[1,4]Zhengping Ji

1]Key Laboratory of Systems and Control, Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, Beijing, 100190, P. R. China

2]Department of Control Science and Engineering, School of Electronics and
Information Engineering, Tongji University, Shanghai, 201210, P. R. China

3]National Center for Mathematics and Interdisciplinary Sciences, Chinese
Academy of Sciences, Beijing, 100190, P. R. China

4]School of Mathematical Sciences, University of Chinese Academy of Sciences,
Beijing, 100149, P. R. China

# Contracted Product of Hypermatrices via STP of Matrices

Daizhan Cheng<EMAIL_ADDRESS>Min Meng<EMAIL_ADDRESS>Xiao Zhang
<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  [  [

###### Abstract

An equivalent definition of hypermatrices is introduced. The matrix expression
of hypermatrices is proposed. Using permutation matrices, the conversion of
different matrix expressions is revealed. The various contracted products of
hypermatrices are realized by semi-tensor products (STP) of matrices via
matrix expressions of hypermatrices.

###### keywords:

$d$-hypermatrix, matrix expression, permutation matrix, contracted product,
semi-tensor product (STP)

**footnotetext: This work is supported partly by the National Natural Science
Foundation of China (NSFC) under Grant 62073315 and 62103305, and China
Postdoctoral Science Foundation 2021M703423 and 2022T150686.

##  1 Introduction

The hypermatrix is an extension of the matrix to the higher order case (order
$d\geq 3$). It has found wide application in many fields, including computer
science [24], signal processing [17], statistics [16], etc. We refer to [21]
for a systematic introduction to the hypermatrices, and to [1, 18] for some
later developments.

A generalized matrix product, called the semi-tensor product (STP), was
proposed two decades ago [2]. Since then it has been rapidly developed both in
theoretical aspects and in various applications [4, 5]. For example, it has
been applied to the study of Boolean and finite-valued networks (see survey
papers [19, 20, 22, 23]); finite games (see survey paper [11]); finite
automata (see survey paper [25]); dimension-varying systems [7, 8], etc.

Later, in addition to the original matrix-matrix STP, the matrix-vector STP
was proposed [9] and applied to dimension-varying dynamic (control) systems
[8]. Recently, the STP of hypermatrices has also been introduced [14]. It
seems possible that some new unknown STPs will appear in the future. In the
early days, the (matrix-matrix) STP is said to be a generalization of the
conventional matrix product. With the appearance of new STPs, this explanation
does not seem to be sufficient.

The purpose of this article is to show that in essence, STPs are
representations of multilinear mappings over hypermatrices. More precisely,
the hypermatrices are first expressed in their matrix forms. Then the
multilinear mappings over them can be executed by STP over their matrix
expressions. For this purpose, the conversion of different matrix expressions
of hypermatrices plays an important role.

The rest of this paper is organized as follows: Section 2 considers how to
transform a hypermatrix into its distinct matrix expressions. The permutation
matrix is also introduced for transforming among different matrix expressions.
Section 3 shows how to realize the contracted product of hypermatrices over
their matrix expressions. In Section 4, the STP is used to perform the
contraction product of hypermatrices, which shows that STPs are essentially
multilinear operators over hypermatrices.

Before ending this section, we give a list of the notations used in this
paper.

  1. 1.

${\mathbb{N}}$: the set of positive integers.

  2. 2.

${\mathbb{R}}$: the set of real numbers.

  3. 3.

${\mathbb{C}}$: the set of complex numbers.

  4. 4.

${\mathbb{F}}^{n}$: the $n$ dimensional Euclidean space over ${\mathbb{F}}$,
where ${\mathbb{F}}$ is a field (Particularly, ${\mathbb{F}}={\mathbb{R}}$ or
${\mathbb{F}}={\mathbb{C}}$).

  5. 5.

$\operatorname{lcm}(a,b)$: the least common multiple of $a$ and $b$,
($a,b\in{\mathbb{N}}$).

  6. 6.

${\mathbb{F}}^{n_{1}\times n_{2}\times\cdots\times n_{d}}$: the $d$-th order
hypermatrices of dimensions $n_{1},n_{2},\cdots,n_{d}$.

  7. 7.

$[a,b]$: the set of integers $a\leq i\leq b$.

  8. 8.

$\langle b\rangle:=[1,b]$.

  9. 9.

$\mathrm{ID}(i_{1},\cdots,i_{d};n_{1},\cdots,n_{d})$: A set of ordered $d$
indices.

  10. 10.

$\delta_{n}^{i}$: the $i$-th column of the identity matrix $I_{n}$.

  11. 11.

$\delta_{n}^{I}:=[(\delta_{n}^{1})^{T},(\delta_{n}^{2})^{T},\cdots,(\delta_{n}^{n})^{T}]^{T}$.

  12. 12.

$\Delta_{n}:=\left\\{\delta_{n}^{i}|i=1,\cdots,n\right\\}$.

  13. 13.

$L\in{\mathbb{R}}^{m\times n}$ is a logical matrix, if its column set
$\operatorname{Col}(L)\subset\Delta_{m}$. Denote by $\mathcal{L}_{m\times n}$
the set of $m\times n$ logical matrices.

  14. 14.

$\delta_{m}[i_{1},\cdots,i_{n}]:=[\delta_{m}^{i_{1}},\cdots,\delta_{m}^{i_{n}}]$.

  15. 15.

${\bf 1}_{\ell}:=(\underbrace{1,1,\cdots,1}_{\ell})^{\mathrm{T}}$.

  16. 16.

${\bf V}_{r}(A)$: the row stacking form of $A$.

  17. 17.

${\bf V}_{c}(A)$: the column stacking form of $A$.

  18. 18.

${\bf V}^{s}_{r}(A)$: the s-row stacking form of $A$.

  19. 19.

${\bf V}^{s}_{c}(A)$: the s-column stacking form of $A$.

  20. 20.

${\bf S}_{d}$: the $d$-th order permutation group.

  21. 21.

$\ltimes$: the STP of matrices.

  22. 22.

$\times^{{\bf r}}_{{\bf s}}$: the contracted product of hypermatrices.

  23. 23.

$\ltimes^{{\bf r}}_{{\bf s}}$: the contracted STP of hypermatrices.

##  2 Matrix Expression of Hypermatrices

###  2.1 Set-based Hypermatrix

######  Definition 1.

[21] For $n_{1},\cdots,n_{d}\in{\mathbb{N}}$, a function $f:\langle
n_{1}\rangle\times\cdots\times\langle n_{d}\rangle\rightarrow{\mathbb{F}}$ is
called a hypermatrix (over ${\mathbb{F}}$), or an order $d$ hypermatrix or a
$d$-hypernatrix. A $d$-hypermatrix is also denoted by
$A=[a_{i_{1},\cdots,i_{d}}]$.

From a set point of view, a hypermatrix consists of two ingredients:

  * (i)

a set of data

$\displaystyle D_{A}:=\\{a_{i_{1},\cdots,i_{d}}\;|\;i_{s}\in\langle
n_{s}\rangle,\;s\in\langle d\rangle\\};$ (1)

  * (ii)

an order over $D_{A}$, which makes $D_{A}$ a totally ordered set. In other
words, each element in $D_{A}$ has a fixed position, just as in the case of a
matrix.

To make the order of $D_{A}$ precise, we introduce a set of ordered indices,
$\mathrm{ID}$, as follows:

$\mathrm{ID}(i_{\alpha_{1}},\cdots,i_{\alpha_{d}};n_{\alpha_{1}},\cdots,n_{a_{d}}),$

which stands for a set of data as in (1), where
$(\alpha_{1},\cdots,\alpha_{d})$ is a permutation of $\langle d\rangle$. The
order of $\mathrm{ID}$ is determined as follows: $a_{p_{1},\cdots,p_{d}}\prec
a_{q_{1},\cdots,q_{d}}$ if and only if there is an $s\in\langle d\rangle$,
such that $p_{i}=q_{i}$, $i<s$ and $p_{s}<q_{s}$.

If $(\alpha_{1},\cdots,\alpha_{d})=(1,2,\cdots,d)$, the corresponding ordered
set of indices is called a natural order, denoted by ${\bf i}_{\langle
d\rangle}$.

Based on the above argument, a hypermatrix can be defined alternatively as
follows.

######  Definition 2.

For $n_{1},\cdots,n_{d}\in{\mathbb{N}}$, a hypermatrix (over ${\mathbb{F}}$),
is a set of order $d$ data as in (1) with an $\mathrm{ID}$ as
$\mathrm{ID}({\bf i}_{\langle d\rangle};n_{1},\cdots,n_{d})$.

To apply classical matrix analysis to hypermatrices, the matrix expression of
hypermatrices is a key issue.

Let ${\bf r}=(r_{1},r_{2},\cdots,r_{d})$ be a set of indices, and

$\displaystyle{\bf
r}=(r_{i_{1}},r_{i_{2}},\cdots,r_{i_{p}})\cup(r_{j_{1}},r_{j_{2}},\cdots,r_{j_{q}}):={\bf
r}_{1}\cup{\bf r}_{2}$ (2)

be a partition, where ${\bf r}_{1}\cap{\bf r}_{2}=\emptyset$, and $p+q=d$. For
ease of notation, we will assume that the elements in two subsets inherit the
element order of the original set, unless otherwise stated.

######  Definition 3.

Given a hypermatrix $A=[a_{r_{1},r_{2},\cdots,r_{d}}]$. For each partition

$\displaystyle{\bf r}_{\langle
d\rangle}=\\{r_{i_{1}},r_{i_{2}},\cdots,r_{i_{p}}\\}\cup\\{r_{j_{1}},r_{j_{2}},\cdots,r_{j_{q}}\\}:={\bf
r}_{1}\cup{\bf r}_{2},$ (3)

there is a matrix expression of $A$, denoted by

$\displaystyle M_{A}^{{\bf r}_{1}\times{\bf r}_{2}}\in{\mathbb{F}}^{s\times
t},$ (4)

where

$s=\prod_{k=1}^{p}n_{i_{k}},\quad t=\prod_{\ell=1}^{q}n_{j_{\ell}}.$

Moreover, the elements in $M_{A}^{{\bf r}_{1}\times{\bf r}_{2}}$ are
$\\{a_{r_{1},r_{2},\cdots,r_{d}}\\}$, which are arranged by $\mathrm{ID}({\bf
r}_{1};n_{r_{i_{1}}},n_{r_{i_{2}}},\cdots,n_{r_{i_{p}}})$ for rows, and by
$\mathrm{ID}({\bf r}_{2};n_{r_{j_{1}}},n_{r_{j_{2}}},\cdots,n_{r_{j_{q}}})$
for columns.

######  Remark 1.

  * (i)

$p$ is called the contra-variant order of $M_{A}^{{\bf r}_{1}\times{\bf
r}_{2}}$ and $q$ is called the co-variant order of $M_{A}^{{\bf
r}_{1}\times{\bf r}_{2}}$.

  * (ii)

If $p=0$, then we set $s=1$, and if $q=0$, then we set $t=1$.

  * (iii)

Particularly, in (3) we require a natural order unless elsewhere is stated.
That is,

$\begin{array}[]{l}r_{i_{1}}<r_{i_{2}}<\cdots<r_{i_{p}},\\\
r_{j_{1}}<r_{j_{2}}<\cdots<r_{j_{q}}.\end{array}$

Thus a $d$-hypermatrix has $2^{d}$ different matrix expressions with the
natural order. The matrix expressions $M_{A}^{{\bf r}_{1}\times{\bf r}_{2}}$
can be briefly expressed as $M_{A}^{{\bf r}_{1}}$, if there is no possible
confusion.

