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\title{Five Dimensional Algebraic Tori}

\begin{document}
\maketitle
\abstract{
  The rationality problem for algebraic tori of dimentions 2, 3 and 4 (with a few unknown cases in dimension 4) is solved.
  The purpose of this work is to solve rationality problem for 5 dimensional algebraic tori in an experimental way. In order to
  do so, we have studied the associated character lattices of the mentioned
  algebraic tori. For each indecomposable character lattice $L$, either we identify the lattice as
  an associated one to a root system (of which the rationality of its corresponding
  algebraic torus is known) or we find a reduced component of $L$ so that we can
  relate rationality of the associated algebraic torus to lower dimensions. Using
  these two main methods from \cite{Nicole1}, we solve rationality problem in some cases.
  For those tori with a decomposable character lattice, we use the known results on the rationality
  of their decomposition components to determine their rationality.
  }

\section{Introduction}
An interesting problem in algebraic geometry is the rationality problem, i.e.
for a given algebraic variety, determine if it is rational or not. Naturally
this is a difficult problem to solve. Hence it makes sense to consider some
relaxed notions of rationality.

An algebraic $F$-variety $X$ (for a field $F$) is called rational if there exists a birational map
form $X$ to $\mathbb{A}^n$ for some $n$. Algebraically, $X$ is rational if
its rational function field is $F$-rational, i.e. it is a purely
transcendental extension of $F$. $X$ is called stably rational if $X \times
\mathbb{A}^n$ is rational for a non-negative integer $n$. Similarly one can say $X$ is stably rational if
a purely transcendental extension of the rational function field of $X$ is
rational. Finally $X$ is called unirational if the rational function field of
$X$ is a subfield of a purely transcendental extension of $F$. Geometrically
this means there exists a dominant rational map from $\mathbb{A}^n$ to $X$ for
some $n$. It is worth mentioning that there is a family of varieties which is
called retract rational which contains the stably rational ones and is contained in
unirational family. However since we have nothing to do with retract rational tori we
do not present their definition (see \cite{Hoshi} for more detail).

It can be seen that rationality implies stable rationality and the later implies
unirationality. It is well-known that none of the converses of these implications
hold in general for varieties.

Assume $X$ is a quasi projective $F$-variety. An $F$-variety $Y$ is called an
$F$-form of $X$ if $X \otimes_F \bar{F} \cong Y \otimes_F \bar{F}$ where $\bar{F}$ is the algebraic closure of $F$. A $d$-dimensional algebraic
$F$-torus is a group $F$-scheme such that it is an $F$-form of $\mathbb{G}_m^d$ (see \cite[Section 1.3]{Voskresenskii} for more detail).

There is a one-to-one correspondence between the isomorphism classes of
$n$ dimensional algebraic $F$-tori and representations of $G_F = \mathrm{Gal}(F_s/F)$
into $\mathrm{GL}(n,\Z)$, where $F_s$ is the separable closure of $F$. We know that
 there exists a finite Galois extension of $K/F$, such that
$T \otimes_F K \cong \mathbb{G}_{m,K}^n$. The smallest such Galois extension
(resp. Galois group) is called the splitting field (resp. splitting group)
 of $T$.

The isomorphism classes of $n$ dimensional algebraic $F$-tori are in bijection
with conjugacy classes of finite subgroups of $\mathrm{GL}(n,\Z)$. By a theorem of Jordan
 we know that the number
of these finite subgroups up to conjugacy is finite. If $G \leq \mathrm{GL}(n,\Z)$ then
the standard lattice, $L = \langle e_i: 1 \leq i \leq n \rangle_\Z$ where $e_i = [\delta_{1i}]$,
with the action (right multiplication) of $G$ defines a $G$-lattice $L_G$ ($\delta_{ij}$ is the Kronecker delta).
Assume $K/F$ is a finite Galois extension with $G = \mathrm{Gal}(K/F)$. Now $T_G = \mathrm{Spec}(K[L_G]^G)$
is an algebraic torus. Note that conjugate groups correspond to isomorphic
lattices and hence are associated to isomorphic algebraic tori. Conversely for
a given $n$-dimensional algebraic torus $T$, which is defined over $F$ and split
by $K$, its character group is a $G$-lattice of rank $n$ and hence associates
to a conjugacy class of subgroups of $\mathrm{GL}(n,\Z)$.

It is well-known that all algebraic tori are unirational. There are unirational algebraic tori which
are not stably rational. Interestingly there exists a famous conjecture by Voskeresenskii,
\begin{center}
 ``any stably rational algebraic torus is rational''.
 \end{center} To the
best of the authors knowledge the conjecture is still open. A substantional amount of work in rationality problem for algebraic tori, is done by Voskeresenskii, Endo, Miyata, Saltman,
Sansuc and colliet-thelene. Their approach to address the problem was the
correspondence between algebraic tori and $G$-lattices. Their efforts led to
characterization of stably rational and retract rational algebraic tori based
on their character lattices.

Voskeresenskii proved that any two dimensional algebraic torus is rational.
In early 90's Kunyavski classified three dimensional algebraic tori up to
rationality. He showed that except for 15 three dimensional algebraic tori
which are not retract rational, the rest of them are rational.

In 2012 Hoshi and Yamasaki \cite{Hoshi}
classified algebraic tori of dimensions 4 and 5 up to stable rationality.
Their classification is based on computing the flasque classes of algebraic
tori in GAP. They showed that there are 3051 algebraic tori of deimension five
which are stably rational and except for these the rest of them are not stably rational.
The rationality of these 3051 cases are not addressed in their work. Hoshi and
Yamasaki's approach to prove stable rationality was different for algebraic tori with
an indecomposable character lattice. They showed that in rank 5, there are exactly 311 indecomposable
$G$-lattices which are stably rational. More precisely they showed their
stable rationality results by finding the maximal groups and proved stable rationality
of their subgroups. The following table presents the maximal groups they found.

\begin{table}\label{tbl:18Max}
\centering
\begin{tabular}{llllll}
Number & CARAT ID & $G$ & $\#G$\\ \hline
 1  & $(5,942,1)$ & ${\rm Imf}(5,1,1)$ & $3840$\\
2  & $(5,953,4)$ & $\mathrm{S}_6$ & $720$ \\
3  & $(5,726,4)$ & $C_2^4\rtimes \mathrm{S}_4$ & $384$  \\
4  & $(5,919,4)$ & $C_2\times \mathrm{S}_5$ & $240$ \\
5 & $(5,801,3)$ & $C_2\times (\mathrm{S}_3^2\rtimes C_2)$ & $144$ \\
6 & $(5,655,4)$ & $D_8^2\rtimes C_2$ & $128$ \\
7  & $(5,911,4)$ & $\mathrm{S}_5$ & $120$ \\
8 & $(5,946,2)$ & $\mathrm{S}_5$ & $120$ \\
9 & $(5,946,4)$ & $\mathrm{S}_5$ & $120$ \\
10 & $(5,947,2)$ & $\mathrm{S}_5$ ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ & $120$ \\
11 & $(5,337,12)$ & $D_8\times \mathrm{S}_3$ & $48$ \\
12  & $(5,341,6)$ & $D_8\times \mathrm{S}_3$ & $48$ \\
13  & $(5,531,13)$ & $C_2\times \mathrm{S}_4$ & $48$ \\
14  & $(5,533,8)$ & $C_2\times \mathrm{S}_4$ & $48$ \\
15 & $(5,623,4)$ & $C_2\times \mathrm{S}_4$ & $48$\\
16   & $(5,245,12)$ & $C_2^2\times \mathrm{S}_3$ & $24$\\
17  & $(5,81,42)$ & $C_2\times D_8$ & $16$ \\
18  & $(5,81,48)$ & $C_2\times D_8$ & $16$ \\
\end{tabular}
\caption{The maximal $18$ groups
in the  $311$ cases found by Hoshi and Yamasaki in \cite{Hoshi}.}
%\label{tbl:18Max}
\end{table}

In 2015 Lemire \cite{Nicole1}, proved that, except for possibly ten of the
4 dimensional stably rational algebraic tori found by Hoshi and Yamasaki,
all of them are rational. The rationality of the ten exceptional cases is
still unknown. The author did not use any computer based arguments except
for finding generating sets of groups and lattices of subgroups in GAP. We
mainly follow the ideas from \cite{Nicole1} in an algorithmic way to make
these ideas computer based and then we use them to do experiments in dimension
five.

We present algorithms which may be applied to character lattices of algebraic
tori, in order to investigate their rationality. These algorithms provide
machinery to reduce the rationality problem in a specific dimension to lower dimensions
provided that the corresponding lattice is reducible.

Our contribution to the rationality problem is an experimental effort to
test Voskeresenskii's conjecture in dimension five. Note that based on the results of
\cite{Nicole1}, the conjecture is almost (since there are a few cases which are unresolved)
true. In dimension five our methods fail to determine the rationality of some algebraic tori.
However, in any case of which our methods work, we got a positive answer.

We partition the set of five dimensional stably rational algebraic tori into the ones with a
decomposable lattice and the ones with and indecomposable lattice. For simplicity we call an algebraic tors from the first family, decomposable and if it is from the second family we call it indecomposable. Both from theoretical and
experimental points of view the indecomposable tori are more interesting. This is due to the
fact that the rationality of decomposable cases can be decided based on their decomposition
 components. This is basically done by forming components and looking up in the
 known rationality tables.

For indecomposable stably rational 5 dimensional algebraic tori,
we consider the ones where their groups are maximal among the 311 cases found by Hoshi and
Yamasaki. These maximal groups are presented in \ref{tbl:18Max} and from now on we will call them
respectively $G_1$ to $G_{18}$.

