\pi^{\prime}=\prod_{v=1}^{z(\pi^{\prime})}\big{(}{\pi^{\prime}}^{(v)}\big{)},~{}~{}~{}~{}where~{}~{}~{}~{}\big{(}{\pi^{\prime}}^{(v)}\big{)}=\prod_{j=1}^{m^{(v)}}t_{h^{(v)}_{j}}^{\imath_{j}^{(v)}},$

by the presentation in the standard $OGS$ canonical form for every $1\leq
v\leq z(\pi)$ and $1\leq j\leq m^{(v)}$ such that:

  * •

$\imath_{j}^{(v)}>0;$

  * •

$\sum_{j=1}^{m^{(1)}}\imath_{j}^{(1)}\leq h^{(1)}_{1}$ i.e.,
$maj\left({\pi^{\prime}}^{(1)}\right)\leq h^{(1)}_{1}$;

  * •

$h^{(v-1)}_{m^{(v-1)}}\leq\sum_{j=1}^{m^{(v)}}\imath_{j}^{(v)}\leq
h^{(v)}_{1}$ for $2\leq v\leq z$

i.e., $h^{(v-1)}_{m^{(v-1)}}\leq
maj\left[\left({\pi^{\prime}}^{(v)}\right)\right]\leq
h^{(v)}_{1}~{}~{}for~{}~{}2\leq v\leq z$.

Then, for every $1\leq v\leq z(\pi)$, let $\dot{\pi}^{(v)}$ be the subword of
$\pi$ such the following holds:

  * •

For every $1\leq v\leq z(\pi)$,
${[{\dot{\pi}}^{(v)}]}^{\prime}={\pi^{\prime}}^{(v)}$.

  * •

For $1\leq j\leq k_{m}$,

$w_{j}\in\dot{\pi}^{(v)}$

if and only if

$\begin{cases}maj\left[\left({\pi^{\prime}}^{(v)}\right)\right]\leq
j<maj\left[\left({\pi^{\prime}}^{(v+1)}\right)\right]&\text{if}\quad 1\leq
v\leq z(\pi)-1\\\ \\\ maj\left[\left({\pi^{\prime}}^{(v)}\right)\right]\leq
j&\text{if}\quad v=z(\pi).\end{cases}$

Then applying Theorem 2.1.17 on $\dot{\pi}^{(v)}$ for every $1\leq v\leq
z(\pi)$.Then we get

$\dot{\pi}^{(v)}=\left(\dot{\pi}^{(v)}\right)^{\bullet}\cdot\left(\dot{\pi}^{(v)}\right)^{\circ}.$

Then by applying Proposition 2.1.12 and Lemma 2.1.15 several times we move the
elements of the form $w_{j}$ to the left hand side, and then we get the
presentation

$\pi=\pi^{\bullet}\cdot\pi^{\circ}$

such that $\pi^{\bullet}\in Id_{n}^{\bullet}$ and $\pi^{\circ}\in
S_{n}^{\circ}$. We explain the process to get the decomposition
$\pi=\pi^{\bullet}\cdot\pi^{\circ}$ step by step in Example 4.0.11 .

##  3 The two parabolic subgroups of $D_{n}$ which are isomorphic to $S_{n}$

In Theorem 2.1.13, we have already defined two subgroups of $D_{n}$, which are
naturally characterized by the generalized standard $OGS$ presentation of
$D_{n}$. The subgroup $S^{\circ}_{n}$ of $D_{n}$, which is isomorphic to
$S_{n}$, and contains the elements of $D_{n}$ such that the generalized
standard $OGS$ presentation of $\pi$ (as it is described in Theorem 2.1.6) is
the same to the standard $OGS$ presentation of $\pi^{\prime}$ (as it is
described in Theorem 1.0.3), and the subgroup $Id^{\bullet}_{n}$ of $D_{n}$,
which is isomorphic to $\mathbb{Z}_{2}^{n-1}$, and contains all the elements
of $D_{n}$ such that $\pi^{\prime}=1$. In this section we show the
presentation of the elements in another interesting parabolic subgroup of
$D_{n}$ which is isomorphic to $S_{n}$ by the generalized standard $OGS$ for
$D_{n}$ . We start with the characterization of the two parabolic subgroups of
$D_{n}$ which are isomorphic to $S_{n}$ (Notice, the first one has been
defined in Theorem 2.1.13).

  * •

The subgroup $S^{\circ}_{n}$ which is generated by
$\\{s_{1},s_{2},\ldots,s_{n-1}\\}.$

  * •

The subgroup $S^{\circ}_{n^{\prime}}$ which is generated by
$\\{s_{1^{\prime}},s_{2},\ldots,s_{n-1}\\}.$

Now, we find the generalized standard $OGS$ presentation for the elements of
the subgroup $S^{\circ}_{n^{\prime}}$.

######  Theorem 3.0.1.

Let $\pi\in S^{\circ}_{n^{\prime}}$ such that $\pi^{\prime}$ is presented in
the standard $OGS$ presentation $\pi^{\prime}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{m}}^{i_{k_{m}}}$ is a standard $OGS$ elementary element (as it is
defined in Definition 1.0.6), Then the decomposition of $\pi$ into the
presentation $\pi=\pi^{\bullet}\cdot\pi^{\circ}$ by Theorem 2.1.17, as follow:

  1. 1.

$\pi^{\bullet}=1$ and $\pi=\pi^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{m}}^{i_{k_{m}}}$ in case $\sum_{j=1}^{m}i_{k_{j}}=k_{1}$.

  2. 2.

$\pi^{\bullet}=w_{maj(\pi^{\prime})}$ and
$\pi^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots t_{k_{m}}^{i_{k_{m}}}$ in case
$\sum_{j=1}^{m}i_{k_{j}}<k_{1}$.

###### Proof.

We divide the proof into the two mentioned cases. First, notice, by Theorem
2.1.17, By considering $\pi^{\circ}$ as an element of $S_{n}$, $\pi^{\circ}$
presents the same element as $\pi^{\prime}$, and the standard $OGS$
presentation (as it presented in Theorem 1.0.3) and the normal form of
$\pi^{\prime}$, is the same to the generalized standard $OGS$ presentation (as
it presented in Theorem 2.1.6) and the normal form of $\pi^{\circ}$. Hence, in
the proof we use $maj(\pi^{\prime})$ and $norm(\pi^{\prime})$ when we deal
with $\pi^{\circ}$. By using the normal form of standard $OGS$ elementary
element of $S_{n}$ as it is described in [9], Theorem 28.

$norm(\pi^{\prime})=\prod_{u=\rho_{1}}^{k_{1}-1}\prod_{r=0}^{\rho_{1}-1}s_{u-r}\cdot\prod_{u=k_{1}}^{k_{2}-1}\prod_{r=0}^{\rho_{2}-1}s_{u-r}\cdot\prod_{u=k_{2}}^{k_{3}-1}\prod_{r=0}^{\rho_{3}-1}s_{u-r}\cdots\prod_{u=k_{m-1}}^{k_{m}-1}\prod_{r=0}^{\rho_{m}-1}s_{u-r},$
(3)

where, $\rho_{j}=\sum_{x=j}^{m}i_{k_{x}}$ for $1\leq j\leq m$.

$\bullet$ We start with the first case where
$\pi^{\prime}=t_{k_{1}}^{i_{k_{1}}}\cdots t_{k_{m}}^{i_{k_{m}}}$ is a standard
OGS elementary element and $\sum_{j=1}^{m}i_{k_{j}}=k_{1}$.
Since $\rho_{1}=\sum_{j=1}^{m}i_{k_{j}}=k_{1}$, Equation 3 has the following
form:

$\text{norm}(\pi^{\prime})=\prod_{u=k_{1}}^{k_{2}-1}\prod_{r=0}^{\rho_{2}-1}s_{u-r}\cdot\prod_{u=k_{2}}^{k_{3}-1}\prod_{r=0}^{\rho_{3}-1}s_{u-r}\cdots\prod_{u=k_{m-1}}^{k_{m}-1}\prod_{r=0}^{\rho_{m}-1}s_{u-r},$

Notice, the minimal value for $u-r$ in the formula of $norm(\pi^{\prime})$
occurs where $u=k_{1}$ and $r=\rho_{2}-1$. Then, $u-r=k_{1}-(\rho_{2}-1).$
Since $k_{1}=\sum_{x=1}^{m}i_{k_{x}}=i_{1}+\rho_{2}$, we get
$\rho_{2}=k_{1}-i_{k_{1}}$. Hence,
$k_{1}-(\rho_{2}-1)=k_{1}-(k_{1}-i_{k_{1}}-1)=i_{k_{1}}+1$. Now, since
$i_{k_{1}}\geq 1$, we have that $i_{k_{1}}+1\geq 2$. Hence, the minimal value
for $u-r$ in the formula for $norm(\pi^{\prime})$ is $u-r=2$, in case where
$\sum_{j=1}^{m}i_{k_{j}}=k_{1}$. Therefore, there is no occurrence of $s_{1}$
in the normal form presentation of $\pi^{\prime}$ (which is by Theorem 2.1.17
is the same to the normal form of $\pi^{\circ}$). Hence,

$\pi=\pi^{\circ}.$

$\bullet$ Now, we turn to the second case:
$\pi^{\prime}=t_{k_{1}}^{i_{k_{1}}}\cdots t_{k_{m}}^{i_{k_{m}}}$ and
$\sum_{j=1}^{m}i_{k_{j}}<k_{1}$.
Consider the normal form of $\pi^{\prime}$: Since
$\rho_{1}=\sum_{j=1}^{m}i_{k_{j}}<k_{1}$, Equation 3 has the following form:

$\text{norm}(\pi^{\prime})=s_{maj(\pi^{\prime})}\cdots s_{1}\cdot
s_{maj(\pi^{\prime})+1}\cdots s_{2}\cdots$

Notice, that by the definition of $t_{j}$ for $2\leq j\leq n$, as it is
defined in Theorem 1.0.3 as follow:

$t_{maj(\pi^{\prime})+1}=s_{1}\cdot s_{2}\cdots s_{maj(\pi^{\prime})}.$

Hence,

$t_{maj(\pi^{\prime})+1}\cdot\pi^{\circ}=t_{maj(\pi^{\prime}+1)}^{1}\cdot
t_{k_{1}}^{i_{k_{1}}}\cdots t_{k_{m}}^{i_{k_{m}}}$ $=(s_{1}\cdot s_{2}\cdots
s_{maj(\pi^{\prime})})\cdot(s_{maj(\pi^{\prime})}\cdots s_{1})\cdot
s_{maj(\pi^{\prime})+1}\cdots s_{2}\cdots$ $=s_{maj(\pi^{\prime})+1}\cdots
s_{2}\cdots$