######  Example 1.

Given $A=[a_{i_{1}i_{2}i_{3}}]\in{\mathbb{F}}^{2\times 3\times 2}$. Then

  * (i)

$\begin{array}[]{ccl}M_{A}^{\emptyset}=[a_{111}~{}a_{112}~{}a_{121}~{}a_{122}~{}a_{131}~{}a_{132}~{}a_{211}~{}a_{212}~{}a_{221}~{}a_{222}~{}a_{231}~{}a_{232}];\end{array}$

  * (ii)

$M_{A}^{(1)}=\begin{bmatrix}a_{111}&a_{112}&a_{121}&a_{122}&a_{131}&a_{132}\\\
a_{211}&a_{212}&a_{221}&a_{222}&a_{231}&a_{232}\end{bmatrix};$
$M_{A}^{(2)}=\begin{bmatrix}a_{111}&a_{112}&a_{211}&a_{212}\\\
a_{121}&a_{122}&a_{221}&a_{222}\\\
a_{131}&a_{132}&a_{231}&a_{232}\end{bmatrix};\quad\mbox{etc.}$

  * (iii)

$M_{A}^{(1,2)}=\begin{bmatrix}a_{111}&a_{112}\\\ a_{121}&a_{122}\\\
a_{131}&a_{132}\\\ a_{211}&a_{212}\\\ a_{221}&a_{222}\\\
a_{231}&a_{232}\end{bmatrix};\quad\mbox{etc.}$
$M_{A}^{(1,3)}=\begin{bmatrix}a_{111}&a_{121}&a_{131}\\\
a_{112}&a_{122}&a_{132}\\\ a_{211}&a_{221}&a_{231}\\\
a_{212}&a_{222}&a_{232}\end{bmatrix};\quad\mbox{etc.}$

  * (iv)

$M_{A}^{(1,2,3)}=\left(M_{A}^{\emptyset}\right)^{\mathrm{T}}.$

######  Definition 4.

  * (i)

$V_{A}:=M_{A}^{\emptyset}$ is called the (row) vector expression of
hypermatrix $A$.

  * (ii)

$M_{A}:=M_{A}^{(1)}$ is called the contra-variant one matrix expression
(briefly, matrix-1 expression) of hypermatrix $A$.

######  Definition 5.

Let $x_{i}\in{\mathbb{F}}^{n_{i}}$, $i\in\langle d\rangle$.

  * (i)

$\displaystyle x:=\ltimes_{i=1}^{d}x_{i}$ (5)

is called a hypervector of order $d$.

  * (ii)

The set of order $d$ hypervectors with corresponding dimensions is denoted by
${\mathbb{F}}^{n_{1}\ltimes\cdots\ltimes n_{d}}$.

Note that the components of $x$ can be expressed as

$\begin{array}[]{ccl}com(x):=\left\\{x_{i_{1},\cdots,i_{d}}=x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots
x_{d}^{i_{d}}\;|\;i_{j}\in\langle n_{j}\rangle,j\in\langle
d\rangle\right\\},\end{array}$

where $x_{r}^{i_{r}}$ is the $i_{r}$-th component of $x_{r}$. Hence it is
clear that $com(x)$ (or briefly hypervector $x$) is a hypermatrix of order
$d$. Moreover, it is obvious that

######  Proposition 1.

$\displaystyle{\mathbb{F}}^{n_{1}\ltimes\cdots\ltimes
n_{d}}\subset{\mathbb{F}}^{n_{1}\times\cdots\times n_{d}}$ (6)

is a subset of hypermatrices.

###  2.2 $\sigma$-transpose of Hypermatrices

######  Definition 6.

[21]

  * (i)

Given a $d$-hypermatrix
$A=\left[a_{j_{1},j_{2},\cdots,j_{d}}\right]\in{\mathbb{F}}^{n_{1}\times
n_{2}\times\cdots\times n_{d}}$, and assume $\sigma\in{\bf S}_{d}$. The
$\sigma$-transpose of $A$ is

$\displaystyle A^{\sigma}:=\left[a_{j_{\sigma(1)}\cdots
j_{\sigma(d)}}\right]\in{\mathbb{F}}^{n_{\sigma(1)}\times\cdots\times
n_{\sigma(d)}}.$ (7)

  * (ii)

If a $d$-hypermatrix $A\in{\mathbb{F}}^{\overbrace{n\times\cdots\times
n}^{d}}$, then $A$ is said to be $d$-hypercubic.

  * (iii)

A $d$-hypercubic $A\in{\mathbb{F}}^{\overbrace{n\times\cdots\times n}^{d}}$ is
said to be symmetric if $\forall\sigma\in{\bf S}_{d}$, $A^{\sigma}=A$; $A$ is
said to be skew-symmetric if $\forall\sigma\in{\bf S}_{d}$,
$A^{\sigma}=\operatorname{sgn}(\sigma)A$.

It is obvious that a matrix is a $2$-hypermatrix, so the above general
definitions coincide with the corresponding definitions for matrices.

######  Proposition 2.

A $2$-hypercubic $A\in{\mathbb{F}}^{n\times n}$ is (skew-)symmetric, if and
only if, $M_{A}$ is (skew-)symmetric.

######  Proposition 3.

Let $A\in{\mathbb{F}}^{n_{1}\times\cdots\times n_{d}}$ and ${\bf r}\subset{\bf
d}=\langle d\rangle$. Then

$\displaystyle\left[M_{A}^{{\bf r}\times({\bf d}\backslash{\bf
r})}\right]^{T}=M_{A}^{({\bf d}\backslash{\bf r})\times{\bf r}}.$ (8)

######  Remark 2.

The above arguments stand true even when ${\mathbb{F}}$ is a set of perfect
hypercomplex numbers (PHNs) [12]. In fact, most of arguments throughout this
paper also hold for PHNs.

Next, we recall the permutation matrix [10], which is a generalization of swap
matrix.

######  Definition 7.

Let $n=\prod_{i=1}^{d}n_{i}$, $n_{i}\geq 2$, $\sigma\in{\bf S}_{n}$. A logical
matrix $W_{[n_{1},n_{2},\cdots,n_{d}]}^{\sigma}\in{\mathcal{L}}_{n\times n}$,
called a $\sigma$-permutation matrix, is constructed as follows:

  * •

Step 1: Define

$\begin{array}[]{l}D=D_{[n_{1},n_{2},\cdots,n_{d}]}^{\sigma}:=\left\\{\delta^{j_{1}}_{n_{\sigma(1)}}\delta^{j_{2}}_{n_{\sigma(2)}}\cdots\delta^{j_{d}}_{n_{\sigma(d)}}\;\Big{|}\;j_{i}\in\langle
n_{\sigma(i)}\rangle,i=1,2,\cdots,d\right\\}.\end{array}$

  * •

Step 2: Arrange $\\{\sigma(i)\;|\;i\in[1,d]\\}$ into an increasing sequence as

$1=\sigma(i_{1})<\sigma(i_{2})<\cdots<\sigma(i_{d})=d.$

That is,

$i_{j}=\sigma^{-1}(j),\quad j\in[1,d].$

Set an index order as

$\begin{array}[]{l}\mathrm{ID}:=\mathrm{ID}\left(j_{i_{1}},j_{i_{2}},\cdots,j_{i_{d}};n_{\sigma(i_{1})},n_{\sigma(i_{2})},\cdots,n_{\sigma(i_{d})}\right)\\\
~{}=\mathrm{ID}\left(j_{\sigma^{-1}(1)},j_{\sigma^{-1}(2)},\cdots,j_{\sigma^{-1}(d)};n_{1},n_{2},\cdots,n_{d}\right).\end{array}$

  * •

Step 3:

$\displaystyle\begin{array}[]{l}W_{[n_{1},n_{2},\cdots,n_{d}]}^{\sigma}:=\left[\delta^{j_{1}}_{n_{\sigma(1)}}\delta^{j_{2}}_{n_{\sigma(2)}}\cdots\delta^{j_{d}}_{n_{\sigma(d)}}\;\Big{|}~{}\mbox{arranged
by the order of}~{}\mathrm{ID}\right].\end{array}$ (10)

######  Example 2.

Consider $d=3$, $n_{1}=2$, $n_{2}=3$, and $n_{3}=5$. We construct
$W^{\sigma}:=W^{\sigma}_{[2,3,5]}$.

  * (1)

$\sigma_{1}=identity$ (i.e., $[1,2,3]\rightarrow[1,2,3]$): We have
$W^{\sigma_{1}}=I_{30}$.

  * (2)

$\sigma_{2}=(2,3)$ (i.e., $[1,2,3]\rightarrow[1,3,2]$): Then

$D=\\{\delta_{2}^{j_{1}}\delta_{5}^{j_{2}}\delta_{3}^{j_{3}}\;|\;j_{1}\in[1,2],j_{2}\in[1,5],j_{3}\in[1,3]\\}$
$\begin{array}[]{l}W^{\sigma_{2}}=\left[\delta_{2}^{1}\delta_{5}^{1}\delta_{3}^{1},\delta_{2}^{1}\delta_{5}^{2}\delta_{3}^{1},\delta_{2}^{1}\delta_{5}^{3}\delta_{3}^{1},\delta_{2}^{1}\delta_{5}^{4}\delta_{3}^{1},\delta_{2}^{1}\delta_{5}^{5}\delta_{3}^{1},\delta_{2}^{1}\delta_{5}^{1}\delta_{3}^{2},\delta_{2}^{1}\delta_{5}^{2}\delta_{3}^{2},\delta_{2}^{1}\delta_{5}^{3}\delta_{3}^{2},\delta_{2}^{1}\delta_{5}^{4}\delta_{3}^{2},\delta_{2}^{1}\delta_{5}^{5}\delta_{3}^{2},\right.\\\
~{}~{}\delta_{2}^{1}\delta_{5}^{1}\delta_{3}^{3},\delta_{2}^{1}\delta_{5}^{2}\delta_{3}^{3},\delta_{2}^{1}\delta_{5}^{3}\delta_{3}^{3},\delta_{2}^{1}\delta_{5}^{4}\delta_{3}^{3},\delta_{2}^{1}\delta_{5}^{5}\delta_{3}^{3},\delta_{2}^{2}\delta_{5}^{1}\delta_{3}^{1},\delta_{2}^{2}\delta_{5}^{2}\delta_{3}^{1},\delta_{2}^{2}\delta_{5}^{3}\delta_{3}^{1},\delta_{2}^{2}\delta_{5}^{4}\delta_{3}^{1},\delta_{2}^{2}\delta_{5}^{5}\delta_{3}^{1},\\\
~{}~{}\left.\delta_{2}^{2}\delta_{5}^{1}\delta_{3}^{2},\delta_{2}^{2}\delta_{5}^{2}\delta_{3}^{2},\delta_{2}^{2}\delta_{5}^{3}\delta_{3}^{2},\delta_{2}^{2}\delta_{5}^{4}\delta_{3}^{2},\delta_{2}^{2}\delta_{5}^{5}\delta_{3}^{2},\delta_{2}^{2}\delta_{5}^{1}\delta_{3}^{3},\delta_{2}^{2}\delta_{5}^{2}\delta_{3}^{3},\delta_{2}^{2}\delta_{5}^{3}\delta_{3}^{3},\delta_{2}^{2}\delta_{5}^{4}\delta_{3}^{3},\delta_{2}^{2}\delta_{5}^{5}\delta_{3}^{3}\right]\\\
~{}~{}=\delta_{30}[1,4,7,10,13,2,5,8,11,14,3,6,9,12,15,\\\
16,19,22,25,28,17,20,23,26,29,18,21,24,27,30].\end{array}$

  * (3)

$\sigma_{3}=(1,2)$ (i.e., $[1,2,3]\rightarrow[2,1,3]$):

Similarly, we have

$D=\\{\delta_{3}^{j_{1}}\delta_{2}^{j_{2}}\delta_{5}^{j_{3}}\;|\;j_{1}\in[1,3],j_{2}\in[1,2],j_{3}\in[1,5]\\}$
$\begin{array}[]{l}W^{\sigma_{3}}=\left[\delta_{3}^{1}\delta_{2}^{1}\delta_{5}^{1},\delta_{3}^{1}\delta_{2}^{1}\delta_{5}^{2},\cdots,\delta_{3}^{2}\delta_{2}^{1}\delta_{5}^{1},\cdots,\delta_{3}^{2}\delta_{2}^{1}\delta_{5}^{5},\cdots,\delta_{3}^{3}\delta_{2}^{2}\delta_{5}^{5}\right]\\\
~{}~{}=\delta_{30}[1,2,3,4,5,11,12,13,14,15,21,22,23,24,\\\
25,6,7,8,9,10,16,17,18,19,20,26,27,28,29,30].\end{array}$