There are two
main methods that we apply to $G_1$ to $G_{18}$ (and their subgroups), both of which were used in \cite{Nicole1}. The
first method is reducing the rationality of a five dimensional torus to rationality
in lower dimensions using a reducible component of corresponding lattices. More precisely, assuming $L_G$ is the corresponding lattice to a given indecomposable torus, we try to find a $G$-lattice, $M\subset
L_G$ so that we get the exact sequence
$$0 \longrightarrow M \longrightarrow L_G \longrightarrow P \longrightarrow 0, $$
where $P$ is a permutation $G$-lattice (see Section \ref{sec:g-lattices}) for a definition).

The second method is to identify character lattices of indecomposable tori as lattices of which their rationality is known such as the root lattice of the root system $B_n$ or a sign permutation lattice.

\todo{give detail on results}

Definitions and useful facts about $G$-lattices are provided in Section \ref{sec:g-lattices}. Section
\ref{sec:gap} gives brief information on computer algebra system GAP and relevant computational
notations for groups based on GAP package, Carat. Section \ref{sec:ratres} is devoted to present
theoretical known results on rationality problem for algebraic tori and rational families of algebraic
tori. Our algorithms to find exact sequence of lattices are introduced in Section \ref{sec:redalg}. Finally Section \ref{sec:results} contains tables of results of our experiments.

\section{$G$-Lattices}\label{sec:g-lattices}
It is known that there is a duality between algebraic tori and \glat s. In order to be able to discuss the duality, we briefly introduce \glat s in this section. For a more detailed discussion see \cite[Chapters 1 and 2]{Lorenz}.\\ \\
A lattice is a free $\mathbb{Z}$-module of finite rank. As a $\Z$-module, it is isomorphic to $\mathbb{Z}^n$ for some $n$. If we have a group $G$, we can endow the lattice with an action of $G$. If $G$ acts on a lattice $L$ by automorphisms i.e. there exists $$ G \longrightarrow \mathrm{GL}(M) \cong \mathrm{GL}(n,\mathbb{Z}),$$
we say $L$ is a $G$-lattice. We can equivalently say that $\G$-lattices are free $\Z$-modules of finite rank which is also $\mathbb{Z}[G]$-modules, for $\mathbb{Z}[G]$ the integral group ring over $G$. A $G$-equivariant, $ \mathbb{Z}$-linear map between $G$-lattices is called a homomorphism of $G$-lattices.\\
\\For a given \glat \, $L$ we define $$ L^\G = \lbrace l \in L : g.l=l,  \,\, \,\,  \forall g \in \G \rbrace. $$

Now let $R$ be an arbitrary commutative (unital) ring. For any \glat  \, $L$ we can form the group algebra $R[L]$ and the action of $G$ on $L$ can be extended to an action on $R[L]$. The subalgebra of all $G$-invariant elements in $R[L]$ $$ R[L]^ \G = \lbrace l \in R[L] : g.l = l, \,\,\, \forall g \in \G \rbrace$$ is called the multiplicative invariant algebra. Studying this algebra is the subject of multiplicative invariant theory.
\\
\\
The group algebra $R[L]$ of a \glat , is isomorphic to the Laurent polynomial algebra \linebreak $R[x_1^{\pm 1}, x_2^{\pm 1}, \ldots , x_n^{\pm 1}] $ and the lattice itself becomes a multiplicative subgroup of all monomials.
\\
\\
If $L$ and $L'$ are two \glat  s then $\mathrm{Hom}_{\Z}(L,L')$, the set of all $\Z$-linear maps from $L$ to $L'$, is a \glat \,\, with the following action $$ (g.f)(m) = g.(f(g^{-1}.m)).$$
For $L'=\Z$ (with trivial action of $G$), we get the \glat \,\,  $L^* = \mathrm{Hom}_{\Z}(L,\Z)$ which is called the dual lattice of $L$.\\
\\Suppose $L$ is a $G$-lattice and $H$ is a subgroup of $G$. $L$ can be considered as an $H$-lattice. The new $H$-lattice is called restricted and is denoted by $L\downarrow^G_H$.Assume $H$ is a finite index subgroup of $G$ and $L$ is an $H$-lattice. The $G$ module $\Z[H] \otimes_{\Z[G]} L$ is a $G$-lattice which is called the induced $G$-lattice and is denoted by $L\uparrow_H^G$.

A \glat \,\, $L$ is called a permutation lattice if there exists a $\Z$ -basis $X$ for $L$ such that, $G$ acts as permutation group on $X$. In this case $L$ is sometimes denoted by $\Z[X]$.
We call a \glat \,$L$, decomposable if there are nontrivial \glat s $L_1$ and $ L_2$ such that $ L\cong L_1 \oplus L_2$. $L$ is called indecomposable if it is not decomposable.
If $L$ is decomposable then it is reducible, but the converse is not true for lattices.
A \glat \,$L$ is called reducible if $L$ has a nontrivial $G$-invariant subspace $M$ such that $L/M$ is torsion free. $L$ is called irreducible if it is not reducible.

$L_G$ represents the corresponding $G$-lattice
to a finite subgroup of $\mathrm{GL}(n,\Z)$, $G$, as defined in Definition
\ref{Assumption}. When we say a group or a lattice is rational we mean their
corresponding algebraic torus is rational. By a decomposable (matrix) group,
we mean its corresponding lattice is decomposable. We say $G'$ is the dual
group of $G$ if $G'$ is the corresponding group to the dual of $L_G$.

\section{GAP: Carat and CrystCat}\label{sec:gap}
GAP \cite{GAP4} stands for Groups, Algorithms, Programming, and is a computer
algebra system for computations in discrete algebra with emphasis on computations
in group theory. GAP is an open source system which is accessible directly or
in SAGE \cite{sagemath}. GAP provides various packages for computations in matrix
groups and representation theory. For our purposes we need Carat and CrystCat packages of GAP.\\
\\
The GAP package Carat provides functions of the stand-alone programs of $CARAT$,
which is a package for the computations related to crystallographic groups. Carat
contains the catalog of all conjugacy classes of finite subgroups of $\mathrm{GL}(n,\Z)$
for $n$ up to six. More precisely the Carat package gives access to all
$\Q$-classes and $\Z$-classes and maximal classes over $\Z$.
\begin{remark}
The $\Q$-classes are conjugacy classes over $\Q$. We note that some $\Z$-classes
may belong to the same conjugacy class over the rationals.
\end{remark}

It is worth mentioning that Carat contains information about crystallographic groups
which we will not use. The CrystCat Package in GAP also provides a catalog of
crystallographic groups up to dimension 4. The catalog mostly covers the data in
\cite{Crystallography}. CrystCat and Carat are complement of each other. \\
\\
The GAP ID, $(n,m,l,k)$ of a finite subgroup $G$ of $\mathrm{GL}(n,\Z)$ means that
$G$ is of rank $n$ and belongs to $k$-th $\Z$ class of the $l$-th $\Q$-class of the
$m$-th crystal system. This works for $2\leq n \leq 4$. Hoshi and Yamasaki wrote
a GAP code using the Carat package to have easy access to the $j$-th $\Z$-class
of the $i$-th $\Q$-class group of rank $n$. They called this Carat ID. The GAP
scripts written by Hoshi and Yamasaki are available from
\begin{center}
http://www.math.h.kyoto-u.ac.jp/~yamasaki/Algorithm/
\end{center}
The algorithms introduced in this paper are implemented in GAP (needs some
functions from the codes written by Hoshi and Yamasaki) and the code is available from
\begin{center}
https://github.com/armin-jamshidpey/Algebraic-Tori
\end{center}
Since the actions of matrix groups in GAP are considered from right, throughout
this chapter we work with row vectors instead of columns. One may also use the
columns by considering the dual groups.\\
\\
We call $\G \leq \mathrm{GL}(n,\Z)$ irreducible (resp. indecomposable), if the corresponding
lattice to $G$ be irreducible (resp. indecomposable).

\section{Well-Known Rationality Results}\label{sec:ratres}
In this section we present some important results about rationality problem for
algebraic tori. Before presenting the results we need the following definition
to avoid repeating the same assumptions.
\begin{definition}\label{Assumption}
If $G$ is a finite subgroup of $\mathrm{GL}(n,\Z)$, then the corresponding lattice
to $G$ which is denoted by $L_G$ is the rank $n$ lattice generated by the standard
basis, i.e. $L_G = \langle e_i : i  = 1, \ldots, n \rangle_\Z$ where $(e_i)_j = \delta_{ij}$.
The action of $G$ on $L_G$ is given by multiplication from right on the $e_i$'s.
 Moreover, if $G \cong \mathrm{Gal}(K/F)$ for some finite Galois extension $K/F$
 then $K[L_G] \cong K[x^{\pm 1}_1, \ldots , x^{\pm 1}_n]$, that is the Laurent
 polynomial ring, and $K(L_G)$ which is the quotient field of $K[L_G]$ is isomorphic
 to $K(x_1, \ldots , x_n)$ ($x_i$'s are algebraically independent over $K$) are equipped
 with an action of $G$ as
\begin{itemize}
\item $G$ acts as Galois group on $K$
\item $\forall g \in G, \,\,\, g(x_i) = \prod_{j=1}^{j=n} x_j^{a_{ij}}$  where $a_{ij}$'s
are given by $g(e_i) = \sum_{j=1}^{j=n} a_{ij}e_j$.
\end{itemize}
Also $T_G$ is the corresponding algebraic torus to $L_G$ i.e. $T_G$ is an algebraic
torus defined on $F$ which splits over $K$, with character lattice $L_G$.
\end{definition}
\noindent
By the duality explained before, $K(L_G)^G$ is the rational function field of $T_G$.
From now on we work with finite subgroups of $\mathrm{GL}(n,\Z)$ (up to conjugacy) and
when we consider their corresponding lattice (or algebraic torus), $L_G$ ($T_G$), we
mean the lattice (algebraic torus) defined in Definition \ref{Assumption}.