Notice, that $\sum_{j=1}^{m}i_{k_{j}}+1=maj(\pi^{\prime})+1$. Hence, by the
proof of the proposition for case 1, there is no occurrence of $s_{1}$ in the
normal form of $t_{maj(\pi^{\prime})+1}\cdot\pi^{\prime}.$

Then by multiplying $\pi^{\prime}$ by $w_{{maj}(\pi^{\prime})}$ on the left
hand side, we have:

$w_{maj(\pi^{\prime})}\cdot\pi^{\circ}=(s_{maj(\pi^{\prime})}\cdots s_{2}\cdot
s_{1^{\prime}}\cdot s_{1}\cdot s_{2}\cdots s_{maj(\pi^{\prime})})\cdot
t_{maj(\pi^{\prime})+1}\cdot\pi^{\circ}$ $=s_{maj(\pi^{\prime})}\cdots
s_{1^{\prime}}\cdot t_{maj(\pi^{\prime})+1}\cdot\pi^{\circ}$

Hence, we get

$\pi=w_{maj(\pi^{\prime})}\cdot\pi^{\circ}.$

∎

######  Example 3.0.2.

Consider

$\pi=t_{5}^{2}\cdot t_{6}^{3}$

Then, by [9], Theorem 28, the normal form of $\pi$ as follow:

$\text{norm}(\pi)=s_{5}\cdot s_{4}\cdot s_{3}.$

Since there is no occurrence of $s_{1}$ in $\text{norm}(\pi)$, we get

$\pi\in S^{\circ}_{n^{\prime}}.$

######  Example 3.0.3.

Consider

$\pi=w_{4}\cdot t_{5}^{2}\cdot t_{6}^{2}$

Then,

$\pi^{\prime}=t_{5}^{2}\cdot t_{6}^{2}.$

Then, by [9], Theorem 28, the normal form of $\pi^{\prime}$ as follow:

$\text{norm}(\pi^{\prime})=(s_{4}\cdot s_{3}\cdot s_{2}\cdot
s_{1})\cdot(s_{5}\cdot s_{4})$

Notice,

$maj(\pi^{\prime})=4$

Hence, by Theorem 3.0.1

$\pi\in S^{\circ}_{n^{\prime}}.$

Indeed,

$\pi=w_{4}\cdot t_{5}^{2}\cdot t_{6}^{2}=(s_{4}\cdot s_{3}\cdot s_{2}\cdot
s_{1^{\prime}}\cdot s_{1}\cdot s_{2}\cdot s_{3}\cdot s_{4})\cdot(s_{4}\cdot
s_{3}\cdot s_{2}\cdot s_{1})\cdot(s_{5}\cdot s_{4})$ $=(s_{4}\cdot s_{3}\cdot
s_{2}\cdot s_{1^{\prime}})\cdot(s_{5}\cdot s_{4})=norm(\pi).$

######  Theorem 3.0.4.

Let $\pi$ be an element of $S^{\circ}_{n^{\prime}}$ such that $\pi^{\prime}$
is not necessarily a standard $OGS$ elementary element. Consider the standard
$OGS$ elementary factorization of $\pi^{\prime}$ as it is defined in
Definition 1.0.8 and then get
$\pi^{\prime}=\pi_{1}^{\prime}\cdots\pi_{z}^{\prime}$, where the presentation
of an element $\pi\in S^{\circ}_{n^{\prime}}$ by the generalized standard
$OGS$ presentation as follow:

  * •

$\pi=\pi_{1}\cdot\pi_{2}\cdots\pi_{z}$;

  * •

$\pi_{v}=\pi_{v}^{\prime}$ , where
$\pi_{v}^{\prime}=t_{h_{v,1}}^{i_{h_{v,1}}}\cdot
t_{h_{v,2}}^{i_{h_{v,2}}}\cdots t_{h_{v,m(\pi_{1})}}^{i_{h_{v,m(\pi_{1})}}}$ ,
where $\text{maj}(\pi_{v}^{\prime})=h_{v,1}$.

  * •

$\pi_{v}=w_{maj(\pi_{v}^{\prime})}\cdot\pi_{v}^{\prime}$ ,where
$\pi_{v}^{\prime}=t_{h_{v,1}}^{i_{h_{v,1}}}\cdot
t_{h_{v,2}}^{i_{h_{v,2}}}\cdots t_{h_{v,m(\pi_{1})}}^{i_{h_{v,m(\pi_{1})}}}$
in case $\text{maj}(\pi_{v}^{\prime})<h_{v,1}$.

###### Proof.

Look at the standard $OGS$ elementary factorization of $\pi^{\prime}$ (as it
defined in Definition 1.0.8). Then we have,

$\pi^{\prime}=\pi_{1}^{\prime}\cdots\pi_{z}^{\prime}$

where $\pi_{v}^{\prime}$ is a standard $OGS$ ekementary factor of
$\pi^{\prime}$. By applying Theorem 3.0.1 on every standard $OGS$ elementary
factor $\pi_{v}^{\prime}$ (for $1\leq v\leq z$), we get the desired result. ∎

######  Example 3.0.5.

Consider:

$\pi=w_{3}\cdot t_{4}^{2}\cdot t_{5}\cdot t_{6}^{3}\cdot t_{9}\cdot
t_{11}^{2}$

Then,

$\pi^{\prime}=t_{4}^{2}\cdot t_{5}\cdot t_{6}^{3}\cdot t_{9}\cdot t_{11}^{2}.$

By considering the standard $OGS$ elementary factorization for $\pi^{\prime}$,
as it is defined in Definition 1.0.8, we get:

$\pi_{1}^{\prime}=t_{4}^{2}\cdot
t_{5}\quad\quad\pi_{2}^{\prime}=t_{6}^{3}\cdot t_{9}\cdot t_{11}^{2}$

$maj(\pi_{1}^{\prime})<h_{1},(3<4)\quad\quad
maj(\pi_{2}^{\prime})=h_{1},(6=6).$

Hence, by Theorem 3.0.1

$w_{maj(\pi_{1}^{\prime})}\cdot\pi_{1}^{\prime}=w_{3}\cdot t_{4}^{2}\cdot
t_{5}\in S^{\circ}_{n^{\prime}}\quad\quad\pi_{2}^{\prime}=t_{6}^{3}\cdot
t_{9}\cdot t_{11}^{2}\in S^{\circ}_{n^{\prime}}.$

Finally, by Theorem 3.0.4 the following holds:

$\pi=(w_{3}\cdot t_{4}^{2}\cdot t_{5})\cdot(t_{6}^{3}\cdot t_{9}\cdot
t_{11}^{2})=w_{maj(\pi_{1}^{\prime})}\cdot\pi_{1}^{\prime}\cdot\pi_{2}^{\prime}\in
S^{\circ}_{n^{\prime}}.$

##  4 The Coxeter length of elements of $D_{n}$

In this section we give a method to find the Coxeter length of elements of
$D_{n}$ by using the generalized standard $OGS$ as it is defined in Definition
2.1.6. We start with some lemmas which explain specific relations between the
Coxeter generators $\\{s_{1^{\prime}},s_{1},s_{2}\ldots s_{n-1}\\}$ of $D_{n}$
and the elements
$w_{k}=s_{k}\cdot s_{k-1}\cdots s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots
s_{k}$ for $1\leq k\leq n-1$, which are used to find the Coxeter length of
elements in $D_{n}$.

######  Lemma 4.0.1.

For every $1\leq k\leq n-1$ let

$w_{k}=s_{k}\cdot s_{k-1}\cdots s_{2}\cdot s_{1}\cdot s_{1^{\prime}}\cdot
s_{2}\cdots s_{k-1}\cdot s_{k}.$ (4)

as it is defined in Definition 2.1.3. Then

  * •

$\ell(w_{k})=2\cdot k$.

  * •

$\ell(w_{j}\cdot w_{k})=\ell(w_{j})+\ell(w_{k})=2\cdot(j+k)$
for $1\leq j,k\leq n-1$ and $j\neq k$.

###### Proof.

The presentation of $w_{k}$ as it is presented in Equation 4, is a
presentation by the normal form of $D_{n}$ (See Definition 2.0.2). Therefore,
the presentation is reduced which contains $2\cdot k$ Coxeter generators.
Hence, $\ell(w_{k})=2\cdot k$. Now, consider $\ell(w_{j}\cdot w_{k})$. Since
$j\neq k$ and by Proposition 2.1.12, $w_{j}\cdot w_{k}=w_{k}\cdot w_{j}$ for
every $1\leq j,k\leq n-1$, without loss of generality, we may assume $j<k$.
Then by Definition 2.0.2, the presentation:

$w_{j}\cdot w_{k}=(s_{j}\cdot s_{j-1}\cdots s_{2}\cdot s_{1}\cdot
s_{1^{\prime}}\cdot s_{2}\cdots s_{j-1}\cdot s_{j})\cdot(s_{k}\cdot
s_{k-1}\cdots s_{2}\cdot s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots
s_{k-1}\cdot s_{k})$

is a presentation by the normal form. Hence, we get the result

$\ell(w_{j}\cdot w_{k})=\ell(w_{j})+\ell(w_{k})=2\cdot(j+k),$

for $1\leq j,k\leq n-1$ and $j\neq k$. ∎

######  Lemma 4.0.2.

Consider the group $D_{n}$ with the set of Coxeter generators
$\\{s_{1^{\prime}},s_{1},s_{2}\ldots s_{n-1}\\}$ and for every $1\leq k\leq
n-1$ let $w_{k}$ be the element of $D_{n}$ as it is defined in Definition
2.1.3 then the following relations holds.