  * (4)

$\sigma_{4}=(1,2,3)$ (i.e., $[1,2,3]\rightarrow[2,3,1]$):

We have

$\begin{array}[]{l}W^{\sigma_{4}}=\left[\delta_{3}^{1}\delta_{5}^{1}\delta_{2}^{1},\cdots,\delta_{3}^{3}\delta_{5}^{5}\delta_{2}^{2}\right]\\\
~{}~{}=\delta_{30}[1,3,5,7,9,11,13,15,17,19,21,23,25,27,\\\
29,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30].\end{array}$

  * (5)

$\sigma_{5}=(1,3,2)$ (i.e., $[1,2,3]\rightarrow[3,1,2]$):

Then

$\begin{array}[]{l}W^{\sigma_{5}}=\left[\delta_{5}^{1}\delta_{2}^{1}\delta_{3}^{1},\cdots,\delta_{5}^{5}\delta_{2}^{2}\delta_{3}^{3}\right]\\\
~{}=\delta_{30}[1,7,13,19,25,2,8,14,20,26,3,9,15,21,27\\\
4,10,16,22,28,5,11,17,23,29,6,12,18,24,30].\end{array}$

  * (6)

$\sigma_{6}=(1,3)$ (i.e., $[1,2,3]\rightarrow[3,2,1]$):

Then

$\begin{array}[]{l}W^{\sigma_{6}}=\left[\delta_{5}^{1}\delta_{3}^{1}\delta_{2}^{1},\cdots,\delta_{5}^{5}\delta_{3}^{3}\delta_{2}^{2}\right]\\\
~{}~{}=\delta_{30}[1,7,13,19,25,3,9,15,21,27,5,11,17,23,\\\
29,2,8,14,20,26,4,10,16,22,28,6,12,18,24,30].\end{array}$

When $n_{1}=n_{2}=\cdots=n_{d}:=n$ the corresponding permutation is briefly
denoted by

$W^{\sigma}_{n}:=W^{\sigma}_{[n_{1},n_{2},\cdots,n_{d}]}.$

Since this kind of permutation matrices are of particular importance, some of
them are listed in the Appendix.

Some basic properties of permutation matrices are presented in the following
proposition, which follows from definition immediately.

######  Proposition 4.

  * (i)

$\displaystyle\left[W^{\sigma}_{[n_{1},\cdots,n_{d}]}\right]^{T}=\left[W^{\sigma}_{[n_{1},\cdots,n_{d}]}\right]^{-1}=W^{\sigma^{-1}}_{[n_{1},\cdots,n_{d}]}.$
(11)

  * (ii)

Let $\sigma,\mu\in{\bf S}_{d}$. Then

$\displaystyle
W^{\sigma}_{[n_{1},n_{2},\cdots,n_{d}]}W^{\mu}_{[n_{1},n_{2},\cdots,n_{d}]}=W^{\sigma\circ\mu}_{[n_{1},n_{2},\cdots,n_{d}]}.$
(12)

The following proposition shows the basic function of permutation matrices.

######  Proposition 5.

[10] Assume $x_{i}\in{\mathbb{F}}^{n_{i}}$, $i\in\langle d\rangle$,
$\sigma\in{\bf S}_{d}$. Then

$\displaystyle\ltimes_{i=1}^{d}x_{\sigma(i)}=W^{\sigma}_{[n_{1},n_{2},\cdots,n_{d}]}\ltimes_{i=1}^{d}x_{i}.$
(13)

As an immediate consequence we have the following result, which shows how to
calculate $A^{\sigma}$.

######  Proposition 6.

Let $A\in{\mathbb{F}}^{n_{1}\times\cdots\times n_{d}}$ be a hypermatrix of
order $d$. Then

$\displaystyle
V_{A^{\sigma}}=V_{A}\left[W_{[n_{1},\cdots,n_{d}]}^{\sigma}\right]^{T}=V_{A}W_{[n_{1},\cdots,n_{d}]}^{\sigma^{-1}}.$
(14)

Proof. Proposition 5 implies that

$V^{T}_{A^{\sigma}}=W_{[n_{1},\cdots,n_{d}]}^{\sigma}V_{A}.$

Taking transpose on both sides yields (14). $\Box$

###  2.3 Conversion of Matrix Expressions

######  Definition 8.

  * (i)

Let $A=[a_{i,j}]\in{\mathbb{F}}^{m\times n}$ be a matrix. Then

$\displaystyle{\bf
V}_{r}(A):=[a_{1,1},a_{1,2},\cdots,a_{1,n},a_{2,1},\cdots,a_{m,n}]$ (15)

is called the row stacking form of $A$;

$\displaystyle{\bf
V}_{c}(A):=[a_{1,1},a_{2,1},\cdots,a_{m,1},a_{1,2},\cdots,a_{m,n}]$ (16)

is called the column stacking form of $A$.

  * (ii)

Let $x\in{\mathbb{F}}^{n}$ and $s|n$. Say, $n=st$. Then

$\displaystyle{\bf V}^{s}_{r}(x):=\begin{bmatrix}x_{1}&x_{2}&\cdots&x_{s}\\\
x_{s+1}&x_{s+2}&\cdots&x_{2s}\\\ \vdots&~{}&\ddots&\vdots\\\
x_{(t-1)s+1}&x_{(t-1)s+2}&\cdots&x_{ts}\\\ \end{bmatrix}$ (17)
$\displaystyle{\bf
V}^{s}_{c}(x):=\begin{bmatrix}x_{1}&x_{s+1}&\cdots&x_{(t-1)s+1}\\\
x_{2}&x_{s+2}&\cdots&x_{(t-1)s+2}\\\ \vdots&~{}&\ddots&\vdots\\\
x_{s}&x_{2s}&\cdots&x_{ts}\\\ \end{bmatrix}$ (18)

  * (iii)

Let $A\in{\mathbb{F}}^{m\times n}$ and $s|(mn)$. Then

$\displaystyle{\bf V}^{s}_{r}(A):={\bf V}^{s}_{r}\left({\bf V}_{r}(A)\right)$
(19)

is called the $s$-row stacking form.

$\displaystyle{\bf V}^{s}_{c}(A):={\bf V}^{s}_{c}\left({\bf V}_{c}(A)\right)$
(20)

is called the $s$-column stacking form.

######  Proposition 7.

[3] Let $A\in{\mathbb{F}}^{m\times n}$, $X\in{\mathbb{F}}^{n\times q}$, and
$Y\in{\mathbb{F}}^{p\times m}$. Then

$\displaystyle{\bf V}_{r}(AX)=A\ltimes{\bf V}_{r}(X),$ (21) $\displaystyle{\bf
V}_{c}(YA)=A^{T}\ltimes{\bf V}_{c}(Y).$ (22)

Denote by

$\delta_{n}^{I}:={\bf V}_{r}(I_{n})={\bf
V}_{c}(I_{n})=[(\delta_{n}^{1})^{T},(\delta_{n}^{2})^{T},\cdots,(\delta_{n}^{n})^{T}]^{T}.$

######  Proposition 8.

Let $A\in{\mathbb{F}}^{m\times n}$. Then

$\displaystyle{\bf V}_{r}(A)=A\ltimes\delta_{n}^{I}.$ (23)

$\displaystyle{\bf V}_{c}(A)=A^{T}\ltimes\delta_{m}^{I}.$ (24)

Conversely,

$\displaystyle A={\bf V}_{r}^{n}({\bf V}_{r}(A))={\bf V}_{c}^{m}({\bf
V}_{c}(A)).$ (25)

Proof. (23) and (24) come from Proposition 7 immediately. (25) follows from
the definition. $\Box$

By definition and (25), we have

$\displaystyle A={\bf V}_{r}^{n}(A)={\bf V}_{c}^{m}(A).$ (26)

Set ${\bf i_{r}}=(i_{1},\cdots,i_{r})\subset{\bf d}=\langle d\rangle$, and
denote

$\displaystyle\sigma_{{\bf i_{r}}}:{\bf d}\rightarrow({\bf i_{r}},{\bf
d}\backslash{\bf i_{r}}).$ (27)

The following proposition shows how to convert a matrix expression to a vector
form and vise versa for a hypermatrix.

######  Proposition 9.

Given $A=[a_{i_{1},\cdots,i_{d}}]\in{\mathbb{F}}^{n_{1}\times\cdots\times
n_{d}}$, ${\bf i_{r}}=(i_{1},\cdots,i_{r})\subset{\bf d}=\langle d\rangle$,
and $\sigma_{{\bf i_{r}}}$ is as in (27),

$n_{{\bf i_{r}}}=\prod_{s=1}^{r}n_{i_{s}},\quad n_{{\bf d}\backslash{\bf
i_{r}}}=\prod_{i_{j}\in{\bf d}\backslash{\bf i_{r}}}n_{i_{j}}.$

Then

  * (i)

(Vector Form to Matrix Form)

$\displaystyle M_{A}^{{\bf i_{r}}\times({\bf d}\backslash{\bf i_{r}})}={\bf
V}_{r}^{n_{{\bf d}\backslash{\bf i_{r}}}}\left(V_{A}W^{\sigma^{-1}_{{\bf
i_{r}}}}_{[n_{1},\cdots,n_{d}]}\right).$ (28)

  * (ii)

(Matrix Form to Vector Form)

$\displaystyle V_{A}=\left(M_{A}^{{\bf i_{r}}\times({\bf d}\backslash{\bf
i_{r}})}\ltimes\delta^{I}_{n_{{\bf d}\backslash{\bf
i_{r}}}}\right)^{T}W_{[n_{1},\cdots,n_{d}]}^{\sigma_{{\bf i_{r}}}}.$ (29)

Proof. For (i), we have

$\begin{array}[]{l}M_{A}^{{\bf i_{r}}\times({\bf d}\backslash{\bf
i_{r}})}={\bf V}_{r}^{n_{{\bf d}\backslash{\bf i_{r}}}}(V_{A^{\sigma_{\bf
i_{r}}}})={\bf V}_{r}^{n_{{\bf d}\backslash{\bf
i_{r}}}}\left(V_{A}W^{\sigma^{-1}_{{\bf
i_{r}}}}_{[n_{1},\cdots,n_{d}]}\right).\end{array}$

As for (ii), using (14) and (23), we have

$\begin{array}[]{l}V_{A}=V_{A^{\sigma_{{\bf
i_{r}}}}}W_{[n_{1},\cdots,n_{d}]}^{\sigma_{{\bf i_{r}}}}=\left(M_{A}^{{\bf
i_{r}}\times({\bf d}\backslash{\bf i_{r}})}\ltimes\delta^{I}_{n_{{\bf
d}\backslash{\bf i_{r}}}}\right)^{T}W_{[n_{1},\cdots,n_{d}]}^{\sigma_{{\bf
i_{r}}}}.\end{array}$

$\Box$

Using (28) and (29), the formula transforming one matrix form to another can
be obtained.