The first family of of rational algebraic tori is given by the following theorem which
shows that any algebraic torus with a permutation character group is rational.

\begin{theorem}\cite{Speiser}\label{thm:no-name lemma}
(no-name lemma) Let $M$ be a permutation \glat \,\,and $K$ be a \G-field. Then $K(M)^\G$
is rational over $K^\G$.
\end{theorem}

Note that if $\G = \mathrm{Gal} (K/F)$, then $K^\G = F$. For a constructive proof see
\cite{JamLemSch2019}, also a more general version of the no-name lemma see \cite{Domokos}. The following
theorem generalizes the previous result to sign permutation lattices. A $G$-lattice is called
sign permutation if it has a basis $X$ which is permuted up to sign changes under the action of $G$.
\begin{theorem}\cite[Proposition 9.5.1]{Lorenz}\label{thm:signpermrat}
Assume $L$ is a sign permutation (permutation up to sign changes) $G$-lattice and $K$ is a $G$-field. Then $K(M)^G$
is rational over $K^G$.
\end{theorem}
The rationality problem for algebraic tori of dimension one was concrete. For
dimension two Voskresenskii used a geometric method to prove the below result.
\begin{theorem}
\cite{Vos67} Any 2 dimensional algebraic torus over k, is k-rational.
\end{theorem}
The following two theorems classifies algebraic tori of dimension 4 and  5 up to stable
rationality. In \cite{Hoshi} the authors gave a complete classification of mentioned tori,
however they did not say anything about rationality of tori of dimension 4 and 5. The
main idea of their work was to investigate the last 3 results above, by means of computer
algebra system, GAP.
\begin{theorem}\cite[Theorem 1.9]{Hoshi}
Let $K/F$ be a finite Galois extension with Galois Group $\G \leqslant \mathrm{GL}(4, \Z)$.
Assume $G$ acts on $L = K(x_1,x_2,x_3,x_4)$ as above. (For tables of the below subgroups
see \cite[Page 4]{Hoshi})
\\
(i) $L^\G$ is stably $F$-rational if $G$ is (up to conjugacy) one of a list of 487
subgroups of $\mathrm{GL}(4,\Z)$.
\\
(ii)  $L^\G$ is not stably but retract $F$-rational if $G$ is (up to conjugacy) one
of a list of 7 subgroups of $\mathrm{GL}(4,\Z)$.
\\
(iii) $L^\G$ is not retract $F$-rational if $G$ is (up to conjugacy) one of a list of
216 subgroups of $\mathrm{GL}(4,\Z)$.
 \end{theorem}
In 2015, Lemire showed that except for possibly ten, all stably rational groups found
by Hoshi and Yamasaki are rational (see \cite{Nicole1}).
\begin{theorem}\label{thm:Hoshi}\cite[Theorem 1.12]{Hoshi}
Let $K/F$ be a finite Galois extension with Galois Group $\G  \leqslant \mathrm{GL}(5, \Z)$.
Assume G acts on $L = K(x_1,x_2,x_3,x_4,x_5)$ as above. (for tables of below subgroups see
\cite[Pages 134-144]{Hoshi})
\\
(i) $L^\G$ is stably $F$-rational if $G$ is (up to conjugacy) one of a list 3051 subgroups
of $\mathrm{GL}(5,\Z)$.
\\
(ii)  $L^\G$ is not stably but retract $F$-rational if $G$ is (up to conjugacy) one of a list
25 subgroups of $\mathrm{GL}(5,\Z)$.
\\
(iii) $L^\G$ is not retract $F$-rational if $G$ is (up to conjugacy) one of a list of 3003
subgroups of $\mathrm{GL}(5,\Z)$.
 \end{theorem}

There are examples of varieties which are stably rational but not rational. So in general
being stably rational is not the same as being rational. However, there is a conjecture
about the equivalence of stable rationality and rationality for algebraic tori.\\ \\
 \textbf{Conjecture}. \cite[Section 2.6.1]{Voskresenskii} Any stably rational algebraic
 torus is rational.

According to Theorem \ref{thm:Hoshi} we know all stably rational algebraic tori of dimension 5.
However, the theorem does not say anything about rationality of those tori. An interesting
problem is to find all rational tori between 3051 mentioned tori in the theorem.

Later on, we investigate the rationality of stably rational algebraic tori
of dimension 5.  We will try to reduce their rationality to the rationality of some
well understood algebraic torus. Here we present some families of
algebraic tori which are rational, so that we can relate our algebraic tori to one
of these families. It is already mentioned that every $n$ dimensional algebraic torus
has a corresponding finite subgroup of $\mathrm{GL}(n,\Z)$. In order to study the
rationality of algebraic tori, we study its corresponding group. We would rather to
consider maximal groups and prove rationality for their subgroups, instead of proving
it case by case.

Let $L$ be a $G$-lattice for $G\leq \mathrm{GL}(n,\Z)$. Following \cite{Nicole1}, if all algebraic tori with character lattice $L\downarrow_H^G$ and splitting group $H$ are rational, for any
subgroup $H \leq G$, then we call $L$ hereditarily rational.
Moreover, if $T$ is an algebraic torus and $L$ is its corresponding lattice, then $T$ is called
hereditarily rational if $L$ is hereditarily rational.

By Theorem \ref{thm:no-name lemma}, a quasi-split torus is rational. For a permutation
$G$-lattice $L$ and any subgroup $H\leq G$, since $L\downarrow_H^G$ is a permutation
lattice, the corresponding torus to $L\downarrow_H^G$ is rational. In other words a
quasi split torus is hereditarily rational. Simillarly by Theorem \ref{thm:signpermrat}
and above argument for (a sign permutation lattice), we conclude that any algebraic
torus with a sign permutation character lattice is hereditarily rational.
In particular this is true for an algebraic torus with character lattice the root
lattice $\Z B_n$ as an $W(B_n)$-lattice. It is also known that any rank $n$ sign
permutation lattice, is isomorphic to the restriction of $\Z B_n$ to a subgroup
of $W(B_n)$.

The following theorem provides an important tool for our algorithm to determine rationality of a
reducible torus.

\begin{theorem}\cite[Proposition 1.6]{Lenstra}\label{thm:lenstra}
Suppose $P$ is a permutation $G$-lattice and $G$ is the Galois group of a finite
Galois extension, $K/F$.  If
$$0 \longrightarrow M \longrightarrow L \longrightarrow P \longrightarrow 0 $$ is
an exact sequence of $G$-lattices, then $K(L)^G$ is rational over $K(M)^G$.
\end{theorem}

An important corollary of the above theorem will be used frequently in chapter 3,
in order to prove the rationality of algebraic tori.
\begin{corollary}\label{permcoker}
Suppose $P$ is a permutation $G$-lattice and $G$ is the Galois group of a finite
Galois extension, $K/F$.  If $$0 \longrightarrow M \longrightarrow L \longrightarrow P \longrightarrow 0 $$
is an exact sequence of $G$-lattices and $K(M)^G$ is rational over $F$, then $K(L)^G$ is rational over $F$.
\end{corollary}

In \cite[Section 2.4.8]{Voskresenskii} the author has shown that any algebraic
torus with an augmentation ideal lattice is hereditarily rational. More precisely
let $T$ be an algebraic torus defined over $F$ and splits over $K$ and $G = \mathrm{Gal}(K/F)$.
Assume the character lattice of $T$ is $I_X$ (the kernel of the augmentation map),
and $\Z [X]$ is a $G$-permuattion lattice, where
\begin{equation}\label{Chevalley}
0 \longrightarrow I_X \longrightarrow \Z [X] \overset{\varepsilon} \longrightarrow  \Z \longrightarrow 0
\end{equation}
is an exact sequence and $\varepsilon : \Z [X]  \rightarrow \Z $, $x \rightarrow 1$
is the augmentation map. The exact sequence (\ref{Chevalley}) corresponds to the exact
sequence of $F$ algebraic tori
$$0 \longrightarrow \mathbb{G}_m \longrightarrow R_{K_1/F}(\mathbb{G}_m) \times \cdots \times R_{K_t/F}(\mathbb{G}_m) \longrightarrow  T \longrightarrow 0$$
where $K_i/F$ (for $i = 1, \ldots ,t $) are intermediate fields of $K/F$ and $K/K_i$ is
Galois. Now $T = \prod^t_{i = 1}R_{K_i/F}(\mathbb{G}_m)/\mathbb{G}_m$ and is rational.
We note that for any subgroup $H$ of $G$, ${I_X}\downarrow^G_H$ is also an augmentation
ideal. Hence an algebraic tori with augmentation ideal character lattice is hereditarily
rational.

It is worth mentioning that passing to dual lattices in \ref{Chevalley} we get
$$0 \longrightarrow \Z \longrightarrow \Z [X] \overset{\varepsilon} \longrightarrow  J_{X} \longrightarrow 0$$
where $J_{X}= I^{\ast}_X$ is called Chevalley module. The corresponding algebraic
torus to $J_X$ has interesting properties and is called norm one torus. Chevalley
was the first one who discovered that norm one torus is not necessarily rational.