  * •

$w_{L}\cdot s_{1}=s_{1^{\prime}}\cdot w_{L}\quad\text{for}\quad 2\leqslant
L\leqslant n-1.$

  * •

$w_{L}\cdot s_{1^{\prime}}=s_{1}\cdot w_{L}\quad\text{for}\quad 2\leqslant
L\leqslant n-1.$

  * •

$w_{L}\cdot s_{k}=s_{k}\cdot w_{L}\quad\text{for}\quad 2\leqslant k\leqslant
L-1\leqslant n-2.$

###### Proof.

  * •

Consider the first case, where $w_{L}\cdot s_{1}=s_{1^{\prime}}\cdot w_{L}.$

$w_{L}\cdot s_{1}=s_{L}\cdots s_{2}\cdot s_{1}\cdot s_{1^{\prime}}\cdot
s_{2}\cdots s_{L}\cdot s_{1}$ $=s_{L}\cdots s_{2}\cdot s_{1}\cdot
s_{1^{\prime}}\cdot s_{2}\cdot s_{1}\cdot s_{3}\cdots s_{L}$ $=s_{L}\cdots
s_{2}\cdot s_{1^{\prime}}\cdot s_{1}\cdot s_{2}\cdot s_{1}\cdot s_{3}\cdots
s_{L}$ $=s_{L}\cdots s_{2}\cdot s_{1^{\prime}}\cdot s_{2}\cdot s_{1}\cdot
s_{2}\cdots s_{L}$ $=s_{L}\cdots s_{3}\cdot s_{1^{\prime}}\cdot s_{2}\cdot
s_{1^{\prime}}\cdot s_{1}\cdot s_{2}\cdots s_{L}$ $=s_{1^{\prime}}\cdot
s_{L}\cdots s_{2}\cdot s_{1^{\prime}}\cdot s_{1}\cdot s_{2}\cdots s_{L}$
$=s_{1^{\prime}}\cdot w_{L}$

  * •

Consider the second case, where $w_{L}\cdot s_{1^{\prime}}=s_{1}\cdot w_{L}.$

$w_{L}\cdot s_{1^{\prime}}=s_{L}\cdots s_{2}\cdot s_{1}\cdot
s_{1^{\prime}}\cdot s_{2}\cdots s_{L}\cdot s_{1^{\prime}}$ $=s_{L}\cdots
s_{2}\cdot s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdot s_{1^{\prime}}\cdots
s_{L}$ $=s_{L}\cdots s_{2}\cdot s_{1}\cdot s_{2}\cdot s_{1^{\prime}}\cdot
s_{2}\cdots s_{L}$ $=s_{L}\cdots s_{1}\cdot s_{2}\cdot s_{1}\cdot
s_{1^{\prime}}\cdot s_{2}\cdots s_{L}$ $=s_{1}\cdot s_{L}\cdots s_{2}\cdot
s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots s_{L}$ $=s_{1}\cdot w_{L}$

  * •

Consider the last case, where $w_{L}\cdot s_{k}=s_{k}\cdot w_{L}.$

$w_{L}\cdot s_{k}=s_{L}\cdots s_{2}\cdot s_{1}\cdot s_{1}^{\prime}\cdot
s_{2}\cdots s_{k-1}\cdot s_{k}\cdot s_{k+1}\cdots s_{L}\cdot s_{k}.$
$=s_{L}\cdots s_{k}\cdots s_{2}\cdot s_{1}\cdot s_{1}^{\prime}\cdot
s_{2}\cdots s_{k-1}\cdot s_{k}\cdot s_{k+1}\cdot s_{k}\cdots s_{L}.$
$=s_{L}\cdots s_{k}\cdots s_{2}\cdot s_{1}\cdot s_{1}^{\prime}\cdot
s_{2}\cdots s_{k-1}\cdot s_{k+1}\cdot s_{k}\cdot s_{k+1}\cdots s_{L}.$
$=s_{L}\cdots s_{k+1}\cdot s_{k}\cdot s_{k+1}\cdot s_{k-1}\cdots s_{2}\cdot
s_{1}\cdot s_{1}^{\prime}\cdot s_{2}\cdots s_{k-1}\cdot s_{k}\cdot
s_{k+1}\cdots s_{L}.$ $=s_{L}\cdots s_{k}\cdot s_{k+1}\cdot s_{k}\cdot
s_{k-1}\cdots s_{2}\cdot s_{1}\cdot s_{1}^{\prime}\cdot s_{2}\cdots
s_{k-1}\cdot s_{k}\cdot s_{k+1}\cdots s_{L}.$ $=s_{k}\cdot s_{L}\cdots
s_{k+1}\cdot s_{k}\cdot s_{k-1}\cdots s_{2}\cdot s_{1}\cdot
s_{1}^{\prime}\cdot s_{2}\cdots s_{k-1}\cdot s_{k}\cdot s_{k+1}\cdots s_{L}.$
$=s_{k}\cdot w_{L}.$

∎

######  Lemma 4.0.3.

Let define $\pi_{j,r}$ to be :

$\pi_{j,r}=s_{j}\cdots s_{j-r}$

Then,

$w_{j}\cdot\pi_{j,r}=\pi_{j,r}\cdot w_{j-r-1}.$

$\ell(w_{j}\cdot\pi_{j,r})=\ell(\pi_{j,r})+2\cdot(j-r-1).$

###### Proof.

First, we recall $w_{j}$

$w_{j}=s_{j}\cdot s_{j-1}\cdots s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots
s_{j}$

Then,

$w_{j}\cdot\pi_{j,r}=s_{j}\cdots s_{1}\cdot s_{1^{\prime}}\cdots
s_{j-r-1}\cdot s_{j-r}\cdots s_{j}\cdot s_{j}\cdots s_{j-r}.$ $=s_{j}\cdots
s_{j-r}\cdot(s_{j-r-1}\cdots s_{1}\cdot s_{1^{\prime}}\cdots s_{j-r-1}).$

Now, we look at $\pi_{j,r}\cdot w_{j-r-1}$

$\pi_{j,r}\cdot w_{j-r-1}=(s_{j}\cdots s_{j-r})\cdot(s_{j-r-1}\cdots
s_{1}\cdot s_{1^{\prime}}\cdots s_{j-r-1})$ $=norm(\pi_{j,r}\cdot
w_{j-r-1})=norm(\pi_{j,r}\cdot norm(w_{j-r-1}).$

Hence, we get:

$\ell(w_{j}\cdot\pi_{j,r})=\ell(\pi_{j,r})+\ell(w_{j-r-1})=\ell(\pi_{j,r})+2\cdot(j-r-1).$

∎

Now, we develop an algorithm for the Coxeter length of an element $\pi\in
D_{n}$ such that $\pi^{\prime}$ is a standard $OGS$ elementary element by
using its generalized standard $OGS$ presentation.

######  Proposition 4.0.4.

Let $\pi\in D_{n}$ such that $\pi^{\prime}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{m}}^{i_{k_{m}}}$ be a standard $OGS$ elementary element (i.e.,
$\sum_{j=1}^{m}i_{k_{j}}\leq k_{1}$) as it is defined in Definition 1.0.6.
Assume, the decomposition of $\pi$ in the form
$\pi=\pi^{\bullet}\cdot\pi^{\circ}$ (as it is described in Theorem 2.1.17) is
as follow:

$\pi=w_{L}\cdot\pi^{\circ},$

where $L$ is a positive integer, such that $1\leq L\leq n-1$, and the element
$w_{L}$ as it is defined in Definition 2.1.3. Then the following holds:

  * •

If $\pi=w_{L}\cdot\pi^{\circ}$ and $L<\sum_{j=1}^{m}i_{k_{j}}$
Then,

$\ell(\pi)=2L+\ell(\pi^{\prime}).$

  * •

If $\pi=w_{L}\cdot\pi^{\circ}$ and $L=\sum_{j=1}^{m}i_{k_{j}}<k_{1}$
Then,

$\ell(\pi)=\ell(\pi^{\prime}).$

  * •

If $\pi=w_{L}\cdot\pi^{\circ}$ and $\sum_{j=1}^{m}i_{k_{j}}<L<k_{1}$
Then,

$\ell(\pi)=2\cdot(L-\text{maj}(\pi^{\prime}))+\ell(\pi^{\prime}).$

  * •

If $\pi=w_{L}\cdot\pi^{\circ}$ and $k_{r}\leq L<k_{r+1}$ for $1\leq r\leq m-1$
Then

$\ell(\pi)=2\cdot(L-\sum_{j=r+1}^{m}i_{k_{j}})+\ell(\pi^{\prime}).$

  * •

If $\pi=w_{L}\cdot\pi^{\circ}$ and $L\geq k_{m}$
Then

$\ell(\pi)=2L+\ell(\pi^{\prime}).$

###### Proof.

Since $\pi^{\prime}$ is a standard $OGS$ elementary element (as it is defined
in Definition 1.0.6), by [9] Theorem 28,

$norm(\pi^{\prime})=\prod_{u=\rho_{1}}^{k_{1}-1}\prod_{r=0}^{\rho_{1}-1}s_{u-r}\cdot\prod_{u=k_{1}}^{k_{2}-1}\prod_{r=0}^{\rho_{2}-1}s_{u-r}\cdot\prod_{u=k_{2}}^{k_{3}-1}\prod_{r=0}^{\rho_{3}-1}s_{u-r}\cdots\prod_{u=k_{m-1}}^{k_{m}-1}\prod_{r=0}^{\rho_{m}-1}s_{u-r},$

where, $\rho_{j}=\sum_{x=j}^{m}i_{k_{x}}$ for $1\leq j\leq m$.

Notice, by Theorem 2.1.17, the standard $OGS$ presentation of $\pi^{\circ}$ is
the same tho the standard $OGS$ presentation of $\pi^{\prime}$, and
$\pi^{\circ}$ and $\pi^{\prime}$ present the same element of $S_{n}$, by
considering $\pi^{\circ}$ an element of $S_{n}$. Hence,
$norm(\pi^{\circ})=norm(\pi^{\prime})$. Therefore, by considering the length
of elements, we write $\ell(\pi^{\prime})$ instead of $\ell(\pi^{\circ})$.

Notice, $norm(\pi^{\prime})$ has one of the following form, which depends on
either $maj(\pi^{\prime})<k_{1}$ or $maj(\pi^{\prime})=k_{1}$.

  * •

If $maj(\pi^{\prime})<k_{1}$ then

$norm(\pi^{\prime})=(s_{maj(\pi^{\prime})}\cdot s_{maj(\pi^{\prime})-1}\cdots
s_{1})\cdot(s_{maj(\pi^{\prime})+1}\cdots)\cdots(s_{k_{m}-1}\cdots
s_{k_{m}-i_{k_{m}}}).$

  * •

If $maj(\pi^{\prime})=k_{1}$ then

$norm(\pi^{\prime})=(s_{k_{1}}\cdot s_{k_{1}-1}\cdots
s_{k_{1}-\sum_{j=2}^{m}i_{k_{j}}})\cdot(s_{k_{1}+1}\cdots)\cdots(s_{k_{m}-1}\cdots
s_{k_{m}-i_{k_{m}}}).$

where $\sum_{j=2}^{m}i_{k_{j}}\geq 2$.

Then, we divide the proof into the 5 different cases of the value of $L$
compare to the value of $maj(\pi^{\prime})$ and $k_{r}$ for $1\leq r\leq n$.