######  Corollary 10.

Let ${\bf i_{r}}$, $\sigma_{{\bf i_{r}}}$ be as in Proposition 9, and ${\bf
j_{s}}=(j_{1},\cdots,j_{s})$ and $\sigma_{{\bf j_{s}}}:{\bf d}\rightarrow({\bf
j_{s}},{\bf d}\backslash{\bf j_{s}})$. Then

$\displaystyle\begin{array}[]{ccl}M_{A}^{{\bf j_{s}}\times({\bf
d}\backslash{\bf j_{s}})}&=&{\bf V}_{r}^{n_{{\bf d}\backslash{\bf
j_{s}}}}\left[\left(M_{A}^{{\bf i_{r}}\times({\bf d}\backslash{\bf
i_{r}})}\ltimes\delta_{{\bf d}\backslash{\bf i_{r}}}^{I}\right)^{T}\times
W^{\sigma_{{\bf i_{r}}}}_{[n_{1},\cdots,n_{d}]}W^{\sigma^{-1}_{{\bf
j_{s}}}}_{[n_{1},\cdots,n_{d}]}\right].\end{array}$ (31)

##  3 Contracted Product of Hypermatrices

######  Definition 9.

[21] Let $A=[a_{i_{1},\cdots,i_{d}}]\in{\mathbb{F}}^{n_{1}\times\cdots\times
n_{d-1}\times n}$, $B=[b_{j_{1},\cdots,j_{r}}]\in{\mathbb{F}}^{n\times
m_{2}\times\cdots\times m_{r}}$. Then the contracted product of $A$ and $B$,
denoted by

$\displaystyle C=A\times^{i_{d}}_{j_{1}}B,$ (32)

is s hypermatrix
$C=[c_{i_{1},\cdots,i_{d-1},j_{2},\cdots,j_{r}}]\in{\mathbb{F}}^{n_{1}\times\cdots\times
n_{d-1}\times m_{2}\times\cdots\times m_{r}}$, where

$\displaystyle
c_{i_{1},\cdots,i_{d-1},j_{2},\cdots,j_{r}}=\mathop{\sum}\limits_{k=1}^{n}a_{i_{1},\cdots,i_{d-1},k}b_{k,j_{2},\cdots,j_{r}}.$
(33)

######  Proposition 11.

Let $M_{A}^{(i_{1},\cdots,i_{d-1})\times i_{d}}$ and
$M_{B}^{j_{1}\times(j_{2},\cdots,j_{r})}$ be the corresponding matrix
expressions of $A$ and $B$ respectively. Then

$\displaystyle
M_{C}^{(i_{1},\cdots,i_{d-1})\times(j_{2},\cdots,j_{r})}=M_{A}^{(i_{1},\cdots,i_{d-1})\times
i_{d}}\times M_{B}^{j_{1}\times(j_{2},\cdots,j_{r})},$ (34)

where the $\times$ between $M_{A}$ and $M_{B}$ is the conventional matrix
product.

Definition 9 can be extended to the case of multiple common indices.

######  Definition 10.

Let $A=[a_{i_{1},\cdots,i_{d}}]\in{\mathbb{F}}^{n_{1}\times\cdots\times
n_{d}}$, $B=[b_{j_{1},\cdots,j_{r}}]\in{\mathbb{F}}^{m_{1}\times\cdots\times
m_{r}}$ with

$n_{\alpha_{t}}=m_{\beta_{t}}:=\ell_{t},\quad\alpha_{t}\in\langle
d\rangle,\beta_{t}\in\langle r\rangle,t\in\langle s\rangle,$

where $s\leq\min(d,r)$ is the number of equal dimension indexes. Define

$\displaystyle\begin{array}[]{l}A\times^{i_{\alpha_{1}},\cdots,i_{\alpha_{s}}}_{j_{\beta_{1}},\cdots,j_{\beta_{s}}}B:=C\\\
~{}~{}\in{\mathbb{F}}^{n_{1}\times\cdots\times\hat{n}_{\alpha_{1}}\times\cdots\times\hat{n}_{\alpha_{s}}\times\cdots\times
n_{d}\times
m_{1}\times\cdots\times\hat{m}_{\beta_{1}}\times\cdots\times\hat{m}_{\beta_{s}}\times\cdots\times
m_{r}},\end{array}$ (37)

where

$\begin{array}[]{l}c_{i_{1},\cdots,\hat{i}_{\alpha_{1}},\cdots,\hat{i}_{\alpha_{s}},\cdots,i_{d},j_{1},\cdots,\hat{j}_{\beta_{1}},\cdots,\hat{j}_{\beta_{s}},\cdots,j_{r}}\\\
=\mathop{\sum}\limits_{k_{1}=1}^{\ell_{1}}\cdots\mathop{\sum}\limits_{k_{s}=1}^{\ell_{s}}a_{i_{1}\cdots
k_{1}\cdots k_{s}\cdots i_{d}}b_{j_{1}\cdots k_{1}\cdots k_{s}\cdots
j_{d}},\end{array}$

where a caret over any entry means that the respective entry is omitted.

######  Proposition 12.

$\displaystyle\begin{array}[]{l}M_{C}^{(i_{1},\cdots,\hat{i}_{\alpha_{1}},\cdots,\hat{i}_{\alpha_{s}},\cdots,i_{d})\times(j_{1},\cdots,\hat{j}_{\beta_{1}},\cdots,\hat{j}_{\beta_{s}},\cdots,j_{r})}\\\
=M_{A}^{(i_{1},\cdots,\hat{i}_{\alpha_{1}},\cdots,\hat{i}_{\alpha_{s}},\cdots,i_{d})\times(i_{\alpha_{1}},\cdots,i_{\alpha_{s}})}\\\
~{}~{}\times
M_{B}^{(j_{\beta_{1}},\cdots,j_{\beta_{s}})\times(j_{1},\cdots,\hat{j}_{\beta_{1}},\cdots,\hat{j}_{\beta_{s}},\cdots,j_{r})},\end{array}$
(41)

where the $\times$ between $M_{A}$ and $M_{B}$ is the conventional matrix
product.

######  Example 3.

Assume $A=[a_{i_{1},i_{2},i_{3}}]\in{\mathbb{F}}^{2\times 3\times 4}$,
$B=[b_{j_{1},j_{2},j_{3}}]\in{\mathbb{F}}^{4\times 5\times 3}$ with natural
ID, calculate

$C=A\times^{(i_{2},i_{3})}_{(j_{3},j_{1})}B.$

We have

$\begin{array}[]{l}M_{A}^{i_{1}\times(i_{2},i_{3})}=\left[\begin{array}[]{llllllllllll}a_{111}&a_{112}&a_{113}&a_{114}&a_{121}&a_{122}&a_{211}&a_{212}&a_{213}&a_{214}&a_{221}&a_{222}\\\
a_{123}&a_{124}&a_{131}&a_{132}&a_{133}&a_{134}&a_{223}&a_{224}&a_{231}&a_{232}&a_{233}&a_{234}\\\
\end{array}\right]\\\ \end{array}$
$\begin{array}[]{l}M_{B}^{(j_{3},j_{1})\times
j_{2}}=\begin{bmatrix}b_{111}&b_{121}&b_{131}&b_{141}&b_{151}\\\
b_{211}&b_{221}&b_{231}&b_{241}&b_{251}\\\
b_{311}&b_{321}&b_{331}&b_{341}&b_{351}\\\
b_{411}&b_{421}&b_{431}&b_{441}&b_{451}\\\
b_{112}&b_{122}&b_{132}&b_{142}&b_{152}\\\
b_{212}&b_{222}&b_{232}&b_{242}&b_{252}\\\
b_{312}&b_{322}&b_{332}&b_{342}&b_{352}\\\
b_{412}&b_{422}&b_{432}&b_{442}&b_{452}\\\
b_{113}&b_{123}&b_{133}&b_{143}&b_{153}\\\
b_{213}&b_{223}&b_{233}&b_{243}&b_{253}\\\
b_{313}&b_{323}&b_{333}&b_{343}&b_{353}\\\
b_{413}&b_{423}&b_{433}&b_{443}&b_{453}\\\ \end{bmatrix}\end{array}$

Then $C\in{\mathbb{F}}^{2\times 5}$ with

$M_{C}^{i_{1}\times
j_{2}}=M_{A}^{i_{1}\times(i_{2},i_{3})}M_{B}^{(j_{3},j_{1})\times j_{2}}.$

A particular case is
$A=[a_{i_{1},\cdots,i_{d}}]\in{\mathbb{F}}^{n_{1}\times\cdots\times n_{d}}$,
$B=[b_{i_{r_{1}},\cdots,i_{r_{s}}}]\in{\mathbb{F}}^{n_{r_{1}}\times\cdots\times
n_{r_{s}}}$, and ${\bf r_{s}}:=\\{r_{1},\cdots,r_{s}\\}\subset{\bf d}:=\langle
d\rangle$. Then we briefly denote

$A\times_{{\bf r_{s}}}B:=A\times^{{\bf r_{s}}}_{\bf r_{s}}B.$

We call this kind of product the onto contracted product. They are of
particular importance.

In the onto contracted product, $A$ can be considered as a multilinear mapping
from ${\mathbb{F}}^{n_{r_{1}}\times\cdots\times n_{r_{s}}}$ to
${\mathbb{F}}^{n_{1}\times\cdots\times\hat{n}_{r_{1}}\times\cdots\times\hat{n}_{r_{s}}\times\cdots\times
n_{d}}$.

######  Proposition 13.

Assume $A\in{\mathbb{F}}^{n_{1}\times\cdots\times n_{d}}$ and
$B\in{\mathbb{F}}^{n_{r_{1}}\times\cdots\times n_{r_{s}}}$, where ${\bf
r_{s}}:=\\{r_{1},\cdots,r_{s}\\}\subset{\bf d}$. The contracted product of $A$
and $B$, denoted by

$C=A\times_{{\bf r_{s}}}B,$

can be obtained by one of the following two equivalent formulas:

  * (i)

Let

$M^{A}:=M_{A}^{({\bf d}\backslash{\bf r_{s}})\times{\bf r_{s}}}.$

Then

$\displaystyle V_{C}^{T}=M^{A}V_{B}^{T}.$ (42)

  * (ii)

Let $\sigma\in{\bf S}_{d}$ be the permutation

$\sigma:{\bf d}\rightarrow({\bf d}\backslash{\bf r_{s}},\bf{r_{s}}).$

Then

$\displaystyle V_{C}=V_{A}W^{\sigma^{-1}}V_{B}^{T}.$ (43)

######  Definition 11.

  * (i)

Let $A\in{\mathbb{F}}^{n_{1}\times\cdots\times n_{d}\times
n_{d+1}\times\cdots\times n_{2d}}$, $B\in{\mathbb{F}}^{n_{1}\times\cdots\times
n_{d}}$, where $n_{d+i}=n_{i}$, $i\in\langle d\rangle$. Then
$A:{\mathbb{F}}^{n_{1}\times\cdots\times
n_{d}}\rightarrow{\mathbb{F}}^{n_{1}\times\cdots\times n_{d}}$ is defined by

$\displaystyle
A(B):=A\times^{d+1,\cdots,2d}_{1,2,\cdots,d}B\in{\mathbb{F}}^{n_{1}\times\cdots\times
n_{d}}.$ (44)

This contracted product is called a unary operator on
${\mathbb{F}}^{n_{1}\times\cdots\times n_{d}}$.

  * (ii)

Let $A\in{\mathbb{F}}^{n_{1}\times\cdots\times n_{3d}}$,
$B,C\in{\mathbb{F}}^{n_{1}\times\cdots\times n_{d}}$, where
$n_{2d+i}=n_{d+i}=n_{i}$, $i\in\langle d\rangle$. Then
$A:{\mathbb{F}}^{n_{1}\times\cdots\times
n_{d}}\times{\mathbb{F}}^{n_{1}\times\cdots\times
n_{d}}\rightarrow{\mathbb{F}}^{n_{1}\times\cdots\times n_{d}}$ is defined by

$\displaystyle\begin{array}[]{ccl}A(B,C):=[A\times^{2d+1,\cdots,3d}_{1,2,\cdots,d}B]\times^{d+1,\cdots,2d}_{1,\cdots,d}C\in{\mathbb{F}}^{n_{1}\times\cdots\times
n_{d}}.\end{array}$ (46)

This contracted product is called a binary operator on
${\mathbb{F}}^{n_{1}\times\cdots\times n_{d}}$.

  * (iii)

Similarly, we can define $k$ argument (i.e., $k$ hypermatrix) operators for
$k\geq 3$.

##  4 STP Realization of Contracted Product of Hypermatrices

HypermatrixContracted ProductResulting HypermatrixMatrixSTPResulting
MatrixMatrixExpressionConverse Figure 1: STP-Realization of Contracted Product

This section shows that a fundamental role of STP is to realize the contracted
product of hypermatrices. This factor may help us to understand what is the
essential meaning of STP. For this purpose, the hypermatrices are first
expressed in their matrix expressions. The STPs are operators on matrices.
When the hypermatrices are transformed into their matrix forms, the action of
operators (especially certain contracted products) on their objective
hypermatrices can be realized by the action of STPs on the matrix expressions
of the corresponding hypermatrices. Figure 1 shows this process.