The following lemma was used in \cite{Nicole1} to show that a given $G$-lattice
is isomorphic to $J_{G/H}$.
\begin{lemma}\cite[Remark 4.1]{Nicole1}
Let $L$ be a $G$-lattice. If there exist $x\in L$ such that,
\begin{itemize}
\item $\langle G.x \rangle_{\Z} = L$
\item $\mathrm{Stab}_G(x) = H$
\item $\sum_{g \in G}gx = 0$,
\end{itemize}
then $L \cong J_{G/H}$.
\end{lemma}

\section{Reduction Algorithms}\label{sec:redalg}
Assume $$0 \rightarrow M  \rightarrow L_G \rightarrow N \rightarrow 0$$ is a short
exact sequence of $G$-lattices such that $N$ is a permutation projective $G$-lattice.
If $K/F$ is a finite Galois extension with $G \cong \mathrm{Gal}(K/F)$, then by
Theorem (\ref{thm:lenstra}), $K(L_G)^G$ is rational over $K(M)^G$. Thus, rationality
of $K(M)^G$ over $F$ implies rationality of $K(L_G)^G$ over $F$.

Suppose $L_G$ is an indecomposable $G$-lattice. In this section, we present methods
to examine the possibility of existence of such a short exact sequence for $L_G$, with
 $N$ a permutation $G$-lattice.

Although sign permutation lattices are not permutation projective, constructing a
short exact sequence of $G$-lattices $$0 \rightarrow M  \rightarrow L_G \rightarrow N \rightarrow 0$$
where $N$ is a rank one sign permutation $G$-lattice might help to determine
rationality of the associated algebraic torus to $L_G$. Note that existence of
such a sequence does not directly imply rationality. However, under some conditions
the rationality may be concluded.

The goal of this section is to provide tools to get exact sequences mentioned above
for a given indecomposable $G$-lattice. The idea behind all of the methods is a simple
fact which we explain briefly here.

A lattice $L_G$, is reducible as a $G$-lattice if and only if $\Q L_G = L_G \otimes_{\Z} \Q$
has a proper $\Q[G]$-submodule $W$ of dimension $0 < m < n$. Let $L_G$ be a $G$-lattice of
rank $n$ and $W$ is an $m$ dimensional proper $\Q[G]$-submodule of $\Q L_G$. Then $L_G \cap W$
is a sublattice of $L_G$ of rank $m$ such that $\Q (L_G \cap W) = W$. Then
$$0 \rightarrow L_G \cap W  \rightarrow L_G \rightarrow L_G/(L_G \cap W) \rightarrow 0$$
 is a short exact sequence of $G$-lattices. Note that this implies in particular that $L_G/(L_G \cap W)$
 is torsion free so that a $\Z$-basis of $L_G \cap W$ can be extended to a $\Z$-basis of $L_G$.

In the next paragraphs we are specifically looking for an $n-1$ dimensional proper
$\Q[G]$-submodule of $\Q L_G$.

If we start with the dual lattice $L^*_G$, and we are able to find a rank 1 permutation
sublattice of $L^*_G$, we get $$0 \rightarrow \Z  \rightarrow L^*_G \rightarrow M \rightarrow 0,$$
where $M = L^*_G/\Z$ is of rank $n-1$. Then by dualizing the sequence we have
$$0 \rightarrow M^*  \rightarrow L_G \rightarrow \Z \rightarrow 0$$ as desired.

Now, we explain how to find a permutation rank one sublattice of $L^*_G$. In order
to get a one dimensional $\Q[G]$-submodule of $\Q L^*_G$, we use the eigenspaces of
the transposes of a generating set of $G$. Let  $\lbrace \sigma_1, \ldots, \sigma_m\rbrace$
be the transposes of a generating set of $G$ and let  $G^* = \langle \sigma_1, \ldots, \sigma_m\rangle$.
Suppose $E_{1,\sigma_i}$ is the left nullspace of $\sigma_i-I$ over $\Q$.
We define $$E_1 = E_{1,\sigma_1}\cap \cdots \cap E_{1,\sigma_n}. $$
Note that $G^*$ acts trivially on $E_1$. If $E_1\neq {0}$ then we can choose a
nonzero vector $u \in E_1$. Let $u = (\frac{a_1}{b_1}, \frac{a_2}{b_2}, \ldots , \frac{a_n}{b_n})\in E_1$
such that $\gcd(a_i, b_i) = 1$. If $m = \mathrm{lcm}(b_1, \ldots, b_n)$ then
$m u = (a'_1, \ldots, a'_n) \in\Z^n$. If $\gcd (a'_1, \ldots, a'_n) = d$ then
$v =\frac{m}{d} u \in L_G \cap E_1$ and the $\gcd$ of its entries is 1.

As a consequence, we can extend $\lbrace v \rbrace$ to a $\Z$-basis of $L^*_G$.
A general algorithm to do this extension is given by Magliveras et al in
\cite{LatticeBase}; GAP also has a function which does the job. In most of the
cases that we will see in the next section, $v$ had a $\pm 1$ as an entry, which makes
the basis extension so simple: if $v_j$ is $\pm 1$ then
$\lbrace e_1, \ldots , e_{j-1}, e_{j+1}, \ldots, e_n , v \rbrace$ forms a $\Z$-basis for $L^*_G$.

Since it is possible to extend $v$ to a basis for the lattice $L^*_G$, there exists a change of basis matrix $T$ in $\mathrm{GL}(n,\Z)$ such that
$$T \sigma_i T^{-1} = \left[ \begin{array}{c|c}
\delta_i & \ast\\
\hline 0 & 1
\end{array} \right]
$$
for some $\delta_i \in \mathrm{GL}(n-1,\Z).$ Since we consider the finite subgroups of $\mathrm{GL}(n,\Z)$ up to conjugacy, we can work with $G' = TG^*T^{-1}$. By considering the first $n-1$ vectors of the standard basis of the $G'$-lattice $L_{G'}$ (which is isomorphic to $L^*_G$), can form the $G'$-lattice $M$ such that $L_{G'}/\Z = M$ and we get
$$0 \longrightarrow \Z \longrightarrow L_{G'} \longrightarrow M \longrightarrow 0.$$
By dualizing the sequence we get  $$0 \longrightarrow M^* \longrightarrow L^*_{G'} \longrightarrow \Z \longrightarrow 0.$$
Note that $L^*_{G'}$ is isomorphic to $L_G$.