First consider the case $\pi=w_{L}\cdot\pi^{\circ}\quad\text{where}\quad
L<maj(\pi^{\prime})$:

$w_{L}\cdot\pi^{\circ}=(s_{L}\cdot s_{L-1}\cdots s_{2}\cdot
s_{1^{\prime}}\cdot s_{1}\cdots
s_{L})\cdot(s_{maj(\pi^{\prime})}\cdots)\cdots(s_{k_{m}-1}\cdots
s_{k_{m}-i_{k_{m}}})=norm(\pi).$

Therefore, in case $L<maj(\pi^{\prime})$ we conclude:

$\ell(\pi)=\ell(w_{L})+\ell(\pi^{\prime})=2L+\ell(\pi^{\prime}).$

Now, consider the case $\pi=w_{L}\cdot\pi^{\circ}\quad\text{where}\quad
L=maj(\pi^{\prime})<k_{1}.$ Then,

$\pi=(s_{maj(\pi^{\prime})}\cdots s_{2}\cdot s_{1^{\prime}}\cdot s_{1}\cdot
s_{2}\cdots s_{maj(\pi^{\prime})})\cdot(s_{maj(\pi^{\prime})}\cdots
s_{1})\cdot(s_{maj(\pi^{\prime})+1}\cdots$ $=(s_{maj(\pi^{\prime})}\cdots
s_{2}\cdot
s_{1^{\prime}})\cdot(s_{maj(\pi^{\prime})+1}\cdots)\cdots(s_{k_{m}-1}\cdots
s_{k_{m}-i_{k_{m}}})=norm(\pi).$

Therefore, in case $L=maj(\pi^{\prime})<k_{1}$, we conclude that :

$\ell(\pi)=\ell(\pi^{\prime}).$

Now, Consider the case $\pi=w_{L}\cdot\pi^{\prime}\quad\text{where}\quad
maj(\pi^{\circ})<L<k_{1}$

$\pi=w_{L}\cdot\pi^{\circ}=(s_{L}\cdots s_{maj(\pi^{\prime})}\cdots s_{2}\cdot
s_{1^{\prime}}\cdot s_{1}\cdots s_{maj(\pi^{\prime})}\cdots
s_{L})\cdot(s_{maj(\pi^{\prime})}\cdots s_{1})\cdots$ $\cdots(s_{L-1}\cdots
s_{L-maj(\pi^{\prime})})\cdot(s_{L}\cdots
s_{L-maj(\pi^{\prime})+1})\cdot(s_{L+1}\cdots$

By Lemma 4.0.2:

$w_{L}\cdot\pi^{\circ}=(s_{maj(\pi^{\prime})}\cdots
s_{1^{\prime}})\cdots(s_{L-1}\cdots s_{L-maj(\pi^{\prime})})\cdot(s_{L}\cdots
s_{2}\cdot s_{1^{\prime}}\cdot s_{1}\cdots s_{L-maj(\pi^{\prime})+1}\cdots
s_{L})$ $\cdot(s_{L}\cdots
s_{L-maj(\pi^{\prime})+1})\cdot(s_{L+1}\cdots)\cdots(s_{k_{m}-1}\cdots
s_{k_{m}-i_{k_{m}}}).$

By Lemma 4.0.3,

$w_{L}\cdot(s_{L}\cdots s_{L-maj(\pi^{\prime})+1})=(s_{L}\cdots
s_{L-maj(\pi^{\prime})+1})\cdot w_{maj(\pi^{\prime})}.$

Hence,

$w_{L}\cdot\pi^{\circ}=(s_{maj(\pi^{\prime})}\cdots
s_{1^{\prime}})\cdots(s_{L-1}\cdots s_{L-maj(\pi^{\prime})})\cdot(s_{L}\cdots
s_{2}\cdot s_{1^{\prime}}\cdot s_{1}\cdots
s_{L-maj(\pi^{\prime})})\cdot(s_{L+1}\cdots)\cdots$ $\cdots(s_{k_{m}-1}\cdots
s_{k_{m}-i_{k_{m}}})=norm(\pi).$

Therefore, in case $maj(\pi^{\prime})\leq L<k_{1}$ we conclude :

$\ell(\pi)=2\cdot(L-\text{maj}(\pi^{\prime}))+\ell(\pi^{\prime}).$

Now, Consider the case $\pi=w_{L}\cdot\pi^{\circ}\quad\text{where}\quad
k_{r}\leq L<k_{r+1}$
$\text{for}\quad 1\leq r\leq m-1$

$\pi=w_{L}\cdot\pi^{\circ}=(s_{L}\cdots s_{2}\cdot s_{1^{\prime}}\cdot
s_{1}\cdots s_{L})\cdot(s_{maj(\pi^{\prime})}\cdots)\cdots$
$\cdots(s_{L-1}\cdots)\cdot(s_{L}\cdots
s_{L-\sum_{j=r+1}^{m}i_{k_{j}}+1})\cdot(s_{L+1}\cdots$

By Lemma 4.0.2:

$w_{L}\cdot\pi^{\circ}=(s_{maj(\pi^{\prime})}\cdots)\cdots(s_{L-1}\cdots)\cdot(s_{L}\cdots
s_{2}\cdot s_{1^{\prime}}\cdot s_{1}\cdots s_{L})$ $\cdot(s_{L}\cdots
s_{L-\sum_{j=r+1}^{m}i_{k_{j}}+1})\cdot(s_{L+1}\cdots)\cdots(s_{k_{m}-1}\cdots
s_{k_{m}-i_{k_{m}}}).$

By Lemma 4.0.3,

$w_{L}\cdot(s_{L}\cdots s_{L-\sum_{j=r+1}^{m}i_{k_{j}}+1})=(s_{L}\cdots
s_{L-\sum_{j=r+1}^{m}i_{k_{j}}+1})\cdot w_{\sum_{j=r+1}^{m}i_{k_{j}}}.$

Hence,

$=(s_{maj(\pi^{\prime})}\cdots)\cdots(s_{L-1}\cdots)\cdot(s_{L}\cdots
s_{2}\cdot s_{1^{\prime}}\cdot s_{1}\cdots
s_{L-\sum_{j=r+1}^{m}i_{k_{j}}})\cdot(s_{L+1}\cdots)\cdots$
$\cdots(s_{k_{m}-1}\cdots s_{k_{m}-i_{k_{m}}})=norm(\pi).$

Therefore, in case $k_{r}\leq L<k_{r+1}$ we conclude :

$\ell(\pi)=2\cdot(L-\sum_{j=r+1}^{m}i_{k_{j}})+\ell(\pi^{\prime}).$

Finally, consider the case $\pi=w_{L}\cdot\pi^{\circ}\quad\text{where}\quad
L\geq k_{m}$:

$w_{L}\cdot\pi^{\circ}=(s_{L}\cdot s_{L-1}\cdots s_{2}\cdot
s_{1^{\prime}}\cdot s_{1}\cdots
s_{L})\cdot(s_{maj(\pi^{\prime})}\cdots)\cdots(s_{k_{m}-1}\cdots
s_{k_{m}-i_{k_{m}}}).$

By Lemma 4.0.2:

$w_{L}\cdot\pi^{\circ}=(s_{maj(\pi^{\prime})}\cdots)\cdots(s_{k_{m}-1}\cdots
s_{k_{m}-i_{k_{m}}})\cdot(s_{L}\cdot s_{L-1}\cdots s_{2}\cdot
s_{1^{\prime}}\cdot s_{1}\cdots s_{L}).$

Therefore, in case $L\geq k_{m}$ we conclude:

$\ell(\pi)=\ell(w_{L})+\ell(\pi^{\prime})=2L+\ell(\pi^{\prime}).$

∎

######  Example 4.0.5.

Consider

$\pi=w_{5}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3}$

$\bullet$ $L<\text{maj}(\pi^{\prime})$,
$L=5<7=2+2+3=\text{maj}(\pi^{\prime})$.

$\pi=(s_{5}\cdot s_{4}\cdot s_{3}\cdot s_{2}\cdot s_{1}\cdot
s_{1}^{\prime}\cdot s_{2}\cdot s_{3}\cdot s_{4}\cdot s_{5})\cdot(s_{7}\cdot
s_{6}\cdot s_{5}\cdot s_{4}\cdot s_{3})$ $\cdot(s_{8}\cdot s_{7}\cdot
s_{6}\cdot s_{5}\cdot s_{4})\cdot(s_{9}\cdot s_{8}\cdot
s_{7})\cdot(s_{10}\cdot s_{9}\cdot s_{8})\cdot(s_{11}\cdot s_{10}\cdot
s_{9})=\text{norm}(\pi).$

Hence,

$\ell(\pi)=2\cdot 5+7\cdot 2+9\cdot 2+12\cdot 3-7^{2}=29.$

######  Example 4.0.6.

Consider

$\pi=w_{4}\cdot t_{5}^{2}\cdot t_{6}^{2}$

$\bullet$ $L=\text{maj}(\pi^{\prime})=4$.

$\pi=(s_{4}\cdot s_{3}\cdot s_{2}\cdot s_{1}^{\prime}\cdot s_{1}\cdot
s_{2}\cdot s_{3}\cdot s_{4})\cdot(s_{4}\cdot s_{3}\cdot s_{2}\cdot
s_{1})\cdot(s_{5}\cdot s_{4})$ $=(s_{4}\cdot s_{3}\cdot s_{2}\cdot
s_{1}^{\prime})\cdot(s_{5}\cdot s_{4})=\text{norm}(\pi).$

Hence,

$\ell(\pi)=5\cdot 2+6\cdot 2-4^{2}=6.$

######  Example 4.0.7.

Consider

$\pi=w_{5}\cdot t_{7}^{2}\cdot t_{9}.$

$\bullet$ $\text{maj}(\pi^{\prime})<L<k_{1}$,
$\text{maj}(\pi^{\prime})=3<L=5<k_{1}=7$.

$\pi=(s_{5}\cdot s_{4}\cdots s_{1}\cdot s_{1}^{\prime}\cdot s_{2}\cdot
s_{3}\cdot s_{4}\cdot s_{5})\cdot(s_{3}\cdot s_{2}\cdot s_{1})$
$\cdot(s_{4}\cdot s_{3}\cdot s_{2})\cdot(s_{5}\cdot s_{4}\cdot
s_{3})\cdot(s_{6}\cdot s_{5}\cdot s_{4})\cdot s_{7}\cdot s_{8}$ $=(s_{3}\cdot
s_{2}\cdot s_{1})\cdot(s_{4}\cdot s_{3}\cdot s_{2})\cdot(s_{5}\cdot s_{4}\cdot
s_{3}\cdot s_{2}\cdot s_{1}\cdot s_{1}^{\prime}\cdot s_{2}\cdot s_{3}\cdot
s_{4}\cdot s_{5})\cdot(s_{5}\cdot s_{4}\cdot s_{3})\cdot(s_{6}\cdot s_{5}\cdot
s_{4})\cdot s_{7}\cdot s_{8}$ $=(s_{3}\cdot s_{2}\cdot s_{1})\cdot(s_{4}\cdot
s_{3}\cdot s_{2})\cdot(s_{5}\cdot s_{4}\cdot s_{3}\cdot s_{2}\cdot s_{1}\cdot
s_{1}^{\prime}\cdot s_{2})\cdot(s_{6}\cdot s_{5}\cdot s_{4})\cdot s_{7}\cdot
s_{8}=\text{norm}(\pi).$

Hence,

$\ell(\pi)=2\cdot(5-3)+14=18.$

Now, we consider the case of $\pi$, where $\pi^{\prime}$ is a standard OGS
elementary element, but $\pi$ not necessarily of the form
$w_{L}\cdot\pi^{\prime}$. In Theorem 4.0.8 we consider the length of $\pi$
where $\pi$ is presented in the form $\pi=\pi^{\bullet}\cdot\pi^{\circ}$, as
it is described in Theorem 2.1.17, and in Theorem 4.0.9 we consider the length
of $\pi$ where $\pi$ is presented in the generalized standard $OGS$
presentation, as it is described in Theorem 2.1.6.