Since there are many possible multilinear operators, including contracted
products, over different hypermatrices, the corresponding operators over their
matrix expressions are also diverse. To express these operators, the STPs are
also diverse.

###  4.1 Classical STPs

From linear algebra one sees that the matrix product has two fundamental
types: (1) Matrix-Matrix Product: It represents the composition of two linear
mappings. (2) Matrix-Vector Product: It realizes a linear mapping over a
vector space (or between two vector spaces). Fortunately, when the dimension-
matching condition is satisfied, the classical matrix product can realize
these two functions simultaneously.

When the conventional matrix product is extended to STP, where the dimension-
matching condition is not satisfied, an STP cannot realize these two functions
simultaneously. Therefore, we need to distinguish between M-M STP and M-V
products. First, we review the classical STPs [8]:

######  Definition 12.

  * (i)

Let $A\in{\mathbb{F}}^{m\times n}$, $B\in{\mathbb{F}}^{p\times q}$ and
$t=\operatorname{lcm}(n,p)$. The matrix-matrix (M-M) STP of $A$ and $B$ is
defined as

$\displaystyle A\ltimes B:=\left(A\otimes I_{t/n}\right)\left(B\otimes
I_{t/p}\right).$ (47)

  * (ii)

Let $A\in{\mathbb{F}}^{m\times n}$, $x\in{\mathbb{F}}^{p}$ and
$t=\operatorname{lcm}(n,p)$. The matrix-vector (M-V) STP of $A$ and $x$ is
defined as

$\displaystyle A\vec{\ltimes}x:=\left(A\otimes
I_{t/n}\right)\left(x\otimes{\bf 1}_{t/p}\right).$ (48)

  * (iii)

Let $x\in{\mathbb{F}}^{m}$, $y\in{\mathbb{F}}^{n}$ and
$t=\operatorname{lcm}(m,n)$. The vector-vector (V-V) STP of $x$ and $y$ is
defined as

$\displaystyle x\vec{\cdot}y:=(x\otimes{\bf 1}_{t/m})^{T}(y\otimes{\bf
1}_{t/n})\in{\mathbb{F}},$ (49)

Define

${\mathcal{M}}=\bigcup_{m=1}^{\infty}\bigcup_{n=1}^{\infty}{\mathcal{M}}_{m\times
n},\quad{\mathbb{R}}^{\infty}=\bigcup_{n=1}^{\infty}{\mathbb{R}}^{n}.$

Then the M-M STP can be considered as the product over ${\mathcal{M}}$; the
M-V STP can be considered as the action of ${\mathcal{M}}$ on
${\mathbb{R}}^{\infty}$; and the V-V STP can be considered as an inner product
over ${\mathbb{R}}^{\infty}$. It is obvious that they are the generalizations
of the corresponding matrices with matrix-matrix or matrix-vector product, or
vectors with vector product. Furthermore, they satisfy the following general
properties.

######  Proposition 14.

  * (i)

(Associativity)

$(A\ltimes B)\ltimes C=A\ltimes(B\ltimes C),\quad A,B,C\in{\mathcal{M}}.$
$(A\ltimes B)\vec{\ltimes}x=A\vec{\ltimes}(B\vec{\ltimes}x),\quad
x\in{\mathbb{R}}^{\infty}.$

  * (ii)

(Distributivity) For $A,B,C\in{\mathcal{M}}$, $x,y,z\in{\mathbb{R}}^{\infty},$

$\begin{array}[]{l}(A+B)\ltimes C=A\ltimes C+B\ltimes C,\\\
C\ltimes(A+B)=C\ltimes A+C\ltimes B;\\\
(A+B)\vec{\ltimes}x=A\vec{\ltimes}x+B\vec{\ltimes}x,\\\
A\vec{\ltimes}(x+y)=A\vec{\ltimes}x+A\vec{\ltimes}y;\\\
(x+y)~{}\vec{\cdot}~{}z=x~{}\vec{\cdot}~{}z+y~{}\vec{\cdot}~{}z,\\\
z~{}\vec{\cdot}~{}(x+y)=z~{}\vec{\cdot}~{}x+z~{}\vec{\cdot}~{}y.\\\
\end{array}$

######  Remark 3.

M-V and V-V STPs are not so commonly used as M-M STP. We explain it a little
bit more.

  * •

Topology on ${\mathbb{R}}^{\infty}$:

Consider $x\in{\mathbb{R}}^{p}$, $y\in{\mathbb{R}}^{q}$,
($x,y\in{\mathbb{R}}^{\infty}$), $t=\operatorname{lcm}(p,q)$. Define

$x~{}\vec{\pm}~{}y:=\left(x\otimes{\bf 1}_{t/p}\right)\pm\left(y\otimes{\bf
1}_{t/q}\right),$

then ${\mathbb{R}}^{\infty}$ becomes a (pseudo-) vector space.

Furthermore, we define

  * (i)

(Inner Product):

$\langle x,y\rangle_{{\mathcal{V}}}:=\frac{1}{t}(x~{}\vec{\cdot}~{}y).$

  * (ii)

(Norm):

$\|x\|_{{\mathcal{V}}}:=\sqrt{\langle x,x\rangle_{{\mathcal{V}}}}.$

  * (iii)

(Distance):

$d(x,y)=\|x~{}\vec{-}~{}y\|_{{\mathcal{V}}}.$

With this distance ${\mathbb{R}}^{\infty}$ becomes a topological space [8].

  * •

Linear Dynamic Systems over ${\mathbb{R}}^{\infty}$:

A linear dynamic system over ${\mathbb{R}}^{\infty}$ is defined by

$x(t+1)=A\vec{\ltimes}x(t).$

It is a cross-dimensional system [13].

###  4.2 STP for Vector Expression of Hypermatrices

Consider a multilinear mapping
$\pi:\prod_{i=1}^{d}V_{i}\rightarrow{\mathbb{F}}$, where
$V_{i}={\mathbb{F}}^{n_{i}}$, $i\in\langle d\rangle$. Assume

$\begin{array}[]{l}\pi\left(\delta_{n_{1}}^{i_{1}},\cdots,\delta_{n_{d}}^{i_{d}}\right)=c_{i_{1},\cdots,i_{d}},\quad
i_{s}\in\langle n_{s}\rangle,\;s\in\langle d\rangle.\end{array}$

Let $\Pi:=[c_{i_{1},\cdots,i_{d}}]$ with
$\mathrm{ID}=\mathrm{ID}(i_{1},\cdots,i_{d};n_{1},\cdots,n_{d})$ be a
hypermatrix of order $d$. Then the following result is well known.

######  Proposition 15.

The multilinear mapping $\pi$ can be calculated by

$\displaystyle\pi(x_{1},\cdots,x_{d})=V_{\Pi}\ltimes_{i=1}^{d}x_{i},\quad
x_{i}\in V_{i},~{}i\in\langle d\rangle.$ (50)

######  Example 4.

Consider a finite game of $n$ players [6]. Assume player $i$ has $k_{i}$
strategies, denoted by $\delta_{k_{i}}^{j_{i}}$, $j_{i}\in\langle
k_{i}\rangle$, $i\in\langle n\rangle$. Let

$c_{i}(\delta_{k_{1}}^{j_{1}},\cdots,\delta_{k_{n}}^{j_{n}})=r^{i}_{j_{i},\cdots,j_{n}},\quad
i\in\langle n\rangle.$

Then

$D_{i}=\\{r^{i}_{j_{i},\cdots,j_{n}}\\}\in{\mathbb{R}}^{k_{1}\times\cdots
k_{n}}.$

Finally, we have

$c_{i}=V_{D_{i}}\ltimes_{t=1}^{n}x_{t},\quad i\in\langle n\rangle.$

###  4.3 STP for Matrix-1 Expression of Hypermatrices

Consider a multilinear mapping $\pi:\prod_{i=1}^{d}V_{i}\rightarrow V_{0}$,
where $V_{i}={\mathbb{F}}^{n_{i}}$, $i\in[0,d]$. Assume

$\pi\left(\delta_{n_{1}}^{i_{1}},\cdots,\delta_{n_{d}}^{i_{d}}\right)=\mathop{\sum}\limits_{i_{0}=1}^{n_{0}}c^{i_{0}}_{i_{1},\cdots,i_{d}}\delta_{n_{0}}^{i_{0}},\quad
i_{s}\in\langle n_{s}\rangle,\;s\in\langle d\rangle.$

Then $\Pi:=[c^{i_{0}}_{i_{1},\cdots,i_{d}}]$ is a hypermatrix of order $d+1$
with matrix expression $M_{\Pi}^{\mathrm{ID}}$, where
$\mathrm{ID}=\mathrm{ID}(i_{0};n_{0})\times\mathrm{ID}(i_{1},\cdots,i_{d};n_{1},\cdots,n_{d})$.
Then the following result is well known.

######  Proposition 16.

The multilinear mapping $\pi$ can be calculated by

$\displaystyle\pi(x_{1},\cdots,x_{d})=M_{\Pi}^{\mathrm{ID}}\ltimes_{i=1}^{d}x_{i},\quad
x_{i}\in V_{i},~{}i\in\langle d\rangle.$ (51)

######  Example 5.

  * (i)

Cross Product on ${\mathbb{R}}^{3}$:

Consider the cross product on $R^{3}$, denoted by $\vec{\times}$. Denote by

$\delta_{3}^{i}\vec{\times}\delta_{3}^{j}=\mathop{\sum}\limits_{k=1}^{3}c_{i,j}^{k},\quad
i,j\in\langle 3\rangle.$

Then we have

$M_{C}^{3\times(1,2)}=\begin{bmatrix}0&0&0&0&0&1&0&-1&0\\\
0&0&-1&0&0&0&1&0&0\\\ 0&1&0&-1&0&0&0&0&0\\\ \end{bmatrix}$

It follows that

$Z=X\vec{\times}Y=M_{C}^{3\times(1,2)}\ltimes X\ltimes Y,\quad
X,Y\in{\mathbb{R}}^{3}.$

  * (ii)

General Linear Algebra $\operatorname{gl}(2,{\mathbb{R}})$. Denote by

$\begin{array}[]{l}A_{1}=\begin{bmatrix}1&0\\\ 0&0\\\
\end{bmatrix}\sim\vec{A}_{1}=\delta_{4}^{1}\\\ A_{2}=\begin{bmatrix}0&1\\\
0&0\\\ \end{bmatrix}\sim\vec{A}_{2}=\delta_{4}^{2}\\\
A_{3}=\begin{bmatrix}0&0\\\ 1&0\\\
\end{bmatrix}\sim\vec{A}_{3}=\delta_{4}^{3}\\\ A_{4}=\begin{bmatrix}0&0\\\
0&1\\\ \end{bmatrix}\sim\vec{A}_{4}=\delta_{4}^{4}\\\ \end{array}$

Then $\delta_{4}^{i}$, $i\in\langle 4\rangle$ form a basis of
$\operatorname{gl}(2,{\mathbb{R}})$. Denote by

$[A_{i},A_{j}]=\mathop{\sum}\limits_{k=1}^{4}c_{i,j}^{k}A_{k},\quad
i,j\in\langle 4\rangle.$

Then $C=[c_{i,j}^{k}]\in{\mathbb{R}}^{4\times 4\times 4}$. Express $C$ into
matrix form

$\begin{array}[]{ccl}M_{C}^{k,(i.j)}&=&\delta_{4}[0,2,-3,0,-2,0,1-4,0,\\\
{}\hfil&{}\hfil&3,-1+4,0,-3,0,0,3,0],\end{array}$

where

$\begin{array}[]{l}\delta_{4}^{0}=[0,0,0,0]^{T},\\\
\delta_{4}^{-3}=[0,0,-1,0]^{T},\\\ \delta_{4}^{1-4}=[1,0,0,-4]^{T},\quad
etc.\\\ \end{array}$

Let $X\in\operatorname{gl}(2,R)$ with $\vec{X}={\bf V}_{c}(X)$. Then

$\overrightarrow{[X,Y]}=M_{C}^{k,(i.j)}\ltimes\vec{X}\ltimes\vec{Y}.$

######  Remark 4.

  * (i)

Roughly speaking, in classical sense, an STP of matrices is a multilinear
operator over hypermatrices. To be precise, when the hypermatrices are
expressed into their matrix forms, the STP works as a matrix product.