The explained method above is presented as an algorithm here.
\begin{algorithm}[H]
	\caption{Fixed Point Algorithm}
	\label{alg:Fixed Point}
	\begin{algorithmic}[1]
		\REQUIRE A finite subgroup $G$ of $\mathrm{GL}(n,\Z)$, given by its generators $\lbrace \sigma_1, \ldots , \sigma_m \rbrace$.
		\ENSURE A matrix $T \in \mathrm{GL}(n,\Z)$ such that $T \sigma^t_i T^{-1}= \left[ \begin{array}{c|c}
		\delta_i & \ast \\
		\hline
		0 &1
		\end{array} \right]
		$ where $\delta_i \in \mathrm{GL}(n-1,\Z)$, and sublattices $M, N$ \\
		\hspace{0.8cm}such that $0\longrightarrow N \longrightarrow L^*_G \longrightarrow M \longrightarrow 0$
		is an exact sequence of lattices.
		\bigskip
		\STATE $E \gets \left[ \begin{array}{c|c|c|c}
		\sigma^t_1-I & \sigma^t_2 -I & \cdots & \sigma^t_n -I
		\end{array} \right]$
	    \STATE $W \gets \mathrm{LeftNullspace}(E)$
	    \STATE if $W $ is not zero then\\
	    \hspace{0.5in} choose a nonzero $v \in W$\\
	    \hspace{0.5 in} if $v \notin \Z^n$ \\
	    \hspace{1in} find $c \in \Z$ s.t $cv \in \Z^n$ and $\gcd(cv) = 1$\\
	    \hspace{1in} $v \gets cv$\\
	    \hspace{0.5 in} end if\\
	    \hspace{0.5in} apply the algorithm in \cite{LatticeBase} to extend $v$ to a basis $B = \lbrace \beta_1, \ldots, \beta_{n-1},v\rbrace$ for $L_G$\\
	    \hspace{0.5in} $T \gets \begin{bmatrix}
	    \beta_1 &  \cdots &  \beta_{n-1}& v
	    \end{bmatrix}^t$\\
		\hspace{0.5in} $N \gets \Z v$\\
		\hspace{0.5in} $M \gets L/N$\\
		\hspace{0.5in} \textbf{return} $M,N, T$\\
	  \hspace{0.5in} end if\\
	  else\\
	     \hspace{0.5in}  \textbf{return} fail\\
	     end if
	 	\end{algorithmic}
\end{algorithm}
\begin{remark}
If the algorithm returns a matrix $T$ then for $\sigma \in \lbrace \sigma^t_1, \ldots , \sigma^t_m \rbrace$,
$$T \sigma T^{-1} = \sigma'$$ where $$\sigma' = \left[ \begin{array}{c|c}
\delta & \ast \\
\hline
0 &1
\end{array}\right].
$$
for some $\delta \in \mathrm{GL}(n-1, \Z)$. More precisely $$T \sigma = \sigma'T$$ and the last row of $T \sigma$ is nothing but $v \sigma = v$. This implies that the last row of $\sigma' = [0 \,\, \ldots \,\, 0 \,\, 1]$.
\end{remark}
\begin{example}
Let $G\leq \mathrm{GL}(4,\Z)$ be generated by
$$ \left[ \begin {array}{cccc} 0&-1&0&0\\ -1&-1&-1&-1
\\ 1&1&1&0\\ 1&1&0&1\end {array}
 \right]
\tand
\left[ \begin {array}{cccc} 0&1&0&0\\ -1&0&0&0
\\ 0&-1&0&-1\\ 0&-1&-1&0
\end {array} \right].
$$
The transposes are
$$
 \sigma = \left[ \begin {array}{cccc} 0&-1&1&1\\ -1&-1&1&1
\\ 0&-1&1&0\\ 0&-1&0&1\end {array}
 \right]
 \tand
\tau =  \left[ \begin {array}{cccc} 0&-1&0&0\\ 1&0&-1&-1
\\ 0&0&0&-1\\ 0&0&-1&0\end {array}
 \right].
$$
Then the $E_{1,\sigma}$ is the left nullspace of $\sigma - I_4$.
One can verify that
\begin{displaymath}
\lbrace \begin{bmatrix}
1& -1 &1 &0
\end{bmatrix}, \begin{bmatrix}
1& -1 &0 &1
\end{bmatrix}
\rbrace
\end{displaymath}
is a basis for $E_{1,\sigma}$. Similarly $E_{1,\tau}$ is the left nullspace of $\tau - I_4$
 which is generated by $\begin{bmatrix}
0& 0 &-1 &1
\end{bmatrix}$.\\
\\It is not hard to see $\begin{bmatrix}
0& 0 &-1 &1
\end{bmatrix}\in E_{1,\sigma}$. Thus $$E_1 = E_{1,\sigma} \cap E_{1,\tau} = \langle \begin{bmatrix}
0& 0 &-1 &1
\end{bmatrix} \rangle$$
and $ \begin{bmatrix}
0& 0 &-1 &1
\end{bmatrix}\in L$. Now $$\lbrace  \begin{bmatrix}
1& 0 &0 &0
\end{bmatrix} ,  \begin{bmatrix}
0& 1 &0 &0
\end{bmatrix},  \begin{bmatrix}
0& 0 &1 &0
\end{bmatrix},  \begin{bmatrix}
0& 0 &-1 &1
\end{bmatrix} \rbrace$$ forms a $\Z$-basis for $L$. The change of basis matrix $T$ is
$$
\begin{bmatrix}
1 & 0 &0 &0\\
0 & 1 &0 &0\\
0 & 0 &1 &0\\
0 & 0 &-1 &1
\end{bmatrix}
$$ Hence
$$
T \sigma T^{-1}= \left[ \begin {array}{ccc|c} 0&-1&2&1\\ -1&-1&2&1
\\ 0&-1&1&0\\ \hline 0&0&0&1\end {array}
 \right], \,\,\,
T\tau T^{-1}=  \left[ \begin {array}{ccc|c} 0&-1&0&0\\ 1&0&-2&-1
\\ 0&0&-1&-1\\ \hline 0&0&0&1\end {array}
 \right]
$$
Now by dualizing we get
$$
 \left[ \begin {array}{ccc|c} 0&-1&0&0\\ -1&-1&-1&0
\\ 2&2&1&0\\ \hline 1&1&0&1\end {array}
 \right]
 \tand
 \left[ \begin {array}{ccc|c} 0&1&0&0\\ -1&0&0&0
\\ 0&-2&-1&0\\ \hline 0&-1&-1&1
\end {array} \right]
$$
as a generating set for a conjugate of $G$. By defining $M\subset L_G$ generated by $e_1, e_2$ and $e_3$ the following exact sequence will be obtained
$$
0 \longrightarrow M \longrightarrow L_G \longrightarrow \Z \longrightarrow 0
$$
\end{example}
We call the above process the fixed point algorithm. One can generalize it as follows. Assume $E_{\lambda,\sigma}$ be the left kernel of $\sigma-\lambda I$ over the  rationals. Now define $E_{\pm 1,\sigma}$ to be  the set $\lbrace E_{1,\sigma}, E_{-1,\sigma}\rbrace$. Assume $E_{\pm 1, G}$ is the Cartesian product of $E_{\pm1, \sigma}$ for all $\sigma \in G^*$, i.e.
$$E_{\pm 1,G} = E_{\pm 1, \sigma_1} \times \cdots \times E_{\pm 1, \sigma_n}.$$
If there exist $A\in E_{\pm 1,  G}$ such that $W = \bigcap\limits_{B \in A} B \neq 0$, then we can find a nonzero $v \in L^*_G \cap W$. As we have seen in the fixed point algorithm, we can extend a multiple of $v$ to a $\Z$-basis for $L^*_G$. Then we can get a change of basis matrix $T$, such that
$$T \sigma_i T^{-1} = \left[ \begin{array}{c|c}
\delta_i & \ast\\
\hline
0 & \pm 1
\end{array}\right]
$$
for some $\delta_i \in \mathrm{GL}(n-1,\Z).$ Thus by a similar argument we can use the new representative of the conjugacy class of $G^*$ to form an equivalent lattice and similarly by choosing the first $n-1$ elements of the standard basis of $L_{G'}$ we can produce
$$0 \longrightarrow \Z^- \longrightarrow L_{G'} \longrightarrow M \longrightarrow 0.$$
By dualizing the sequence we get
$$0 \longrightarrow M^* \longrightarrow L^*_{G'} \longrightarrow \Z^- \longrightarrow 0.$$
Again note that $L^*_{G'}$ is isomorphic to $L_G$.\\
\\
This process will be called the sign fixed point algorithm and it is presented as an algorithm here.
\begin{algorithm}[H]
	\caption{Sign Fixed Point Algorithm}
	\label{alg:Sign Fixed Point}
	\begin{algorithmic}[1]
		\REQUIRE A finite subgroup $G$ of $\mathrm{GL}(n,\Z)$, given by its generators $\lbrace \sigma_1, \ldots , \sigma_m \rbrace$.
		\ENSURE A matrix $T \in \mathrm{GL}(n,\Z)$ such that $T \sigma^t_i T^{-1}= \left[ \begin{array}{c|c}
		\delta_i & \ast \\
		\hline
		0 &1
		\end{array} \right]
		$ where $\delta_i \in \mathrm{GL}(n-1,\Z)$, and sublattices $M, N$\\
		\hspace{.8 cm}such that $0\longrightarrow N \longrightarrow L^*_G \longrightarrow M \longrightarrow 0$ is an exact sequence of lattices.
		\bigskip
		\STATE	 for g in $\lbrace	\sigma^t_1, \ldots , \sigma^t_m \rbrace$ do\\
		\hspace{0.5 in} $E_g \gets $ the set of left nullspaces of $g \pm I$ over $\Q$\\
		end do
		\STATE $E \gets E_{\sigma^t_1} \times E_{\sigma^t_2} \times \cdots \times E_{\sigma^t_m}$
		\STATE $W \gets 0$
		\STATE while $W = 0$ and $E \neq \emptyset$ do\\
		\hspace{0.5in} $A$ $\gets$ a random element of $E$ \\
	    \hspace{0.5in}$W \gets \bigcap\limits_{a\in A}a$\\
	      \hspace{0.5in}$E \gets E \setminus A$\\
	    end do
	    \STATE if $W $ is not zero then\\
	    \hspace{0.5in} choose a nonzero $v \in W$\\
	    \hspace{0.5 in} if $v \notin \Z^n$ \\
	    \hspace{1in} find $c \in \Z$ s.t $cv \in \Z^n$ and $\gcd(cv) = 1$\\
	    \hspace{1in} $v \gets cv$\\
	    \hspace{0.5 in} end if\\
	    \hspace{0.5in} apply the algorithm in \cite{LatticeBase} to extend $v$ to get a basis $B = \lbrace \beta_1, \ldots, \beta_{n-1},v\rbrace$ for $L$\\
	    \hspace{0.5in} $T \gets \begin{bmatrix}
	    \beta_1 &  \cdots &  \beta_{n-1}& v
	    \end{bmatrix}^t$\\
		\hspace{0.5in} $N \gets \Z v$\\
		\hspace{0.5in} $M \gets L/N$\\
		\hspace{0.5in} \textbf{return} $M,N,T$\\
	  \hspace{0.5in} end if\\
	  else\\
	     \hspace{0.5in}  \textbf{return} fail\\
	     end if
	 	\end{algorithmic}
\end{algorithm}