######  Theorem 4.0.8.

Let $\pi\in D_{n}$ presented in the form $\pi=\pi^{\bullet}\cdot\pi^{\circ}$,
as it is described in Theorem 2.1.17. Assume

$\pi^{\bullet}=w_{L_{1}}\cdot w_{L_{2}}\cdots w_{L_{u}},$

for some positive integer $u$, where for $1\leq j\leq u$, $w_{L_{j}}$ as it is
defined in Definition 2.1.3. Assume $\pi^{\circ}$ is a standard $OGS$
elementary element (as it is defined in Definition 1.0.6) by considering it as
an element of $S_{n}$, such that the standard $OGS$ presentation of
$\pi^{\circ}$ as follow:

$\pi^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdot t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{m}}^{i_{k_{m}}}.$

Let $\pi^{\prime}$ be an element of $S_{n}$ as it is defined in Definition
2.1.10 (Notice, by Theorem 2.1.17, the presentation of $\pi^{\prime}$ and
$\pi^{\circ}$ have the same standard $OGS$ presentation, and present the same
element of $S_{n}$, by considering $\pi^{\circ}$ as an element of $S_{n}$).
For every $1\leq j\leq u$, define $\varrho_{L_{j}}(\pi^{\prime})$ to be

$\varrho_{L_{j}}(\pi^{\prime})=\begin{cases}L_{j}&\text{if}\quad
L_{j}<maj(\pi^{\prime})\quad\text{or}\quad Lj\geq k_{m}\\\ \\\
L_{j}-maj(\pi^{\prime})&\text{if}\quad maj(\pi^{\prime})\leq L_{j}<k_{1}\\\
\\\ L_{j}-\sum_{x=q}^{m}i_{k_{x}}\quad\text{for}\quad 2\leq q\leq
m&\text{if}\quad k_{x-1}\leq L_{j}<k_{x}.\end{cases}$

Then

$\ell(\pi)=\ell(\pi^{\prime})+2\cdot\sum_{j=1}^{u}\varrho_{L_{j}}.$

###### Proof.

Consider $\pi=\pi^{\bullet}\cdot\pi^{\circ}$, where

$\pi^{\bullet}=w_{L_{1}}\cdot w_{L_{2}}\cdots w_{L_{u}},$

for some positive integer $u$, where for $1\leq j\leq u$, $w_{L_{j}}$ as it is
defined in Definition 2.1.3. Since by Proposition 2.1.12, $w_{L_{p}}\cdot
w_{L_{q}}=w_{L_{q}}\cdot w_{L_{p}}$ for every $1\leq p,q\leq u$, we may assume

$L_{1}<L_{2}<\ldots<L_{u}.$

Hence, by Definition 2.1.3, the presentation of $\pi^{\bullet}$ by Coxeter
generators

$\pi^{\bullet}=w_{L_{1}}\cdot w_{L_{2}}\cdots w_{L_{u}}$ $=(s_{L_{1}}\cdot
s_{L_{1}-1}\cdots s_{2}\cdot s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots
s_{L_{1}})\cdot(s_{L_{2}}\cdot s_{L_{2}-1}\cdots s_{2}\cdot s_{1}\cdot
s_{1^{\prime}}\cdot s_{2}\cdots s_{L_{2}})\cdots$ $\cdots(s_{L_{u}}\cdot
s_{L_{u}-1}\cdots s_{2}\cdot s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots
s_{L_{u}}).$

Assume $\pi^{\circ}$ is a standard $OGS$ elementary element, by considering it
as an element of $S_{n}$, with the following standard $OGS$ presentation as it
is described in Theorem 1.0.3.

$\pi^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdot t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{m}}^{i_{k_{m}}}.$

Then, by [9], Theorem 28:

$norm(\pi^{\circ})=\prod_{u=\rho_{1}}^{k_{1}-1}\prod_{r=0}^{\rho_{1}-1}s_{u-r}\cdot\prod_{u=k_{1}}^{k_{2}-1}\prod_{r=0}^{\rho_{2}-1}s_{u-r}\cdot\prod_{u=k_{2}}^{k_{3}-1}\prod_{r=0}^{\rho_{3}-1}s_{u-r}\cdots\prod_{u=k_{m-1}}^{k_{m}-1}\prod_{r=0}^{\rho_{m}-1}s_{u-r},$

where, $\rho_{j}=\sum_{x=j}^{m}i_{k_{x}}$ for $1\leq j\leq m$.

Now, we consider $norm(\pi)=norm(\pi^{\bullet}\cdot\pi^{\circ})$ in the
following way.

First, consider $norm(w_{L_{u}}\cdot\pi^{\circ})$. Since, by Theorem 2.1.17,
the standard $OGS$ presentation of $\pi^{\circ}$ is same to the standard $OGS$
presentation of $\pi^{\prime}$, we can consider
$norm(w_{L_{u}}\cdot\pi^{\circ})$ as it is described in the proof of
Proposition 4.0.4. Hence, by Proposition 4.0.4

$\ell(w_{L_{u}}\cdot\pi^{\circ})=2\cdot\dot{\varrho}_{L_{u}}(\pi^{\prime})+\ell(\pi^{\prime}),$

where, for every $1\leq j\leq u$,

$\dot{\varrho}_{L_{j}}(\pi^{\prime})=\begin{cases}L_{j}&\text{if}\quad
L_{j}<maj(\pi^{\prime})\quad\text{or}\quad Lj\geq k_{m}\\\ \\\
L_{j}-maj(\pi^{\prime})&\text{if}\quad maj(\pi^{\prime})\leq L_{j}<k_{1}\\\
\\\ L_{j}-\sum_{x=q}^{m}i_{k_{x}}\quad\text{for}\quad 2\leq q\leq
m&\text{if}\quad k_{x-1}\leq L_{j}<k_{x}.\end{cases}$

Now, we consider $w_{L_{u-1}}\cdot norm(w_{L_{u}}\cdot\pi^{\circ})$. Since,
$L_{u-1}<L_{u}$, for finding $norm(w_{L_{u-1}}\cdot
norm(w_{L_{u}}\cdot\pi^{\circ}))$, we can apply the same algorithm as it is
described in Proposition 4.0.4 for finding $norm(w_{L}\cdot\pi^{\prime})$ and
$\ell(w_{L}\cdot\pi^{\prime})$, and we get

$\ell(w_{L_{u-1}}\cdot(w_{L_{u}}\cdot\pi^{\circ})=2\cdot\dot{\varrho}_{L_{u-1}}(\pi^{\prime})+\ell(w_{L_{u}}\cdot\pi^{\circ})=2\cdot\big{(}\dot{\varrho}_{L_{u-1}}(\pi^{\prime})+\dot{\varrho}_{L_{u}}(\pi^{\prime})\big{)}+\ell(\pi^{\prime}).$

Now, assume by induction on $j$,

$\ell(w_{L_{u-j}}\cdot w_{L_{u-(j-1)}}\cdots
w_{L_{u}}\cdot\pi^{\circ})=2\cdot\sum_{x=0}^{j}\dot{\varrho}_{L_{u-x}}(\pi^{\prime})+\ell(\pi^{\prime}).$

Now, we consider $w_{L_{u-(j+1)}}\cdot norm(w_{L_{u-j}}\cdot
w_{L_{u-(j-1)}}\cdots w_{L_{u}}\cdot\pi^{\circ})$. Since,
$L_{u-(j+1)}<L_{u-j}$, for finding

$norm(w_{L_{u-(j+1)}}\cdot norm(w_{L_{u-j}}\cdot w_{L_{u-(j-1)}}\cdots
w_{L_{u}}\cdot\pi^{\circ})),$

we can apply the same algorithm as it is described in Proposition 4.0.4 for
finding $norm(w_{L}\cdot\pi^{\circ})$ and $\ell(w_{L}\cdot\pi^{\circ})$, and
we get

$\ell(w_{L_{u-(j+1)}}\cdot(w_{L_{u-j}}\cdot w_{L_{u-(j-1)}}\cdots
w_{L_{u}}\cdot\pi^{\circ}))$
$=2\cdot\dot{\varrho}_{L_{u-(j+1)}}(\pi^{\prime})+\ell((w_{L_{u-j}}\cdot
w_{L_{u-(j-1)}}\cdots
w_{L_{u}}\cdot\pi^{\circ})=2\cdot\sum_{x=0}^{j+1}\dot{\varrho}_{L_{u-x}}(\pi^{\prime})+\ell(\pi^{\circ}).$

Hence, the induction assumption holds for every $0\leq u-1$, and we have:

$\ell(\pi)=\ell(w_{L_{1}}\cdot w_{L_{2}}\cdots
w_{L_{u}}\cdot\pi^{\circ})=\ell(\pi^{\circ})+2\cdot\sum_{j=1}^{u}\dot{\varrho}_{L_{j}}(\pi^{\prime}).$

∎

######  Theorem 4.0.9.

Let $\pi$ be an element of $D_{n}$, such that $\pi^{\prime}$ is a standard
$OGS$ elementary element (defined in Definition 1.0.6). Consider the
presentation of $\pi$ as it is presented in Corollary 2.1.14, i.e.,$\pi$ is
presented by the generalized standard $OGS$ as follow:

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\mu-1}^{\circ}\cdot\pi_{\mu-1}^{\bullet}\cdot\pi_{\mu}^{\circ}$
$=\pi_{1}^{\circ}\cdot w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot\pi_{\mu}^{\circ},$

where,

$\text{either}\quad\pi_{1}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\quad\text{or}\quad\pi_{1}^{\circ}=1,$

and,

$\text{either}\quad\pi_{\mu}^{\circ}=t_{k_{r_{\mu-1}+1}}^{i_{k_{r_{\mu-1}+1}}}\cdots
t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}\quad\text{or}\quad\pi_{\mu}^{\circ}=1.$

Then,

$\ell(\pi)=\begin{cases}\sum_{u=1}^{\mu-1}\big{(}maj_{u}(\pi)\cdot\big{(}(-1)^{\nu_{u}}+1\big{)}+2\cdot\sum_{j=1}^{\nu_{u}}\varrho_{L_{u_{j}}}(\pi)\big{)}+\ell(\pi^{\prime})&\text{if}\quad\pi_{\mu}^{\circ}\neq
1\\\ \\\
\sum_{u=1}^{\mu-2}\big{(}maj_{u}(\pi)\cdot\big{(}(-1)^{\nu_{u}}+1\big{)}+2\cdot\sum_{j=1}^{\nu_{u}}\varrho_{L_{u_{j}}}(\pi)\big{)}+\ell(\pi^{\prime})&\text{if}\quad\pi_{\mu}^{\circ}=1.\end{cases}$

where, by Definition 2.1.16,

$maj_{u}(\pi)=\sum_{j=1}^{r_{u}}i_{k_{j}}\quad\rho_{u}(\pi)=\sum_{j=r_{u}+1}^{r_{\mu}}i_{k_{j}},$
$\varrho_{L_{u_{j}}}(\pi^{\circ})=\begin{cases}L_{u_{j}}-\rho_{u}(\pi)&\text{if}\quad
L_{u_{j}}\geq maj(\pi^{\prime})\\\ L_{u_{j}}&\text{if}\quad
L_{u_{j}}<maj(\pi^{\prime})\end{cases}$

###### Proof.

Consider the presentation of $\pi$ as it is presented in Corollary 2.1.14.