  * (ii)

Since multilinear mappings over hypermatrices can be various, there can also
be various STPs.

  * (iii)

When partial arguments are known, an operator becomes a restricted operator
over the remaining arguments. In this case, STP combined with permutation
matrices becomes more powerful.

###  4.4 STP for General Matrix Expression of Hypermatrices

We give two examples for this.

######  Example 6.

Assume $V$ is an $n$-dimensional vector space over ${\mathbb{F}}$.
$T:V^{r}\times(V^{*})^{s}\rightarrow{\mathbb{F}}$ is a tensor of covariant
order $r$ and contra-variant order $s$. Let $\delta_{n}^{i}$, $i\in\langle
n\rangle$ be a basis of $V$ and $\omega_{i}=(\delta_{n}^{i})^{T}$,
$i\in\langle n\rangle$ be the dual basis of dual space $V^{*}$, and set

$\begin{array}[]{l}T(\delta_{n}^{i_{1}},\cdots,\delta_{n}^{i_{r}},(\delta_{n}^{j_{1}})^{*},\cdots,(\delta_{n}^{j_{s}})^{*})=\mu^{i_{1},\cdots,i_{r}}_{j_{1},\cdots,j_{s}},\\\
~{}~{}i_{p}\in\langle n\rangle,p\in\langle r\rangle,j_{q}\in\langle
n\rangle,q\in\langle s\rangle.\end{array}$

Then

$\begin{array}[]{ccl}\Omega&:=&\\{\mu^{i_{1},\cdots,i_{r}}_{j_{1},\cdots,j_{s}}\;|\;i_{p}\in\langle
n\rangle,p\in\langle r\rangle,j_{q}\in\langle n\rangle,q\in\langle
s\rangle\\}\in{\mathbb{R}}^{\overbrace{n\times\cdots\times
n}^{r+s}}\end{array}$

is a hypermatrix of order $r+s$.

Construct $M_{\Omega}^{{\bf j_{s}}\times{\bf i_{r}}}$, where ${\bf
j_{s}}:=(j_{1},\cdots,j_{s})$, ${\bf i_{r}}:=(i_{1},\cdots,i_{r}).$

For $\omega_{1},\cdots,\omega_{s}\in V^{*}$, $x_{1},\cdots,x_{r}\in V$, we
have

$T(x_{1},\cdots,x_{r};\omega_{1},\cdots,\omega_{s})=\ltimes_{j=1}^{q}\omega_{s+1-j}M^{{\bf
j_{s}}\times{\bf i_{r}}}_{\Omega}\ltimes_{i=1}^{p}x_{i}.$

######  Example 7.

[21] In statistical mechanics, the Yang-Baxter equation is given as follows:
Let $R=[r_{i_{1},i_{2},i_{3},i_{4}}]\in{\mathbb{F}}^{N\times N\times N\times
N}$. Then we have (in our notation):

$\displaystyle\left(R\times^{4}_{1}R\right)\times^{(2,4)}_{(3,4)}R=R\times^{(1,2)}_{(3,4)}\times\left(R\times^{4}_{1}R\right).$
(52)

We express (52) into matrix form as follows:

Expressing $R\times^{4}_{1}R$ into matrix form, we have

$\displaystyle M_{T}:=M_{R}^{(1,2,3)\times
4}M_{R}^{1\times(2,3,4)}\in{\mathbb{F}}^{N^{3}\times N^{3}},$ (53)

where its rows are indexed by $\mathrm{ID}(i_{1},i_{2},i_{3};N,N,N)$, which
represents the original $(i_{1},i_{2},i_{3})$ of $R$; its columns are indexed
by $\mathrm{ID}(i_{4},i_{5},i_{6};N,N,N)$, which represents the original
$(i_{2},i_{3},i_{4})$ of $R$.

For the product on the LHS of (52), we convert $M_{T}$ into
$M_{T}^{(i_{1},i_{3},i_{4},i_{5})\times(i_{2},i_{6})}$. Define

$\sigma_{s}:\langle 6\rangle\rightarrow(1,3,4,5,2,6).$

Then we have

$M_{T}^{(1,3,4,5)\times(2,6)}={\bf
V}_{r}^{N^{2}}\left[\left(M_{T}\delta_{N^{3}}^{I}\right)^{T}W^{\sigma^{-1}_{s}}_{\underbrace{[N\cdots
N]}_{6}}\right].$

Hence, the LHS of (52) becomes

$\displaystyle LHS=\left({\bf
V}_{r}^{N^{2}}\left[\left(M_{T}\delta_{N^{3}}^{I}\right)^{T}W^{\sigma^{-1}_{s}}_{\underbrace{[N\cdots
N]}_{6}}\right]\right)M_{R}^{(34)\times(12)}.$ (54)

Similarly, for the product on the LHS of (52), we convert $M_{T}$ into
$M_{T}^{(i_{3},i_{4})\times(i_{1},i_{2},i_{5},i_{6})}$. Define

$\sigma_{t}:\langle 6\rangle\rightarrow(3,4,1,2,5,6).$

Then we have

$M_{T}^{(3,4)\times(1,2,5,6)}={\bf
V}_{r}^{N^{4}}\left[\left(M_{T}\delta_{N^{3}}^{I}\right)^{T}W^{\sigma^{-1}_{t}}_{\underbrace{[N\cdots
N]}_{6}}\right].$

Hence, the RHS of (52) becomes

$\displaystyle RHS=M_{R}^{(34)\times(12)}\left({\bf
V}_{r}^{N^{4}}\left[\left(M_{T}\delta_{N^{3}}^{I}\right)^{T}W^{\sigma^{-1}_{t}}_{\underbrace{[N\cdots
N]}_{6}}\right]\right).$ (55)

Hence, (52) implies that

$\begin{array}[]{l}\left({\bf
V}_{r}^{N^{2}}\left[\left(M_{T}\delta_{N^{3}}^{I}\right)^{T}W^{\sigma^{-1}_{s}}_{\underbrace{[N\cdots
N]}_{6}}\right]\right)M_{R}^{(34)\times(12)}\\\
=M_{R}^{(34)\times(12)}\left({\bf
V}_{r}^{N^{4}}\left[\left(M_{T}\delta_{N^{3}}^{I}\right)^{T}W^{\sigma^{-1}_{t}}_{\underbrace{[N\cdots
N]}_{6}}\right]\right).\end{array}$

##  5 Conclusion

In this paper, the matrix expression of hypermatrices was first proposed. As
an auxiliary tool, the permutation matrices have also been discussed in
detail. It is used to reveal certain properties of the matrix expression of
hypermatrices. Then it was shown that the STPs of matrices are essentially the
multilinear operators of hypermatrices (including hypervectors). The operators
over hypermatrices including contracted products, are realized by STPs through
the matrix expression of hypermatrices. In fact, the actions of STP over
hypermatrices can be considered as a generalization of the actions of the
conventional matrix product over matrices. This fact can be demonstrated by
Figure 2.

$\begin{array}[]{c}\mbox{Hypervector}\\\
\mbox{Hypermatrix}\end{array}$$\begin{array}[]{c}\mbox{Hypervector}\\\
\mbox{Hypermatrix}\end{array}$$\begin{array}[]{c}\mbox{Vector}\\\
\mbox{Matrix}\end{array}$$\begin{array}[]{c}\mbox{Vector}\\\
\mbox{Matrix}\end{array}$$\ltimes$$\times$ Figure 2: $\times$ vs $\ltimes$

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## Appendix

In the following some permutation matrices are listed.

  1. 1.

$d=3$, $n=2$:

     * (i)

$\sigma_{1}:[1,2,3]\rightarrow[1,2,3]$:

$W_{2}^{\sigma_{1}}=\delta_{8}[1,2,3,4,5,6,7,8].$

     * (ii)

$\sigma_{2}:[1,2,3]\rightarrow[1,3,2]$:

$W_{2}^{\sigma_{2}}=\delta_{8}[1,3,2,4,5,7,6,8].$

     * (iii)

$\sigma_{3}:[1,2,3]\rightarrow[2,1,3]$:

$W_{2}^{\sigma_{3}}=\delta_{8}[1,2,5,6,3,4,7,8].$

     * (iv)

$\sigma_{4}:[1,2,3]\rightarrow[2,3,1]$:

$W_{2}^{\sigma_{4}}=\delta_{8}[1,3,5,7,2,4,6,8]..$

     * (v)

$\sigma_{5}:[1,2,3]\rightarrow[3,1,2]$:

$W_{2}^{\sigma_{5}}=\delta_{8}[1,5,2,6,3,7,4,8].$

     * (vi)

$\sigma_{6}:[1,2,3]\rightarrow[3,2,1]$:

$W_{2}^{\sigma_{6}}=\delta_{8}[1,5,3,7,2,6,4,8].$

  2. 2.

$d=3$, $n=3$:

     * (i)

$\sigma_{1}:[1,2,3]\rightarrow[1,2,3]$:

$W_{3}^{\sigma_{1}}=\delta_{27}[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27].$

     * (ii)

$\sigma_{2}:[1,2,3]\rightarrow[1,3,2]$:

$W_{3}^{\sigma_{2}}=\delta_{27}[1,4,7,2,5,8,3,6,9,10,13,16,11,14,17,12,15,18,19,22,25,20,23,26,21,24,27].$

     * (iii)

$\sigma_{3}:[1,2,3]\rightarrow[2,1,3]$:

$W_{3}^{\sigma_{3}}=\delta_{27}[1,2,3,10,11,12,19,20,21,4,5,6,13,14,15,22,23,24,7,8,9,16,17,18,25,26,27].$

     * (iv)

$\sigma_{4}:[1,2,3]\rightarrow[2,3,1]$:

$W_{3}^{\sigma_{4}}=\delta_{27}[1,10,19,2,11,20,3,12,21,4,13,22,5,14,23,6,15,24,7,16,25,8,17,26,9,18,27].$

     * (v)

$\sigma_{5}:[1,2,3]\rightarrow[3,1,2]$:

$W_{3}^{\sigma_{5}}=\delta_{27}[1,4,7,10,13,16,19,22,25,2,5,8,11,14,17,20,23,26,3,6,9,12,15,18,21,24,27].$

     * (vi)

$\sigma_{6}:[1,2,3]\rightarrow[3,2,1]$:

$W_{3}^{\sigma_{6}}=\delta_{27}[1,10,19,4,13,22,7,16,25,2,11,20,5,14,23,8,17,26,3,12,21,6,15,24,9,18,27].$

  3. 3.