\begin{example}
Let $G\leq \mathrm{GL}(4,\Z)$ be generated by
$$
\left[ \begin {array}{cccc} -1&1&0&1\\ 0&0&0&1
\\ -1&0&1&1\\ 0&1&0&0\end {array}
 \right]
\tand
\left[ \begin {array}{cccc} 0&1&0&0\\ 0&0&1&0
\\ 0&0&0&-1\\ -1&1&0&1\end {array}
 \right].
$$
The transposes are
$$
\sigma = \left[ \begin {array}{cccc} -1&0&-1&0\\ 1&0&0&1
\\ 0&0&1&0\\ 1&1&1&0\end {array}
 \right]
\tand
\tau =  \left[ \begin {array}{cccc} 0&0&0&-1\\ 1&0&0&1
\\ 0&1&0&0\\ 0&0&-1&1\end {array}
 \right].
$$
By computing the left nullspaces ($\sigma \pm I_4$ and $\tau \pm I_4$) we get
$$E_{1,\sigma} =\langle [ 0, 0, 1, 0 ], [ 1, 1, 0, 1 ] \rangle, \,\,\,
E_{-1,\sigma} =\langle [ [ 2, 0, 1, 0 ], [ 1, -1, 0, 1 ] ] \rangle $$
$$E_{1,\tau} =\langle [ -1, -1, -1, 1 ]  \rangle, \,\,\,
E_{-1,\tau} =\langle [ 1, -1, 1, 1 ]\rangle. $$
So
$$ E_{1,\sigma} \cap E_{1,\tau}= {0}, \,\,\,
 E_{1,\sigma} \cap E_{-1,\tau}= {0} $$
$$ E_{-1,\sigma} \cap E_{1,\tau}= \langle [ 1, 1, 1, -1 ] \rangle, \,\,\,
 E_{-1,\sigma} \cap E_{-1,\tau}= {0}. $$
Let $W = E_{-1,\sigma} \cap E_{1,\tau}.$ So $[ 1, 1, 1, -1 ] \in L \cap W$ is extendable to a basis for $L$ by vectors
$$[ 1, 0, 0, 0 ] \, , \, [ 0,1,0,0 ] \tand [ 0,0,1,0] $$
and the corresponding transformation is
$$
T = \begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
1 & 1 & 1 & -1
\end{bmatrix}
$$
$$
\sigma' = T\sigma T^{-1} = \left[ \begin {array}{cccc} 0&1&-1&-1\\ 1&0&1&0
\\ 0&0&1&1\\ 0&0&0&-1\end {array}
 \right]
, \,\,\,
\tau' = T \tau T^{-1} =\left[ \begin {array}{cccc} 0&0&-1&0\\ 1&0&1&0
\\ 1&1&0&-1\\ 0&0&0&1\end {array}
 \right].
$$
Now by dualizing we get
$$
 \left[ \begin {array}{ccc|c} 0&1&0&0\\ 1&0&0&0
\\ -1&1&1&0\\ \hline -1&0&1&-1
\end {array} \right]
 \tand
 \left[ \begin {array}{ccc|c} 0&1&1&0\\ 0&0&1&0
\\ -1&1&0&0\\ \hline 0&0&-1&1\end {array}
 \right]
$$
as a generating set for a conjugate of $G$. By defining $M\subset L_G$ generated by $e_1, e_2$ and $e_3$ the following exact sequence will be obtained
$$0 \longrightarrow M \longrightarrow L_G \longrightarrow \Z^- \longrightarrow 0.$$
\end{example}

In general to get a $\Q[G]$-submodule $W$ of $\Q L$, where $L = L_G$ is a $G$-lattice, one can use the decomposition of $\Q L$ (provided that it is decomposable). It is not always easy to find the decomposition of a $\Q[G]$-module. The most well-known tool for module decomposition is the meataxe algorithm. The algorithm was first introduced by Parker in \cite{Parker} in order to check irreducibility of a finite dimensional module over a finite field and finding explicit submodules in case of reducibility. Later on Parker extended the idea of the meataxe algorithm to characteristic zero (see \cite{Parker2}). His algorithm can be used to decompose an integral representation of a finite group. In \cite{Plesken2}, the authors provided machinery which enables us to decompose $\Q[G]$-modules up to dimension 200. So there are algorithms which give the decomposition over $\Q$. We invite the reader to see \cite{Lux} and \cite{Holt} for more details.

Assume $G \leq \mathrm{GL}(n,\Z)$ is finite and $\Q L = L \otimes_{\Z} \Q$ is the $\Q$-vector corresponding space to $L$. If $\Q L$ is a decomposable $\Q[G]$-module, then there exists a change of basis matrix such that generators of the $\Q$-class of $G$ can be written as block diagonal matrices
$$
T \sigma_i T^{-1} =  \left[ \begin{array}{c|c}
\delta_i & 0 \\
\hline
0 & \gamma_i
\end{array} \right]
$$
where $\delta_i	\in \mathrm{GL}(m,\Q)$ and $ \gamma_i \in \mathrm{GL}(m',\Q)$ for some $m, m' \in \Z$. Let $\lbrace e_1, \ldots, e_m, e_{m+1}, \ldots , e_{m+m'}\rbrace$ be the standard basis for $\Q L$. The $\Q M$ and $\Q N$ generated respectively by $\lbrace e_1, \ldots, e_m \rbrace$ and
$\lbrace e_{m+1}, \ldots , e_{m+m'} \rbrace$ are invariant (set wise) under the action of $G$ and $\Q L = \Q M \oplus \Q N$. Now $T^{-1}(\Q M)$ is a $G$-stable subspace and $$M = L \cap T^{-1}(\Q M) $$ is $G$ stable. Then we get
$$0 \longrightarrow M \longrightarrow L \longrightarrow L/M \longrightarrow 0$$
as an exact sequence of lattices.

The above idea can be turned into an algorithm. In order to do so, one need to compute the decomposition of $\Q L$ (meataxe or any other algorithm can be applied). If in the previous step the change of basis, namely $T$, to get the decomposition is not computed, it should be done next. The next step is to choose a component of the decomposition, say $\Q M$ and a basis of it. After that, $M = T^{-1}(\Q M) \cap L$ is a sublattice of $L$. The last step is to extend a basis of $M$ to $L$ (see \cite{LatticeBase} for an algorithm).
\begin{example}\label{exmp:Decompose}
Consider the group $G$ generated by
% [4,149,1]
$$
\sigma = \left[\begin{array}{rrrrr}
0 & 0 & -1 & 0 & 1 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & -1 & -1 \\
0 & 0 & -1 & 0 & 0
\end{array}\right]
 \tand
\tau = \left[\begin{array}{rrrrr}
-1 & 0 & 0 & -1 & 0 \\
0 & 0 & -1 & 0 & 1 \\
0 & 0 & 0 & 0 & -1 \\
1 & -1 & 0 & 0 & -1 \\
-1 & 0 & 0 & 0 & 0
\end{array}\right].
$$
%The $\Q$-class of $G$ is accessible in the list of rank 4 groups in \cite{Hoshi} by the name cryst4[149].
The generators of the $\Q$-class are
$$
\sigma' =\left[\begin{array}{rrr|rr}
0 & 0 & 1 & 0 & 0 \\
1 & -1 & 1 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 \\
\hline
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0
\end{array}\right]
 \tand
\tau' = \left[\begin{array}{rrr|rr}
1 & -1 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 \\
\hline
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & -1 & -1
\end{array}\right].
$$
Considering the orders of matrices we can figure out there exist an invertible integral matrix $T$ such that
$$T\sigma T^{-1}= \sigma'$$
$$T\tau T^{-1} = \tau'.$$
Assume 25 indeterminates $t_{00}  ,\ldots , t_{44}$ and the matrix $T=\left[t_{ij}\right]_{0 \leq i,j \leq 4}$
Then
$$
T\sigma= \sigma' T
, \,\,\,
T\tau= \tau' T
$$
The transformation $T$ can be found by solving (and replacing parameters) the linear system obtained from above equations as
$$
T =\left[\begin{array}{rrrrr}
-1 & 1 & -2 & 1 & 1 \\
-1 & 2 & -1 & 0 & -1 \\
0 & -1 & -1 & -1 & 0 \\
1 & -2 & 1 & 0 & -1 \\
1 & 0 & -1 & 0 & 1
\end{array}\right].
$$
Let $\Q M = \langle e_1, e_2 , e_3 \rangle_\Q$.
Hence $$T^{-1}(\Q M) = \langle   \left[-\frac{1}{4},\,\frac{1}{4},\,-\frac{1}{4},\,\frac{1}{4},\,\frac{3}{4}\right], \left[-\frac{1}{4},\,\frac{1}{4},\,-\frac{1}{4},\,-\frac{1}{4},\,\frac{1}{4}\right],  \left[-\frac{1}{4},\,-\frac{1}{4},\,-\frac{1}{4},\,-\frac{1}{4},\,-\frac{1}{4}\right]\rangle_\Q .$$
Now
$$M = L \cap T^{-1}(\Q M) = \langle \left[\begin{array}{ccccc}1&0&1&0&-1\end{array}\right], \left[\begin{array}{rrrrr}
0 & 1 & 0 & 0 & 1
\end{array}\right], \left[\begin{array}{rrrrr}
0 & 0 & 0 & 1 & 1
\end{array}\right] \rangle_\Z$$
By extending the above basis of $M$ to a basis of $L$ (by adding $[0,0,1,0,0]$ and $[0,0,0,0,1]$) and forming the change of basis matrix we get
$$
S =  \left[\begin{array}{rrrrr}
1 & 0 & 1 & 0 & -1 \\
0 & 1 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1
\end{array}\right]
$$
 which gives
$$
 S
\left[\begin{array}{rrrrr}
0 & 0 & -1 & 0 & 1 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & -1 & -1 \\
0 & 0 & -1 & 0 & 0
\end{array}\right]
S^{-1}
=
\left[\begin{array}{rrr|rr}
0 & 1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 \\
\hline
0 & 1 & 0 & 0 & -1 \\
0 & 0 & 0 & -1 & 0
\end{array}\right]
$$
$$
 S
\left[\begin{array}{rrrrr}
-1 & 0 & 0 & -1 & 0 \\
0 & 0 & -1 & 0 & 1 \\
0 & 0 & 0 & 0 & -1 \\
1 & -1 & 0 & 0 & -1 \\
-1 & 0 & 0 & 0 & 0
\end{array}\right]
S^{-1}
=
\left[\begin{array}{rrr|rr}
0 & 0 & -1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 \\
\hline
0 & 0 & 0 & 0 & -1 \\
-1 & 0 & 0 & 1 & -1
\end{array}\right]
$$
Now we can get the following exact sequence of lattices
$$0 \longrightarrow M \longrightarrow L \longrightarrow L/M \longrightarrow 0.$$
\end{example}