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdots\pi_{\mu-1}^{\circ}\cdot\pi_{\mu-1}^{\bullet}\cdot\pi_{\mu}^{\circ}$
$=\pi_{1}^{\circ}\cdot w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot
t_{k_{r_{1}+1}}^{i_{k_{r_{1}+1}}}\cdots t_{k_{r_{2}}}^{i_{k_{r_{2}}}}\cdot
w_{L_{2_{1}}}\cdots w_{L_{2_{\nu_{2}}}}\cdot
t_{k_{r_{2}+1}}^{i_{k_{r_{2}}+1}}\cdots$ $\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot\pi_{\mu}^{\circ}.$

where,

$\text{either}\quad\pi_{1}^{\circ}=t_{k_{1}}^{i_{k_{1}}}\cdots
t_{k_{r_{1}}}^{i_{k_{r_{1}}}}\quad\text{or}\quad\pi_{1}^{\circ}=1,$

and

$\text{either}\quad\pi_{\mu}^{\circ}=t_{k_{r_{\mu-1}+1}}^{i_{k_{r_{\mu-1}+1}}}\cdots
t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}\quad\text{or}\quad\pi_{\mu}^{\circ}=1.$

Now, we recall the definitions of $maj_{\alpha}(\pi)$, $\rho_{\alpha}(\pi)$
and $\varrho_{L_{\alpha_{j}}}(\pi)$ for every $1\leq\alpha\leq\mu$ and $1\leq
j\leq\nu_{\alpha}$, as it is defined in Definition 2.1.16 :

$maj_{\alpha}(\pi)=\sum_{j=1}^{r_{\alpha}}i_{k_{j}}\quad\quad\rho_{\alpha}(\pi)=maj(\pi^{\prime})-maj_{\alpha}(\pi)=\sum_{j=r_{\alpha}+1}^{r_{\mu}}i_{k_{j}}.$

$\varrho_{L_{u_{j}}}(\pi)=\begin{cases}L_{u_{j}}-\rho_{u}(\pi)&\text{if}\quad
L_{u_{j}}\geq maj(\pi^{\prime})\\\ L_{u_{j}}&\text{if}\quad
L_{u_{j}}<maj(\pi^{\prime})\end{cases}$

By Theorem 2.1.17,

$\pi=\prod_{j=1}^{\mu-1}w_{maj_{j}(\pi)~{}|~{}maj_{j}(\pi)<k_{1}}^{-0.5\cdot(-1)^{\nu_{j}}+0.5}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot w_{L_{2_{1}}}\cdots
w_{L_{2_{\nu_{2}}}}\cdots w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot\pi^{\circ}.$

First, notice by Theorem 2.1.17,

$\pi^{\circ}=\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdots\pi_{\mu}^{\circ}=\begin{cases}t_{k_{1}}^{i_{k_{1}}}\cdot
t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}&\text{if}\quad\pi_{\mu}^{\circ}=1\\\
t_{k_{1}}^{i_{k_{1}}}\cdot t_{k_{2}}^{i_{k_{2}}}\cdots
t_{k_{r_{\mu-1}}}^{i_{k_{r_{\mu-1}}}}\cdot
t_{k_{r_{\mu-1}+1}}^{i_{k_{r_{\mu-1}+1}}}\cdots
t_{k_{r_{\mu}}}^{i_{k_{r_{\mu}}}}&\text{if}\quad\pi_{\mu}^{\circ}\neq
1.\end{cases}$

Now, we consider $L_{\alpha_{j}}$ for $1\leq\alpha\leq\mu-1$ and $1\leq
j\leq\nu_{\alpha}$.

By Corollary 2.1.14,

$\begin{cases}L_{\alpha_{j}}<k_{1}&\text{if}\quad\alpha=1\quad\text{and}\quad\pi_{1}^{\circ}=1\\\
L_{\alpha_{j}}\geq
k_{m-1}&\text{if}\quad\alpha=\mu-1\quad\text{and}\quad\pi_{\mu}^{\circ}=1\\\
k_{r_{\alpha}}\leq
L_{\alpha_{j}}<k_{r_{\alpha}+1}&\text{otherwise}\end{cases}$ (5)

Notice, by Theorem 2.1.17, the standard $OGS$ presentation of $\pi^{\circ}$ by
considering it as an element of $S_{n}$ is same to the standard $OGS$
presentation of $\pi^{\prime}$. Hence, when we calculate length of $\pi$ or
normal form of $\pi$ we often write $\pi^{\prime}$ instead of $\pi^{\circ}$.

Now, consider $\dot{\varrho}_{L_{\alpha_{j}}}(\pi^{\prime})$ for
$1\leq\alpha\leq\mu-1$ and $1\leq j\leq\nu_{\alpha}$, as it is defined in
Theorem 4.0.8.

$\dot{\varrho}_{L_{\alpha_{j}}}(\pi^{\prime})=\begin{cases}L_{\alpha_{j}}&\text{if}\quad
L_{\alpha_{j}}<maj(\pi^{\prime})\quad\text{or}\quad L_{\alpha_{j}}\geq
k_{\mu}\\\ \\\ L_{\alpha_{j}}-maj(\pi^{\prime})&\text{if}\quad
maj(\pi^{\prime})\leq L_{\alpha_{j}}<k_{1}\\\ \\\
L_{\alpha_{j}}-\sum_{x=q}^{m}i_{k_{x}}\quad\text{for}\quad 2\leq
q\leq\mu&\text{if}\quad k_{x-1}\leq L_{\alpha_{j}}<k_{x}.\end{cases}$

Hence, by Equation 5, and by the definition of $\varrho_{\alpha}(\pi)$ for
$1\leq\alpha\leq\mu-1$, the following are satisfied:

For $1\leq\alpha\leq\mu-1$, and $1\leq j\leq\nu_{\alpha}$,

$\dot{\varrho}_{L_{\alpha_{j}}}(\pi^{\prime})=\varrho_{\alpha}(\pi).$ (6)

Now, we consider $maj_{\alpha}(\pi)$ for $1\leq\alpha\leq\mu-1$.

Notice, by the definition of $maj_{\alpha}(\pi)$ for $1\leq\alpha\leq\mu-1$,
the following holds:

$maj_{\alpha}(\pi)=\prod_{j=1}^{\alpha}maj(\pi_{j}^{\circ})\leq\prod_{j=1}^{\mu}maj(\pi_{j}^{\circ})=maj(\pi^{\prime}),$

by considering $\pi^{\circ}$ as an element of $S_{n}$.

where,

$maj_{\alpha}(\pi)=maj(\pi^{\prime})$

if and only if

$\alpha=\mu-1\quad\text{and}\quad\pi_{\mu}^{\circ}=1.$

Therefore, in case:

$\pi_{\mu}^{\circ}\neq 1\quad\text{or}\quad 1\leq\alpha\leq\mu-2$

we have

$maj_{\alpha}(\pi)<maj(\pi^{\prime}).$

Hence, by the definition of $\dot{\varrho}_{maj_{\alpha}(\pi)}(\pi^{\prime})$,
for every $\leq\alpha\leq\mu-1$, such that $maj_{\alpha}(\pi)<k_{1}$, the
following holds:

$\dot{\varrho}_{maj_{\alpha}(\pi)}(\pi^{\prime})=\begin{cases}maj_{\alpha}(\pi)&\text{if}\quad\pi_{\mu}^{\circ}\neq
1\quad\text{or}\quad 1\leq\alpha\leq\mu-2\\\
0&\text{if}\quad\alpha=\mu-1\quad\text{and}\quad\pi_{\mu}^{\circ}=1.\end{cases}$
(7)

Hence, by Theorem 4.0.8, by using Equations 6 and 7 we get the length formula:

$\ell(\pi)=$
$=\ell(\prod_{j=1}^{\mu-1}w_{maj_{j}(\pi)~{}|~{}maj_{j}(\pi)<k_{1}}^{-0.5\cdot(-1)^{\nu_{j}}+0.5}\cdot
w_{L_{1_{1}}}\cdots w_{L_{1_{\nu_{1}}}}\cdot w_{L_{2_{1}}}\cdots
w_{L_{2_{\nu_{2}}}}\cdots w_{L_{{\mu-1}_{1}}}\cdots
w_{L_{{\mu-1}_{\nu_{\mu-1}}}}\cdot\pi^{\circ})$
$=\begin{cases}\sum_{u=1}^{\mu-1}\big{(}maj_{u}(\pi)\cdot\big{(}(-1)^{\nu_{u}}+1\big{)}+2\cdot\sum_{j=1}^{\nu_{u}}\varrho_{L_{u_{j}}}(\pi)\big{)}+\ell(\pi^{\prime})&\text{if}\quad\pi_{\mu}^{\circ}\neq
1\\\ \\\
\sum_{u=1}^{\mu-2}\big{(}maj_{u}(\pi)\cdot\big{(}(-1)^{\nu_{u}}+1\big{)}+2\cdot\sum_{j=1}^{\nu_{u}}\varrho_{L_{u_{j}}}(\pi)\big{)}+\ell(\pi^{\prime})&\text{if}\quad\pi_{\mu}^{\circ}=1.\end{cases}$

∎

######  Example 4.0.10.

Consider

$\pi=t_{10}^{4}\cdot w_{11}\cdot t_{12}^{2}\cdot w_{13}\cdot w_{14}\cdot
w_{15}\cdot t_{16}^{3}\cdot t_{17}.$

Notice,

$\pi=\pi_{1}^{\circ}\cdot\pi_{1}^{\bullet}\cdot\pi_{2}^{\circ}\cdot\pi_{2}^{\bullet}\cdot\pi_{3}^{\circ},$

where,

$\pi_{1}^{\circ}=t_{10}^{4}\quad\pi_{1}^{\bullet}=w_{11}\quad\pi_{2}^{\circ}=t_{12}^{2}\quad\pi_{2}^{\bullet}=w_{13}\cdot
w_{14}\cdot w_{15}\quad\pi_{3}^{\circ}=t_{16}^{3}\cdot t_{17}.$