$d=3$, $n=4$:

     * (i)

$\sigma_{1}:[1,2,3]\rightarrow[1,2,3]$:

$\begin{array}[]{l}W_{4}^{\sigma_{1}}=\delta_{64}[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,\\\
~{}~{}~{}~{}23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,\\\
~{}~{}~{}~{}44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64].\end{array}$

     * (ii)

$\sigma_{2}:[1,2,3]\rightarrow[1,3,2]$:

$\begin{array}[]{l}W_{4}^{\sigma_{2}}=\delta_{64}[1,5,9,13,2,6,10,14,3,7,11,15,4,8,12,16,17,21,25,29,18,22,\\\
~{}~{}~{}~{}26,30,19,23,27,31,20,24,28,32,33,37,41,45,34,38,42,46,35,39,43,\\\
~{}~{}~{}~{}47,36,40,44,48,49,53,57,61,50,54,58,62,51,55,59,63,52,56,60,64].\end{array}$

     * (iii)

$\sigma_{3}:[1,2,3]\rightarrow[2,1,3]$:

$\begin{array}[]{l}W_{4}^{\sigma_{3}}=\delta_{64}[1,2,3,4,17,18,19,20,33,34,35,36,49,50,51,52,5,6,7,8,21,22,\\\
~{}~{}~{}~{}23,24,37,38,39,40,53,54,55,56,9,10,11,12,25,26,27,28,41,42,43,44,\\\
~{}~{}~{}~{}57,58,59,60,13,14,15,16,29,30,31,32,45,46,47,48,61,62,63,64].\end{array}$

     * (iv)

$\sigma_{4}:[1,2,3]\rightarrow[2,3,1]$:

$\begin{array}[]{l}W_{4}^{\sigma_{4}}=\delta_{64}[1,17,33,49,2,18,34,50,3,19,35,51,4,20,36,52,5,21,37,53,6,\\\
~{}~{}~{}~{}22,38,54,7,23,39,55,8,24,40,56,9,25,41,57,10,26,42,58,11,27,43,\\\
~{}~{}~{}~{}59,12,28,44,60,13,29,45,61,14,30,46,62,15,31,47,63,16,32,48,64].\end{array}$

     * (v)

$\sigma_{5}:[1,2,3]\rightarrow[3,1,2]$:

$\begin{array}[]{l}W_{5}^{\sigma_{4}}=\delta_{64}[1,5,9,13,17,21,25,29,33,37,41,45,49,53,57,61,2,6,10,14,\\\
~{}~{}~{}~{}18,22,26,30,34,38,42,46,50,54,58,62,3,7,11,15,19,23,27,31,35,39,\\\
~{}~{}~{}~{}43,47,51,55,59,63,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64].\end{array}$

     * (vi)

$\sigma_{6}:[1,2,3]\rightarrow[3,2,1]$:

$\begin{array}[]{l}W_{6}^{\sigma_{4}}=\delta_{64}[1,17,33,49,5,21,37,53,9,25,41,57,13,29,45,61,2,18,34,50,\\\
~{}~{}~{}~{}6,22,38,54,10,26,42,58,14,30,46,62,3,19,35,51,7,23,39,55,11,27,\\\
~{}~{}~{}~{}43,59,15,31,47,63,4,20,36,52,8,24,40,56,12,28,44,60,16,32,48,64].\end{array}$

  4. 4.

$d=4$, $n=2$:

     * (i)

$\sigma_{1}:[1,2,3,4]\rightarrow[1,2,3,4]$:

$W_{2}^{\sigma_{1}}=\delta_{16}[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16].$

     * (ii)

$\sigma_{2}:[1,2,3,4]\rightarrow[1,2,4,3]$:

$W_{2}^{\sigma_{2}}=\delta_{16}[1,3,2,4,5,7,6,8,9,11,10,12,13,15,14,16].$

     * (iii)

$\sigma_{3}:[1,2,3,4]\rightarrow[1,3,2,4]$:

$W_{2}^{\sigma_{3}}=\delta_{16}[1,2,5,6,3,4,7,8,9,10,13,14,11,12,15,16].$

     * (iv)

$\sigma_{4}:[1,2,3,4]\rightarrow[1,3,4,2]$:

$W_{2}^{\sigma_{4}}=\delta_{16}[1,5,2,6,3,7,4,8,9,13,10,14,11,15,12,16].$

     * (v)

$\sigma_{5}:[1,2,3,4]\rightarrow[1,4,2,3]$:

$W_{2}^{\sigma_{5}}=\delta_{16}[1,3,5,7,2,4,6,8,9,11,13,15,10,12,14,16].$

     * (vi)

$\sigma_{6}:[1,2,3,4]\rightarrow[1,4,3,2]$:

$W_{2}^{\sigma_{6}}=\delta_{16}[1,5,3,7,2,6,4,8,9,13,11,15,10,14,12,16].$

     * (vii)

$\sigma_{7}:[1,2,3,4]\rightarrow[2,1,3,4]$:

$W_{2}^{\sigma_{7}}=\delta_{16}[1,2,3,4,9,10,11,12,5,6,7,8,13,14,15,16].$

     * (viii)

$\sigma_{8}:[1,2,3,4]\rightarrow[2,1,4,3]$:

$W_{2}^{\sigma_{8}}=\delta_{16}[1,3,2,4,9,11,10,12,5,7,6,8,13,15,14,16].$

     * (ix)

$\sigma_{9}:[1,2,3,4]\rightarrow[2,3,1,4]$:

$W_{2}^{\sigma_{8}}=\delta_{16}[1,2,9,10,3,4,11,12,5,6,13,14,7,8,15,16].$

     * (x)

$\sigma_{10}:[1,2,3,4]\rightarrow[2,3,4,1]$:

$W_{2}^{\sigma_{10}}=\delta_{16}[1,9,2,10,3,11,4,12,5,13,6,14,7,15,8,16].$

     * (xi)

$\sigma_{11}:[1,2,3,4]\rightarrow[2,4,1,3]$:

$W_{2}^{\sigma_{11}}=\delta_{16}[1,3,9,11,2,4,10,12,5,7,13,15,6,8,14,16].$

     * (xii)

$\sigma_{12}:[1,2,3,4]\rightarrow[2,4,3,1]$:

$W_{2}^{\sigma_{12}}=\delta_{16}[1,9,3,11,2,10,4,12,5,13,7,15,6,14,8,16].$

     * (xiii)

$\sigma_{13}:[1,2,3,4]\rightarrow[3,1,2,4]$:

$W_{2}^{\sigma_{13}}=\delta_{16}[1,2,5,6,9,10,13,14,3,4,7,8,11,12,15,16].$

     * (xiv)

$\sigma_{14}:[1,2,3,4]\rightarrow[3,1,4,2]$:

$W_{2}^{\sigma_{14}}=\delta_{16}[1,5,2,6,9,13,10,14,3,7,4,8,11,15,12,16].$

     * (xv)

$\sigma_{15}:[1,2,3,4]\rightarrow[3,2,1,4]$:

$W_{2}^{\sigma_{15}}=\delta_{16}[1,5,2,6,9,13,10,14,3,7,4,8,11,15,12,16].$

     * (xvi)

$\sigma_{16}:[1,2,3,4]\rightarrow[3,2,4,1]$:

$W_{2}^{\sigma_{16}}=\delta_{16}[1,9,2,10,5,13,6,14,3,11,4,12,7,15,8,16].$

     * (xvii)

$\sigma_{17}:[1,2,3,4]\rightarrow[3,4,1,2]$:

$W_{2}^{\sigma_{17}}=\delta_{16}[1,5,9,13,2,6,10,14,3,7,11,15,4,8,12,16].$

     * (xviii)

$\sigma_{18}:[1,2,3,4]\rightarrow[3,4,2,1]$:

$W_{2}^{\sigma_{18}}=\delta_{16}[1,9,5,13,2,10,6,14,3,11,7,15,4,12,8,16].$

     * (xix)

$\sigma_{19}:[1,2,3,4]\rightarrow[4,1,2,3]$:

$W_{2}^{\sigma_{19}}=\delta_{16}[1,3,5,7,9,11,13,15,2,4,6,8,10,12,14,16].$

     * (xx)

$\sigma_{20}:[1,2,3,4]\rightarrow[4,1,3,2]$:

$W_{2}^{\sigma_{20}}=\delta_{16}[1,5,3,7,9,13,11,15,2,6,4,8,10,14,12,16].$

     * (xxi)

$\sigma_{21}:[1,2,3,4]\rightarrow[4,2,1,3]$:

$W_{2}^{\sigma_{21}}=\delta_{16}[1,3,9,11,5,7,13,15,2,4,10,12,6,8,14,16].$

     * (xxii)

$\sigma_{22}:[1,2,3,4]\rightarrow[4,2,3,1]$:

$W_{2}^{\sigma_{22}}=\delta_{16}[1,9,3,11,5,13,7,15,2,10,4,12,6,14,8,16].$

     * (xxiii)

$\sigma_{23}:[1,2,3,4]\rightarrow[4,3,1,2]$:

$W_{2}^{\sigma_{23}}=\delta_{16}[1,5,9,13,3,7,11,15,2,6,10,14,4,8,12,16].$

     * (xxiv)

$\sigma_{24}:[1,2,3,4]\rightarrow[4,3,2,1]$:

$W_{2}^{\sigma_{24}}=\delta_{16}[1,9,5,13,3,11,7,15,2,10,6,14,4,12,8,16].$

  5. 5.

$d=4$, $n=3$:

     * (i)

$\sigma_{1}:[1,2,3,4]\rightarrow[1,2,3,4]$:

$\begin{array}[]{l}W_{3}^{\sigma_{1}}=\delta_{81}[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,\\\
~{}~{}21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,\\\
~{}~{}41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,\\\
~{}~{}61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81].\end{array}$

     * (ii)

$\sigma_{2}:[1,2,3,4]\rightarrow[1,2,4,3]$:

$\begin{array}[]{l}W_{3}^{\sigma_{2}}=\delta_{81}[1,4,7,2,5,8,3,6,9,10,13,16,11,14,17,12,15,18,19,22,\\\
~{}~{}25,20,23,26,21,24,27,28,31,34,29,32,35,30,33,36,37,40,43,38,\\\
~{}~{}41,44,39,42,45,46,49,52,47,50,53,48,51,54,55,58,61,56,59,62,\\\
~{}~{}57,60,63,64,67,70,65,68,71,66,69,72,73,76,79,74,77,80,75,78,81].\end{array}$

     * (iii)

$\sigma_{3}:[1,2,3,4]\rightarrow[1,3,2,4]$:

$\begin{array}[]{l}W_{3}^{\sigma_{3}}=\delta_{81}[1,2,3,10,11,12,19,20,21,4,5,6,13,14,15,22,23,24,7,8,\\\
~{}~{}9,16,17,18,25,26,27,28,29,30,37,38,39,46,47,48,31,32,33,40,\\\
~{}~{}41,42,49,50,51,34,35,36,43,44,45,52,53,54,55,56,57,64,65,66,\\\
~{}~{}73,74,75,58,59,60,67,68,69,76,77,78,61,62,63,70,71,72,79,80,81].\end{array}$

     * (iv)

$\sigma_{4}:[1,2,3,4]\rightarrow[1,3,4,2]$:

$\begin{array}[]{l}W_{3}^{\sigma_{4}}=\delta_{81}[1,10,19,2,11,20,3,12,21,4,13,22,5,14,23,6,15,24,7,16,\\\
~{}~{}25,8,17,26,9,18,27,28,37,46,29,38,47,30,39,48,31,40,49,32,\\\
~{}~{}41,50,33,42,51,34,43,52,35,44,53,36,45,54,55,64,73,56,65,74,\\\
~{}~{}57,66,75,58,67,76,59,68,77,60,69,78,61,70,79,62,71,80,63,72,81].\end{array}$

     * (v)

$\sigma_{5}:[1,2,3,4]\rightarrow[1,4,2,3]$:

$\begin{array}[]{l}W_{3}^{\sigma_{5}}=\delta_{81}[1,4,7,10,13,16,19,22,25,2,5,8,11,14,17,20,23,26,3,6,\\\
~{}~{}9,12,15,18,21,24,27,28,31,34,37,40,43,46,49,52,29,32,35,38,\\\
~{}~{}41,44,47,50,53,30,33,36,39,42,45,48,51,54,55,58,61,64,67,70,\\\
~{}~{}73,76,79,56,59,62,65,68,71,74,77,80,57,60,63,66,69,72,75,78,81].\end{array}$

     * (vi)

$\sigma_{6}:[1,2,3,4]\rightarrow[1,4,3,2]$:

$\begin{array}[]{l}W_{3}^{\sigma_{6}}=\delta_{81}[1,10,19,4,13,22,7,16,25,2,11,20,5,14,23,8,17,26,3,12,\\\
~{}~{}21,6,15,24,9,18,27,28,37,46,31,40,49,34,43,52,29,38,47,32,\\\
~{}~{}41,50,35,44,53,30,39,48,33,42,51,36,45,54,55,64,73,58,67,76,\\\
~{}~{}61,70,79,56,65,74,59,68,77,62,71,80,57,66,75,60,69,78,63,72,81].\end{array}$

     * (vii)

$\sigma_{7}:[1,2,3,4]\rightarrow[2,1,3,4]$:

$\begin{array}[]{l}W_{3}^{\sigma_{7}}=\delta_{81}[1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,36,55,56,\\\
~{}~{}57,58,59,60,61,62,63,10,11,12,13,14,15,16,17,18,37,38,39,40,\\\
~{}~{}41,42,43,44,45,64,65,66,67,68,69,70,71,72,19,20,21,22,23,24,\\\
~{}~{}25,26,27,46,47,48,49,50,51,52,53,54,73,74,75,76,77,78,79,80,81].\end{array}$

     * (viii)

$\sigma_{8}:[1,2,3,4]\rightarrow[2,1,4,3]$:

$\begin{array}[]{l}W_{3}^{\sigma_{8}}=\delta_{81}[1,4,7,2,5,8,3,6,9,28,31,34,29,32,35,30,33,36,55,58,\\\
~{}~{}61,56,59,62,57,60,63,10,13,16,11,14,17,12,15,18,37,40,43,38,\\\
~{}~{}41,44,39,42,45,64,67,70,65,68,71,66,69,72,19,22,25,20,23,26,\\\
~{}~{}21,24,27,46,49,52,47,50,53,48,51,54,73,76,79,74,77,80,75,78,81].\end{array}$

     * (ix)

$\sigma_{9}:[1,2,3,4]\rightarrow[2,3,1,4]$:

$\begin{array}[]{l}W_{3}^{\sigma_{8}}=\delta_{81}[1,2,3,28,29,30,55,56,57,4,5,6,31,32,33,58,59,60,7,8,\\\
~{}~{}9,34,35,36,61,62,63,10,11,12,37,38,39,64,65,66,13,14,15,40,\\\
~{}~{}41,42,67,68,69,16,17,18,43,44,45,70,71,72,19,20,21,46,47,48,\\\
~{}~{}73,74,75,22,23,24,49,50,51,76,77,78,25,26,27,52,53,54,79,80,81].\end{array}$

     * (x)

$\sigma_{10}:[1,2,3,4]\rightarrow[2,3,4,1]$:

$\begin{array}[]{l}W_{3}^{\sigma_{10}}=\delta_{81}[1,28,55,2,29,56,3,30,57,4,31,58,5,32,59,6,33,60,7,34,\\\
~{}~{}61,8,35,62,9,36,63,10,37,64,11,38,65,12,39,66,13,40,67,14,\\\
~{}~{}41,68,15,42,69,16,43,70,17,44,71,18,45,72,19,46,73,20,47,74,\\\
~{}~{}21,48,75,22,49,76,23,50,77,24,51,78,25,52,79,26,53,80,27,54,81].\end{array}$

     * (xi)

$\sigma_{11}:[1,2,3,4]\rightarrow[2,4,1,3]$:

$\begin{array}[]{l}W_{3}^{\sigma_{11}}=\delta_{81}[1,4,7,28,31,34,55,58,61,2,5,8,29,32,35,56,59,62,3,6,\\\
~{}~{}9,30,33,36,57,60,63,10,13,16,37,40,43,64,67,70,11,14,17,38,\\\
~{}~{}41,44,65,68,71,12,15,18,39,42,45,66,69,72,19,22,25,46,49,52,\\\
~{}~{}73,76,79,20,23,26,47,50,53,74,77,80,21,24,27,48,51,54,75,78,81].\end{array}$

     * (xii)

$\sigma_{12}:[1,2,3,4]\rightarrow[2,4,3,1]$:

$\begin{array}[]{l}W_{3}^{\sigma_{12}}=\delta_{81}[1,28,55,4,31,58,7,34,61,2,29,56,5,32,59,8,35,62,3,30,\\\
~{}~{}57,6,33,60,9,36,63,10,37,64,13,40,67,16,43,70,11,38,65,14,\\\
~{}~{}41,68,17,44,71,12,39,66,15,42,69,18,45,72,19,46,73,22,49,76,\\\
~{}~{}25,52,79,20,47,74,23,50,77,26,53,80,21,48,75,24,51,78,27,54,81].\end{array}$

     * (xiii)

$\sigma_{13}:[1,2,3,4]\rightarrow[3,1,2,4]$:

$\begin{array}[]{l}W_{3}^{\sigma_{13}}=\delta_{81}[1,2,3,10,11,12,19,20,21,28,29,30,37,38,39,46,47,48,55,56,\\\
~{}~{}57,64,65,66,73,74,75,4,5,6,13,14,15,22,23,24,31,32,56,57,\\\
~{}~{}64,65,66,73,74,75,4,5,6,13,14,15,22,23,24,31,32,16,17,18,\\\
~{}~{}25,26,27,34,35,36,43,44,45,52,53,54,61,62,63,70,71,72,79,80,81].\end{array}$

     * (xiv)

$\sigma_{14}:[1,2,3,4]\rightarrow[3,1,4,2]$:

$\begin{array}[]{l}W_{3}^{\sigma_{14}}=\delta_{81}[1,10,19,2,11,20,3,12,21,28,37,46,29,38,47,30,39,48,55,64,\\\
~{}~{}73,56,65,74,57,66,75,4,13,22,5,14,23,6,15,24,31,40,49,32,\\\
~{}~{}41,50,33,42,51,58,67,76,59,68,77,60,69,78,7,16,25,8,17,26,\\\
~{}~{}9,18,27,34,43,52,35,44,53,36,45,54,61,70,79,62,71,80,63,72,81].\end{array}$

     * (xv)

$\sigma_{15}:[1,2,3,4]\rightarrow[3,2,1,4]$:

$\begin{array}[]{l}W_{3}^{\sigma_{15}}=\delta_{81}[1,10,19,2,11,20,3,12,21,28,37,46,29,38,47,30,39,48,55,64,\\\
~{}~{}73,56,65,74,57,66,75,4,13,22,5,14,23,6,15,24,31,40,49,32,\\\
~{}~{}41,50,33,42,51,58,67,76,59,68,77,60,69,78,7,16,25,8,17,26,\\\
~{}~{}9,18,27,34,43,52,35,44,53,36,45,54,61,70,79,62,71,80,63,72,81].\end{array}$

     * (xvi)

$\sigma_{16}:[1,2,3,4]\rightarrow[3,2,4,1]$:

$\begin{array}[]{l}W_{3}^{\sigma_{16}}=\delta_{16}[1,28,55,2,29,56,3,30,57,10,37,64,11,38,65,12,39,66,19,46,\\\
~{}~{}73,20,47,74,21,48,75,4,31,58,5,32,59,6,33,60,13,40,67,14,\\\
~{}~{}41,68,15,42,69,22,49,76,23,50,77,24,51,78,7,34,61,8,35,62,\\\
~{}~{}9,36,63,16,43,70,17,44,71,18,45,72,25,52,79,26,53,80,27,54,81].\end{array}$

     * (xvii)

$\sigma_{17}:[1,2,3,4]\rightarrow[3,4,1,2]$:

$\begin{array}[]{l}W_{3}^{\sigma_{17}}=\delta_{81}[1,10,19,28,37,46,55,64,73,2,11,20,29,38,47,56,65,74,3,12,\\\
~{}~{}21,30,39,48,57,66,75,4,13,22,31,40,49,58,67,76,5,14,23,32,\\\
~{}~{}41,50,59,68,77,6,15,24,33,42,51,60,69,78,7,16,25,34,43,52,\\\
~{}~{}61,70,79,8,17,26,35,44,53,62,71,80,9,18,27,36,45,54,63,72,81].\end{array}$

     * (xviii)

$\sigma_{18}:[1,2,3,4]\rightarrow[3,4,2,1]$:

$\begin{array}[]{l}W_{3}^{\sigma_{18}}=\delta_{81}[1,28,55,10,37,64,19,46,73,2,29,56,11,38,65,20,47,74,3,30,\\\
~{}~{}57,12,39,66,21,48,75,4,31,58,13,40,67,22,49,76,5,32,59,14,\\\
~{}~{}41,68,23,50,77,6,33,60,15,42,69,24,51,78,7,34,61,16,43,70,\\\
~{}~{}25,52,79,8,35,62,17,44,71,26,53,80,9,36,63,18,45,72,27,54,81].\end{array}$

     * (xix)

$\sigma_{19}:[1,2,3,4]\rightarrow[4,1,2,3]$:

$\begin{array}[]{l}W_{3}^{\sigma_{19}}=\delta_{81}[1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,\\\
~{}~{}61,64,67,70,73,76,79,2,5,8,11,14,17,20,23,26,29,32,35,38,\\\
~{}~{}41,44,47,50,53,56,59,62,65,68,71,74,77,80,3,6,9,12,15,18,\\\
~{}~{}21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81].\end{array}$

     * (xx)

$\sigma_{20}:[1,2,3,4]\rightarrow[4,1,3,2]$:

$\begin{array}[]{l}W_{3}^{\sigma_{20}}=\delta_{81}[1,10,19,4,13,22,7,16,25,28,37,46,31,40,49,34,43,52,55,64,\\\
~{}~{}73,58,67,76,61,70,79,2,11,20,5,14,23,8,17,26,29,38,47,32,\\\
~{}~{}41,50,35,44,53,56,65,74,59,68,77,62,71,80,3,12,21,6,15,24,\\\
~{}~{}9,18,27,30,39,48,33,42,51,36,45,54,57,66,75,60,69,78,63,72,81].\end{array}$

     * (xxi)

$\sigma_{21}:[1,2,3,4]\rightarrow[4,2,1,3]$:

$\begin{array}[]{l}W_{3}^{\sigma_{21}}=\delta_{81}[1,4,7,28,31,34,55,58,61,10,13,16,37,40,43,64,67,70,19,22,\\\
~{}~{}25,46,49,52,73,76,79,2,5,8,29,32,35,56,59,62,11,14,17,38,\\\
~{}~{}41,44,65,68,71,20,23,26,47,50,53,74,77,80,3,6,9,30,33,36,\\\
~{}~{}57,60,63,12,15,18,39,42,45,66,69,72,21,24,27,48,51,54,75,78,81].\end{array}$

     * (xxii)

$\sigma_{22}:[1,2,3,4]\rightarrow[4,2,3,1]$:

$\begin{array}[]{l}W_{3}^{\sigma_{22}}=\delta_{81}[1,28,55,4,31,58,7,34,61,10,37,64,13,40,67,16,43,70,19,46,\\\
~{}~{}73,22,49,76,25,52,79,2,29,56,5,32,59,8,35,62,11,38,65,14,\\\
~{}~{}41,68,17,44,71,20,47,74,23,50,77,26,53,80,3,30,57,6,33,60,\\\
~{}~{}9,36,63,12,39,66,15,42,69,18,45,72,21,48,75,24,51,78,27,54,81].\end{array}$

     * (xxiii)

$\sigma_{23}:[1,2,3,4]\rightarrow[4,3,1,2]$:

$\begin{array}[]{l}W_{3}^{\sigma_{23}}=\delta_{81}[1,10,19,28,37,46,55,64,73,4,13,22,31,40,49,58,67,76,7,16,\\\
~{}~{}25,34,43,52,61,70,79,2,11,20,29,38,47,56,65,74,5,14,23,32,\\\
~{}~{}41,50,59,68,77,8,17,26,35,44,53,62,71,80,3,12,21,30,39,48,\\\
~{}~{}57,66,75,6,15,24,33,42,51,60,69,78,9,18,27,36,45,54,63,72,81].\end{array}$

     * (xxiv)

$\sigma_{24}:[1,2,3,4]\rightarrow[4,3,2,1]$:

$\begin{array}[]{l}W_{3}^{\sigma_{24}}=\delta_{81}[1,28,55,10,37,64,19,46,73,4,31,58,13,40,67,22,49,76,7,34,\\\
~{}~{}61,16,43,70,25,52,79,2,29,56,11,38,65,20,47,74,5,32,59,14,\\\
~{}~{}41,68,23,50,77,8,35,62,17,44,71,26,53,80,3,30,57,12,39,66,\\\
~{}~{}21,48,75,6,33,60,15,42,69,24,51,78,9,36,63,18,45,72,27,54,81].\end{array}$