\section{Results}\label{sec:results}
The 18 maximal indecomposable stably rational lattices found in \cite{Hoshi} are divided into 4 families. The first family are the ones interpreted as lattices of root systems which are hereditarily rational. The second family contains all lattices on which Algorithm \ref{alg:Fixed Point} will not fail. The third family contains lattices on which  Algorithm \ref{alg:Sign Fixed Point} does not fail while Algorithm \ref{alg:Fixed Point} fails. The last family contains all lattices on which either, both Algorithms \ref{alg:Fixed Point} and \ref{alg:Sign Fixed Point} fail but still the general idea of reduction works, or they are irreducible.
All lattices of the first family are hereditarily rational. The second family contains lattices of which after reduction, rationality of the reduced component is unknown. By arguments in the previous sections we have proved the following theorem.
\begin{theorem}
The groups presented in Table \ref{Tbl:Rational} are hereditarily rational.
\end{theorem}
 \begin{table}
\centering
\begin{tabular}{lllll}
CARAT ID & Group Structure & $\#G$ & Description.\\\hline
$(5,942,1)$ & ${\rm Imf}(5,1,1)$ & $3840$ & The root lattice of $B_5$ \\
 $(5,953,4)$ & $\mathrm{S}_6$ & $720$ &  The root lattice of $A_5$ \\
 $(5,726,4)$ & $C_2^4\rtimes \mathrm{S}_4$ & $384$   & reduced component $[4,32,21,1]$  \\
$(5,911,4)$ & $\mathrm{S}_5$ & $120$  & reduced component $[ 4, 31, 4, 1 ]$  \\
 $(5,341,6)$ & $D_8\times \mathrm{S}_3$ & $48$  & reduced component $[ 4, 20, 17, 2 ]$ &\\
 $(5,531,13)$ & $C_2\times \mathrm{S}_4$ & $48$  & reduced component $[ 4, 25, 9, 2 ]$ &\\
 $(5,245,12)$ & $C_2^2\times \mathrm{S}_3$ & $24$ &reduced component $[ 4, 14, 10, 2 ]$&
\end{tabular}
\caption{Hereditarily rational groups among the maximal $18$ groups
found in \cite{Hoshi}.}
\label{Tbl:Rational}
\end{table}
\noindent
The exceptional cases of the second family are presented in Table \ref{Tbl:RationalityUnknown}. In each case the reduced component is stably rational as proved in \cite{Hoshi}. Their rationality is unknown yet. In \cite{Nicole1} the author has proved that subgroups of $[ 4, 25, 8, 5 ]$ are rational except for possibly $$[4, 6, 2, 11], [4, 12, 4, 13], [4, 13, 2, 6], [4, 13, 3, 6], [4, 13, 7, 12], [4, 24, 4, 6], [4, 25, 4, 5], [4, 25, 8, 5].$$
There will be precisely one subgroup of $G_{14}$ with each of the dimension 4 reduced components in the above list, so except for possibly those subgroups of $G_{14}$ the rest are rational. The exceptional cases are presented in Table \ref{ExceptionalG14}.

From the above list $$[4, 6, 2, 11], [4, 12, 4, 13], [4, 13, 2, 6], [4, 13, 3, 6], [4, 13, 7, 12]$$ are subgroups of $[4, 13, 7, 12]$ which means we have the same problem for cases $G_{17}$ and $G_{18}$. Hence except for possibly the subgroups of $G_{17}$ and $G_{18}$ associated to above list their rest of subgroups are rational. For the exceptional cases see Table \ref{ExceptionalG17} and Table \ref{ExceptionalG18}.
 \begin{table}
\centering
\begin{tabular}{lllll}
CARAT ID & Group Structure & $\#G$ & Description.\\\hline
 $(5,533,8)$ & $C_2\times \mathrm{S}_4$ & $48$  & reduced component $[ 4, 25, 8, 5 ]$ &\\
 $(5,81,42)$ & $C_2\times D_8$ & $16$  & reduced component $[ 4, 13, 7, 12 ]$&\\
 $(5,81,48)$ & $C_2\times D_8$ & $16$  & reduced component $[ 4, 13, 7, 12 ]$ &
\end{tabular}
\caption{Groups among 18 maximals which are reduced but rationality of rank 4 sublattice is unknown}
\label{Tbl:RationalityUnknown}
\end{table}
\begin{table}
\centering
\begin{tabular}{llll}
CARAT ID & Group Structure & $\#G$ & Description.\\\hline
$( 5, 32, 52)$ &	$C_2 \times C_2 \times C_2$& 8 &reduced component 	[4, 6, 2, 11]\\
$( 5, 99, 53 )$ &	$D_8$ & 8&	reduced component [4, 12, 4, 13]\\
$( 5, 103, 22 )$ & $C_4 \times C_2$& 8 &reduced component [4, 13, 2, 6]\\
$( 5, 98, 22 )$& 	$D_8$ & 8&	reduced component [4, 13, 3, 6]\\
$( 5, 81, 50 )$ &		$C_2 \times D_8$ & 16& reduced component [4, 13, 7, 12]\\
$( 5, 522, 15)$ &	$\mathrm{S}_4$& 24&	reduced component 	[4, 24, 4, 6]\\
$( 5, 521, 15 )$ &		$\mathrm{S}_4$ &24&		reduced component [4, 25, 4, 5]\\
$( 5, 533, 8 )$&		$C_2 \times	 \mathrm{S}_4$ & 48&		reduced component [4, 25, 8, 5]
\end{tabular}
\caption{Subgroups of $G_{14}$ that have associated tori which are stably rational but whose rationality is unknown.}
\label{ExceptionalG14}
\end{table}
\begin{table}
\centering
\begin{tabular}{llll}
CARAT ID & Group Structure & $\#G$ & Description.\\\hline
$(5, 32, 49)$ &	$C_2 \times C_2 \times C_2$& 8 &reduced component [4, 6, 2, 11]\\
$( 5, 99, 52)$ &	$D_8$ & 8&	reduced component 	 [4, 12, 4, 13]\\
$(  5, 103, 16)$ & $C_4 \times C_2$& 8 &		reduced component  [4, 13, 2, 6]\\
$(  5, 98, 16 )$& 	$D_8$ & 8&		reduced component  [4, 13, 3, 6]\\
$(  5, 81, 42 )$ &		$C_2 \times D_8$ & 16& reduced component  [4, 13, 7, 12]
\end{tabular}
\caption{Subgroups of $G_{17}$ that have associated tori which are stably rational but whose rationality is unknown.}
\label{ExceptionalG17}
\end{table}
\begin{table}
\centering
\begin{tabular}{llll}
CARAT ID & Group Structure & $\#G$ & Description.\\\hline
$(5, 32, 46)$ &	$C_2 \times C_2 \times C_2$& 8 &reduced component [4, 6, 2, 11]\\
$( 5, 99, 54 )$ &	$D_8$ & 8&		reduced component  [4, 12, 4, 13]\\
$( 5, 103, 24 )$ & $C_4 \times C_2$& 8 &	reduced component 	 [4, 13, 2, 6]\\
$( 5, 98, 24)$& 	$D_8$ & 8&		reduced component  [4, 13, 3, 6]\\
$( 5, 81, 48 )$ &		$C_2 \times D_8$ & 16&reduced component  [4, 13, 7, 12]
\end{tabular}
\caption{Subgroups of $G_{18}$ that have associated tori which are stably rational but whose rationality is unknown.}
\label{ExceptionalG18}
\end{table}
\begin{theorem}
All subgroups of $G_{14},G_{17}$ and $G_{18}$ are rational except for possibly the subgroups in Table \ref{ExceptionalG14}, Table \ref{ExceptionalG17}  and Table \ref{ExceptionalG18}.
\end{theorem}
There are 4 cases, namely $G_4, G_5,G_6$ and $G_{11}$, in which after the reduction we get a rank one sign permutation lattice (more information is given in Table \ref{SignPerm}). The same also happened in some subgroups of irreducible lattices (see Table \ref{SubIrreducible}). It is possible that these groups are hereditarily rational, but we do not currently have a proof. One possible approach to prove rationality in these cases may be the following argument.