$maj_{1}(\pi)=maj(\pi_{1}^{\circ})=4\quad
maj_{2}(\pi)=maj(\pi_{1}^{\circ}\cdot\pi_{2}^{\circ})=4+2=6.$
$maj(\pi^{\prime})=maj(\pi_{1}^{\circ}\cdot\pi_{2}^{\circ}\cdot\pi_{3}^{\circ})=4+2+4=10.$

$\rho_{1}(\pi)=maj(\pi^{\prime})-maj_{1}(\pi)=10-4=6\quad\rho_{2}(\pi)=maj(\pi^{\prime})-maj_{2}(\pi)=10-6=4.$

Hence, by Theorem 4.0.9

$\ell(\pi)=2\cdot maj_{1}(\pi)+2\cdot
maj_{2}(\pi)+2\cdot(11-\rho_{1}(\pi))+2\cdot(13-\rho_{2}(\pi))+2\cdot(14-\rho_{2}(\pi))+2\cdot$
$(15-\rho_{2}(\pi))+10\cdot 4+12\cdot 2+16\cdot 3+17-10^{2}$ $=2\cdot 4+2\cdot
6+2\cdot(11-6)+2\cdot(13-4)+2\cdot(14-4)+2\cdot(15-4)+$ $+10\cdot 4+12\cdot
2+16\cdot 3+17-10^{2}=90+29=119.$

Now, consider $\ell(\pi)$ by looking at the presentation of $\pi$ in terms of
Coxeter generators:
First, notice by Theorem 2.1.17

$\pi=w_{4}\cdot w_{6}\cdot w_{11}\cdot w_{13}\cdot w_{14}\cdot w_{15}\cdot
t_{10}^{4}\cdot t_{12}^{2}\cdot t_{16}^{3}\cdot t_{17}.$

Then, the presentation of $\pi$ in terms of Coxeter generators as follow:

$\pi=(s_{4}\cdot s_{3}\cdot s_{2}\cdot s_{1}\cdot s_{1^{\prime}}\cdot
s_{2}\cdot s_{3}\cdot s_{4})\cdot(s_{6}\cdot s_{5}\cdots s_{1}\cdot
s_{1^{\prime}}\cdot s_{2}\cdots s_{6})\cdot(s_{11}\cdot s_{10}\cdots
s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots s_{11})$ $\cdot(s_{13}\cdot
s_{12}\cdots s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots
s_{13})\cdot(s_{14}\cdot s_{13}\cdots s_{1}\cdot s_{1^{\prime}}\cdot
s_{2}\cdots s_{14})\cdot(s_{15}\cdots s_{1}\cdot s_{1^{\prime}}\cdot
s_{2}\cdots s_{15})$ $\cdot(s_{10}\cdot s_{9}\cdot s_{8}\cdot s_{7}\cdot
s_{6}\cdot s_{5})\cdot(s_{11}\cdot s_{10}\cdot s_{9}\cdot s_{8}\cdot
s_{7}\cdot s_{6})\cdot(s_{12}\cdot s_{11}\cdot s_{10}\cdot
s_{9})\cdot(s_{13}\cdot s_{12}\cdot s_{11}\cdot s_{10})$ $\cdot(s_{14}\cdot
s_{13}\cdot s_{12}\cdot s_{11})\cdot(s_{15}\cdot s_{14}\cdot s_{13}\cdot
s_{12})\cdot s_{16}.$

By Lemma 4.0.2, we get :

$\pi=(s_{4}\cdot s_{3}\cdot s_{2}\cdot s_{1}\cdot s_{1^{\prime}}\cdot
s_{2}\cdot s_{3}\cdot s_{4})\cdot(s_{6}\cdot s_{5}\cdots s_{1}\cdot
s_{1^{\prime}}\cdot s_{2}\cdots s_{6})\cdot(s_{10}\cdot s_{9}\cdot s_{8}\cdot
s_{7}\cdot s_{6}\cdot s_{5})\cdot$ $(s_{11}\cdot s_{10}\cdots s_{1}\cdot
s_{1^{\prime}}\cdot s_{2}\cdots s_{11})\cdot(s_{11}\cdot s_{10}\cdots
s_{6})\cdot(s_{12}\cdot s_{11}\cdot s_{10}\cdot s_{9})\cdot(s_{13}\cdot
s_{12}\cdots s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots s_{13})\cdot$
$(s_{13}\cdot s_{12}\cdot s_{11}\cdot s_{10})\cdot(s_{14}\cdot s_{13}\cdots
s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots s_{14})\cdot(s_{14}\cdot
s_{13}\cdot s_{12}\cdot s_{11})\cdot$ $(s_{15}\cdots s_{1}\cdot
s_{1^{\prime}}\cdot s_{2}\cdots s_{15})\cdot(s_{15}\cdot s_{14}\cdot
s_{13}\cdot s_{12})\cdot s_{16}$ $=(s_{4}\cdot s_{3}\cdot s_{2}\cdot
s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdot s_{3}\cdot s_{4})\cdot(s_{6}\cdot
s_{5}\cdots s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots s_{6})\cdot(s_{10}\cdot
s_{9}\cdot s_{8}\cdot s_{7}\cdot s_{6}\cdot s_{5})\cdot(s_{11}\cdot
s_{10}\cdots s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots s_{5})$
$\cdot(s_{12}\cdot s_{11}\cdot s_{10}\cdot s_{9})\cdot(s_{13}\cdot
s_{12}\cdots s_{1}\cdot s_{1^{\prime}}\cdot s_{2}\cdots
s_{9})\cdot(s_{14}\cdot s_{13}\cdots s_{1}\cdot s_{1^{\prime}}\cdot
s_{2}\cdots s_{10})$ $\cdot(s_{15}\cdots s_{1}\cdot s_{1^{\prime}}\cdot
s_{2}\cdots s_{11})\cdot s_{16}=\text{norm}(\pi).$

Thus:

$\ell(\pi)=2\cdot 4+2\cdot
6+2\cdot(11-6)+2\cdot(13-4)+2\cdot(14-4)+2\cdot(15-4)+$ $+10\cdot 4+12\cdot
2+16\cdot 3+17-(10^{2})=90+29=119.$

In the next example, we present an algorithm to find the Coxeter length of an
arbitrary element $\pi$ of $D_{n}$, presented by the generalized standard
$OGS$ of $D_{n}$, where $\pi^{\prime}$ is not necessarily a standard $OGS$
elementary element.

######  Example 4.0.11.

Consider

$\pi=t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{4}\cdot t_{14}^{7}\cdot w_{15}\cdot
t_{16}^{4}\cdot t_{19}^{10}\cdot w_{20}\cdot t_{22}^{5}\cdot t_{23}^{4}.$

Therefore,

$\pi^{\prime}=t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{4}\cdot t_{14}^{7}\cdot
t_{16}^{4}\cdot t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}.$

Notice,

$maj(\pi^{\prime})=2+2+4+7+4+10+5+4=38>k_{1}=7.$

Hence $\pi^{\prime}$ is not a standard $OGS$ elementary element, So, we
consider a standard $OGS$ elementary factorization of $\pi^{\prime}$ as it is
described in Definition 1.0.8:

$\pi^{\prime}=t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3}\quad|t_{12}\cdot
t_{14}^{7}\cdot t_{16}^{4}\quad|t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

Therefore,

$\pi=t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3}\quad|t_{12}\cdot t_{14}^{7}\cdot
w_{15}\cdot t_{16}^{4}\quad|t_{19}^{10}\cdot w_{20}\cdot t_{22}^{5}\cdot
t_{23}^{4}$

Then by Remark 2.1.19:

$\dot{\pi}^{(1)}=t_{7}^{2}\cdot t_{9}^{2}\cdot
t_{12}^{3}\quad\dot{\pi}^{(2)}=t_{12}\cdot t_{14}^{7}\cdot w_{15}\cdot
t_{16}^{4}\quad\dot{\pi}^{(3)}=t_{19}^{10}\cdot w_{20}\cdot t_{22}^{5}\cdot
t_{23}^{4}$

Notice,

$maj\big{[}{\big{(}\pi^{\prime}\big{)}}^{(1)}\big{]}=7,$
$maj\big{[}{\big{(}\pi^{\prime}\big{)}}^{(2)}\big{]}=12\quad
maj_{1}(\dot{\pi}^{(2)})=8,$
$maj\big{[}{\big{(}\pi^{\prime}\big{)}}^{(3)}\big{]}=19\quad
maj_{1}(\dot{\pi}^{(3)})=10.$

Hence, by Theorem 2.1.17 on $\dot{\pi}^{(j)}$ for $1\leq j\leq 3$:

$\pi=t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3}\quad|w_{8}\cdot w_{15}\cdot
t_{12}\cdot t_{14}^{7}\cdot t_{16}^{4}\quad|w_{10}\cdot w_{20}\cdot
t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

By applying Proposition 2.1.12 on $t_{12}^{3}\cdot w_{8}$ and on
$t_{16}^{4}\cdot w_{10}$ we get :

$\pi=t_{7}^{2}\cdot t_{9}^{2}\cdot w_{3}\cdot w_{11}\cdot
t_{12}^{3}\quad|w_{15}\cdot t_{12}\cdot t_{14}^{7}\cdot w_{4}\cdot w_{14}\cdot
t_{16}^{4}\quad|w_{20}\cdot t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

By applying Proposition 2.1.12 on $t_{9}^{2}\cdot w_{3}$ and on
$t_{14}^{7}\cdot w_{4}$ we get :

$\pi=t_{7}^{2}\cdot w_{2}\cdot w_{5}\cdot t_{9}^{2}\cdot w_{11}\cdot
t_{12}^{3}\quad|w_{15}\cdot t_{12}\cdot w_{7}\cdot w_{11}\cdot t_{14}^{7}\cdot
w_{14}\cdot t_{16}^{4}\quad|w_{20}\cdot t_{19}^{10}\cdot t_{22}^{5}\cdot
t_{23}^{4}$

By applying Proposition 2.1.12 on $t_{7}^{2}\cdot w_{2}$ and on $t_{12}\cdot
w_{7}$ we get :

$\pi=w_{2}\cdot w_{4}\cdot t_{7}^{2}\cdot w_{5}\cdot t_{9}^{2}\cdot
w_{11}\cdot t_{12}^{3}\quad|w_{15}\cdot w_{1}\cdot w_{8}\cdot t_{12}\cdot
w_{11}\cdot t_{14}^{7}\cdot w_{14}\cdot t_{16}^{4}\quad|w_{20}\cdot
t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

By applying Proposition 2.1.12 on $t_{7}^{2}\cdot w_{5}$ and on $t_{12}\cdot
w_{11}$ we get :

$\pi=w_{2}\cdot w_{4}\cdot w_{2}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot
w_{11}\cdot t_{12}^{3}\quad|w_{15}\cdot w_{1}\cdot w_{8}\cdot w_{1}\cdot
t_{12}\cdot t_{14}^{7}\cdot w_{14}\cdot t_{16}^{4}\quad|w_{20}\cdot
t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

By applying Proposition 2.1.12 on $w_{4}\cdot w_{2}$ and on $w_{8}\cdot w_{1}$
we get:

$\pi=w_{2}\cdot w_{2}\cdot w_{4}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot
w_{11}\cdot t_{12}^{3}\quad|w_{15}\cdot w_{1}\cdot w_{1}\cdot w_{8}\cdot
t_{12}\cdot t_{14}^{7}\cdot w_{14}\cdot t_{16}^{4}\quad|w_{20}\cdot
t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

Since by applying Lemma 2.1.4, $w_{2}^{2}=w_{1}^{2}=1$:

$\pi=w_{4}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot w_{11}\cdot
t_{12}^{3}\quad|w_{15}\cdot w_{8}\cdot t_{12}\cdot t_{14}^{7}\cdot w_{14}\cdot
t_{16}^{4}\quad|w_{20}\cdot t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

Then, by applying Lemma 2.1.15 on $t_{7}^{2}\cdot t_{9}^{2}\cdot w_{11}$ and
on $t_{12}\cdot t_{14}^{7}\cdot w_{14}$:

$\pi=w_{4}\cdot w_{4}\cdot w_{11}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot
t_{12}^{3}\quad|w_{15}\cdot w_{8}\cdot w_{8}\cdot w_{14}\cdot t_{12}\cdot
t_{14}^{7}\cdot t_{16}^{4}\quad|w_{20}\cdot t_{19}^{10}\cdot t_{22}^{5}\cdot
t_{23}^{4}$

By Lemma 2.1.4, $w_{4}^{2}=w_{8}^{2}=1$:

$\pi=w_{11}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3}\quad|w_{15}\cdot
w_{14}\cdot t_{12}\cdot t_{14}^{7}\cdot t_{16}^{4}\quad|w_{20}\cdot
t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

Then, by applying Proposition 2.1.12 on $w_{15}\cdot w_{14}$ we get:

$\pi=w_{11}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3}\quad|w_{14}\cdot
w_{15}\cdot t_{12}\cdot t_{14}^{7}\cdot t_{16}^{4}\quad|w_{20}\cdot
t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$ (8)

Then,

$\pi=\pi^{(1)}\cdot\pi^{(2)}\cdot\pi^{(3)}$

such that:

$\pi^{(1)}=w_{11}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot
t_{12}^{3}\quad\pi^{(2)}=w_{14}\cdot w_{15}\cdot t_{12}\cdot t_{14}^{7}\cdot
t_{16}^{4}\quad\pi^{(3)}=w_{20}\cdot t_{19}^{10}\cdot t_{22}^{5}\cdot
t_{23}^{4}$

Now, consider $\pi^{(1)}$.

$\pi^{(1)}=w_{11}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3},$

where $t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3}$ is a standard $OGS$
elementary element, with
$k_{1}=7,\quad k_{2}=9,\quad k_{3}=12$, and $i_{k_{1}}=2,\quad
i_{k_{2}}=2,\quad i_{k_{3}}=3$, as it is defined in Definition 1.0.6.

Since, $9<11<12$, we consider Proposition 4.0.4 on $w_{11}\cdot(t_{7}^{2}\cdot
t_{9}^{2}\cdot t_{12}^{3})$, for the case $k_{2}\leq L<k_{3}$.

Then, we get:

$norm(\pi^{(1)})=(s_{7}\cdot s_{6}\cdot s_{5}\cdot s_{4}\cdot
s_{3})\cdot(s_{8}\cdot s_{7}\cdot s_{6}\cdot s_{5}\cdot s_{4})\cdot(s_{9}\cdot
s_{8}\cdot s_{7})\cdot(s_{10}\cdot s_{9}\cdot s_{8})$ $\cdot(s_{11}\cdot
s_{10}\cdots s_{2}\cdot s_{1^{\prime}}\cdot s_{1}\cdot s_{2}\cdots s_{8}).$

The length of $\pi^{(1)}$:

$\ell(\pi^{(1)})=\ell(t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3})+2\cdot(11-3).$

Now, consider $\pi^{(2)}$.

$\pi^{(2)}=w_{14}\cdot w_{15}\cdot t_{12}\cdot t_{14}^{7}\cdot t_{16}^{4},$

where $t_{12}\cdot t_{14}^{7}\cdot t_{16}^{4}$ is a standard $OGS$ elementary
element, with
$k_{1}=12,\quad k_{2}=14,\quad k_{3}=16$, and $i_{k_{1}}=1,\quad
i_{k_{2}}=7,\quad i_{k_{3}}=4$, as it is defined in Definition 1.0.6.

Since, $14<15<16$ and $14\leq 14<16$, we consider Proposition 4.0.4 on
$w_{14}\cdot w_{15}\cdot(t_{12}\cdot t_{14}^{7}\cdot t_{16}^{4})$, for the
case $k_{2}\leq L<k_{3}$.

Then, we get:

$norm(\pi^{(2)})=(s_{12}\cdot s_{11}\cdots s_{2})\cdot(s_{13}\cdot
s_{12}\cdots s_{3})$ $\cdot(s_{14}\cdot s_{13}\cdots s_{2}\cdot
s_{1^{\prime}}\cdot s_{1}\cdot s_{2}\cdots s_{10})\cdot(s_{15}\cdot
s_{14}\cdots s_{2}\cdot s_{1^{\prime}}\cdot s_{1}\cdot s_{2}\cdots s_{11}).$

The length of $\pi^{(2)}$:

$\ell(\pi^{(2)})=\ell(t_{12}\cdot t_{14}^{7}\cdot
t_{16}^{4})+2\cdot(14-4)+2\cdot(15-4).$

Now, consider $\pi^{(3)}$.

$\pi^{(3)}=w_{20}\cdot t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4},$

where $t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$ is a standard $OGS$
elementary element, with
$k_{1}=19,\quad k_{2}=22,\quad k_{3}=23$, and $i_{k_{1}}=10,\quad
i_{k_{2}}=5,\quad i_{k_{3}}=4$, as it is defined in Definition 1.0.6.

Since, $19<20<22$, we consider Proposition 4.0.4 on
$w_{20}\cdot(t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4})$, for the case
$k_{1}\leq L<k_{2}$.

Then, we get:

$norm(\pi^{(3)})=(s_{19}\cdot s_{18}\cdots s_{11})\cdot(s_{20}\cdot
s_{19}\cdots s_{2}\cdot s_{1^{\prime}}\cdot s_{1}\cdot s_{2}\cdots s_{11})$
$\cdot(s_{21}\cdot s_{20}\cdots s_{13})\cdot(s_{22}\cdot s_{21}\cdot
s_{20}\cdot s_{19}).$

The length of $\pi^{(3)}$:

$\ell(\pi^{(3)})=\ell(t_{19}^{10}\cdot t_{22}^{5}\cdot
t_{23}^{4})+2\cdot(20-9).$

Therefore,

$norm(\pi)=norm(\pi^{(1)}\cdot\pi^{(2)}\cdot\pi^{(3)})=norm(\pi^{(1)})\cdot
norm(\pi^{(2)})\cdot norm(\pi^{(3)}).$

Hence,

$\ell(\pi)=\ell(\pi^{(1)}\cdot\pi^{(2)}\cdot\pi^{(3)})=\ell(\pi^{(1)})+\ell(\pi^{(2)})+\ell(\pi^{(3)}).$

Now, we find $\pi=\pi^{\bullet}\cdot\pi^{\circ}$ decomposition (where,
$\pi^{\bullet}\in Id_{n}^{\bullet}$ and $\pi^{\circ}\in S_{n}^{\circ}$) of
$\pi$, as it is described in Remark 2.1.19, by considering Equation 8.

By applying Lemma 2.1.15 on $(t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3})\cdot
w_{14}\cdot w_{15}$ we get:

$\pi=w_{11}\cdot w_{14}\cdot w_{15}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot
t_{12}^{3}\quad|t_{12}\cdot t_{14}^{7}\cdot t_{16}^{4}\quad|w_{20}\cdot
t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

Then, by applying Lemma 2.1.15 on $(t_{12}\cdot t_{14}^{7}\cdot
t_{16}^{4})\cdot w_{20}$.

$\pi=w_{11}\cdot w_{14}\cdot w_{15}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot
t_{12}^{3}\quad|w_{20}\cdot t_{12}\cdot t_{14}^{7}\cdot
t_{16}^{4}\quad|t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

Then, by applying Lemma 2.1.15 on $(t_{7}^{2}\cdot t_{9}^{2}\cdot
t_{12}^{3})\cdot w_{20}$.

$\pi=w_{11}\cdot w_{14}\cdot w_{15}\cdot w_{20}\cdot t_{7}^{2}\cdot
t_{9}^{2}\cdot t_{12}^{3}\quad|t_{12}\cdot t_{14}^{7}\cdot
t_{16}^{4}\quad|t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}$

Hence,

$\pi=\pi^{\bullet}\cdot\pi^{\circ}$

such that:

$\pi^{\bullet}=w_{11}\cdot w_{14}\cdot w_{15}\cdot w_{20},$

$\pi^{\circ}=t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3}\quad|t_{12}\cdot
t_{14}^{7}\cdot t_{16}^{4}\quad|t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}.$

##  5 Conclusions and future plans

This paper is a natural continuation of the paper [9], where there was
introduced a quite interesting generalization of the fundamental theorem for
abelian groups to two important and very elementary families of non-abelian
Coxeter groups, the $I$-type (dihedral groups), and the $A$-type (symmetric
groups). There were introduced canonical forms, with very interesting exchange
laws, and quite interesting properties concerning the Coxeter lengths of the
elements. In this paper we generalized the results of [9] for the $D$-type
Coxeter groups, namely $D_{n}$. The results of the paper motivate us for
further generalizations of the $OGS$ and the arising properties from it for
more families of Coxeter and generalized Coxeter groups, which have an
importance in the classification of Lie algebras and the Lie-type simple
groups, and in other fields of mathematics, such as algebraic geometry for
classification of fundamental groups of Galois covers of surfaces [2].
Furthermore, it is interesting to find generalization of the $OGS$ to the
affine classical families $\tilde{A}_{n}$, $\tilde{B}_{n}$, $\tilde{C}_{n}$,
and $\tilde{D}_{n}$, and also to other generalizations of the mentioned
Coxeter groups, as the complex reflection groups $G(r,p,n)$ [8] or the
generalized affine classical groups, the definition of which is described in
[7], [3].

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