The following table summarizes the information about reduction of lattices in the third family.
 \begin{table}[H]
\centering
\begin{tabular}{lllll}
 CARAT ID & $G$ & $\#G$ & Description.\\\hline
 $(5,919,4)$ & $C_2\times \mathrm{S}_5$ & $240$  &  reduced component $[4,31,7,1]$ \\
 $(5,801,3)$ & $C_2\times (\mathrm{S}_3^2\rtimes C_2)$ & $144$  &reduced component $[ 4, 29, 9, 2 ]$  \\
 $(5,655,4)$ & $D_8^2\rtimes C_2$ & $128$  & reduced component $[4,32,17,1]$    \\
 $(5,337,12)$ & $D_8\times \mathrm{S}_3$ & $48$  &  reduced component $[ 4, 20, 20, 4 ]$   &\\
\end{tabular}
\caption{The groups corresponding to maximal stably rational tori of dimension 5 whose associated lattices are indecomposable and have a rank 1 sign quotient.}
\label{SignPerm}
\end{table}
For the last family we have considered their maximal subgroups and we could not decide about the rationality of the groups presented in the following table.
\begin{theorem}\label{SubG8}
All groups in Table \ref{IrreducibleSubs} are hereditarily rational. That is, all subgroups of $G_8$, $G_9$, $G_{10}$ and $G_{15}$ except for possibly the subgroups in Table \ref{SubIrreducible} are hereditarily rational.
\end{theorem}
A proof of Theorem \ref{SubG8} is provided in Appendix A.
\begin{center}
\begin{table}
\begin{tabular}{cccc}
 Carat ID  & $G$ & $\#G$ & Description\\
\hline
 $[5,173,4]$ & $\mathrm{S}_3$ & 6 &  reduced comp. $[ 4, 17, 1, 1 ]$,  rank 1 sign perm. quot. \\
 $[5,391,4]$ &$D_{12}$ & $12$ &  reduced comp. $ [ 4, 21, 3, 1 ]$  rank 1 sign perm. quot. \\
 $[5,461,4]$ &$C^2_2 \times \mathrm{S}_3$ & 24 & reduced comp.$[3,6,7,1]$, quot $[2,4,4,1]$  \\
 $[5,580,4]$ & $A_4$ & $12$ &  reduced comp.$[3,7,1,1]$, quot $[ 2, 4, 1, 1 ]$\\
 $[5,606,4]$ & $C_2 \times A_4$ & 24 &reduced comp. $[3,7,2,1]$, quot $[ 2, 4, 1, 1 ]$ \\
 $[5,607,4]$ & $\mathrm{S}_4$ & 24 &reduced comp. $[3,7,4,1]$, quot $[ 2, 4, 2, 1 ]$ \\
 $[5,607,9]$ &$\mathrm{S}_4$ & 24 & reduced comp. $ [ 3, 7, 4, 1 ]$, quot $[ 2, 4, 2, 2 ]$ \\
 $[5,608,4]$ &$\mathrm{S}_4$ & 24 & reduced comp.t $[ 3, 7, 3, 1 ]$ , quot $[ 2, 4, 2, 1 ]$ \\
 $[5,917,3]$ &$C_5 \rtimes C_4$ & 20 &  reduced comp.  $[ 4, 31, 1, 1 ]$  rank 1 sign perm. quot. \\
 $[5,917,4]$ &$C_5 \rtimes C_4$ & 20 &  reduced comp.  $[ 4, 31, 1, 1 ]$   rank 1 sign perm. quot. \\
 $[5,623,4]$ &$C_2 \times \mathrm{S}_4$ & 48 & reduced comp. $[3,7,5,1]$, quot $[ 2, 4, 2, 1 ]$ \\
 $[5,952,2]$ &$A_5$ & 60 & absolutely irreducible \\
 $[5,952,4]$ &$A_5$ & 60 & absolutely irreducible \\
 $[5,946,2]$ &$\mathrm{S}_5$ & 120 & absolutely irreducible \\
 $[5,946,4]$ &$\mathrm{S}_5$ & 120 & absolutely irreducible \\
 $[5,947,2]$ &$\mathrm{S}_5$ & 120 & absolutely irreducible
\end{tabular}
\caption{Subgroups of $G_8, G_9, G_{10}$ and $G_{15}$ that have associated tori which are stably rational but whose rationality is unknown.}
\label{SubIrreducible}
\end{table}
\end{center}
The following table present the reduced components of the subgroups mentioned in Theorem \ref{SubG8}.
\begin{table}
\begin{tabular}{llllll}
Number & CARAT ID & $G$ & $\#G$ & Description.\\\hline
1 & $[5,6,3]$ & $C_2$ & 2  & [ 4, 2, 2, 2 ] \\
2 & $[5,18,28]$ & $C_2 \times C_2$  &  4& [ 4, 4, 3, 4 ] \\
3 & $[5,19,14]$ &  $C_2 \times C_2$  &  4  & [ 4, 5, 1, 10 ]  \\
4 & $[5,22,14]$ & $C_2 \times C_2 \times C_2$  & 8 & [ 4, 6, 1, 9 ] \\
5 & $[5,57,8]$ & $C_4$ &  4 & [ 4, 7, 1, 2 ] \\
6 & $[5,81,54]$ & $C_2 \times D_8$  & 16 & [ 4, 13, 7, 5 ]  \\
7 & $[5,98,28]$ & $D_8$ & 8 & [ 4, 13, 3, 3 ] \\
8 & $[5,99,57]$ & $D_8$ & 8  & [ 4, 12, 4, 7 ]   \\
9 & $[5,164,2]$ & $C_3$ & 3 & [ 4, 11, 1, 1 ]  \\
10 & $[5,174,2]$ & $\mathrm{S}_3$ & 6  & [ 4, 17, 1, 3 ] \\
11 & $[5,174,5]$ & $\mathrm{S}_3$ & 6 & [ 4, 17, 1, 2 ] \\
12 & $[5,389,4]$ & $D_{12}$ &  12 &  [ 4, 21, 3, 2 ] \\
13 & $[5,901,3]$ & $D_{10}$  & 10 & [ 4, 27, 3, 1 ] \\
14 & $[5,918,4]$ & $C_5 \rtimes C_4$ & 20 & [4, 31, 1, 2]
\end{tabular}
\caption{Hereditarily rational subgroups of $G_8, G_9, G_{10}$ and $G_{15}$.}
\label{IrreducibleSubs}
\end{table}
\begin{remark}
The union of the set of groups in Table \ref{SubIrreducible} and the set of subgroups of the groups in Table \ref{IrreducibleSubs}, is the set of all subgroups of $G_8, G_9, G_{10}$ and $G_{15}$.
\end{remark}
Appendix B presents tables of conjugacy classes of indecomposable subgroups of $\mathrm{GL}(5,\Z)$ which correspond to stably rational tori of dimension 5 from Hoshi and Yamasaki's list. From this list, those stably rational tori of dimension 5 whose rationality is unknown are listed.

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\begin{document}

\title{Effective action of spinor electrodynamics}

\date{}

%\author[1,4]{Alexander A. Andrianov\thanks{a.andrianov@spbu.ru} }
%\author[2]{Chen Lan\thanks{stlanchen@yandex.ru} }
%\author[1]{Oleg O. Novikov\thanks{o.novikov@spbu.ru}}
%\author[3]{Yi-Fan Wang\thanks{yfwang@thp.uni-koeln.de}}
%\author[5]{Zhongyi Zhang\thanks{zhongyi@th.physik.uni-bonn.de}}

%\affil[1]{Saint-Petersburg State University, St. Petersburg 198504, Russia}
%\affil[2]{ELI-ALPS, ELI-Hu NKft, Dugonics t\'er 13, Szeged 6720, Hungary}
%\affil[3]{Institut f\"ur Theoretische Physik, Universit\"at zu K\"oln,
%Z\"ulpicher Stra\ss e 77, 50937 K\"oln, Germany}
%\affil[4]{Institut de Ci\`encies del Cosmos (ICCUB), Universitat de Barcelona,
%Spain}
%\affil[5]{Physikalisches Institut, Universit\"at Bonn, Nussallee 12, 53115
%Bonn, Germany}

\maketitle

%\input{./sections/0abstract}
%\abstract{}

%\tableofcontents

\section{Spinor electrodynamics in flat space-time}

Maxwell Lagrangian
\begin{equation}
\sfun{S_\text{Maxwell}}{A_\mu} \coloneqq \int\dif^{d+1} x\,\rbr{
-\frac{1}{4} F^{\mu\nu} F_{\mu\nu} }
\end{equation}

Dirac Lagrangian \citep[sec.~11]{weinberg1995}
\begin{equation}
\sfun{S_\text{Dirac}}{\psi, \bar\psi} \coloneqq \int\dif^{d+1} x\,
\cbr{-\bar{\psi} \rbr{\gamma^\mu\partial_\mu + m}\psi}.
\end{equation}

Interaction term
\begin{equation}
\sfun{S_\text{IDM}}{A_\mu, \psi, \bar\psi} \coloneqq \int\dif^{d+1}
x\,\rbr{-\bar{\psi}\gamma^\mu\mi e A_\mu\psi}.
\end{equation}

The total action for spinor electrodynamics reads
\begin{align}
\sfun{S_{1/2}}{A_\mu, \psi, \bar\psi} &\coloneqq
S_\text{Dirac} + S_\text{IDM}+S_\text{Maxwell} \nonumber \\
&= \int\dif^{d+1}x\,\cbr{-\bar\psi\rbr{\gamma^\mu \nabla_\mu+m}\psi
-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}},
\end{align}
where
\begin{align}
\nabla_\mu\psi \coloneqq \rbr{\partial_\mu+\mi e A_\mu}\psi.
\end{align}

Generating functional
\begin{equation}
\sfun{\mathcal{Z}}{j^\mu, \bar\eta, \eta} \coloneqq
\int\Dif A\,\Dif\psi\,\Dif\bar\psi\,\cfun{\exp}{\mi\rbr{S_{1/2}
+\int\dif^{d+1} x\,\rbr{j^\mu A_\mu + \bar\eta \psi+\bar\psi \eta}}}
\end{equation}

Effective action
\begin{equation}
\sfun{\mathcal{Z}}{j^\mu, 0, 0} \eqqcolon
\int\Dif A\,\cfun{\exp}{\mi\rbr{S_\text{Maxwell} +
\sfun{\varGamma_\text{EH}}{A_\mu}
+\int\dif^{d+1} x \,j^\mu A_\mu}}.
\end{equation}
In other words,
\begin{align}
\cfun{\exp}{\mi\sfun{\varGamma_\text{EH}}{A_\mu}} &\equiv \int
\Dif\psi\,\Dif\bar\psi\,\cfun{\exp}{\mi\rbr{S_\text{Dirac}+S_\text{IDM}}}
\nonumber \\
&\equiv \int\Dif\psi\,\Dif\bar\psi\,\cfun{\exp}{\mi\int\dif^{d+1} x\,
\bar{\psi} \rbr{-\slashed{\partial}-\mi e \slashed{A} - m}\psi}
\nonumber \\
&= \int\Dif\psi\,\Dif\bar\psi\,\cfun{\exp}{\mi\int\dif^{d+1} x\,\dif^{d+1} y\,
\rfun{\bar\psi}{x} \rfun{S^{-1}}{x, y} \rfun{\psi}{y}}
\nonumber \\
&= \tilde{\mathcal{N}}\sfun{\det}{-\mi \rfun{S^{-1}}{x, y}},
\end{align}
where
\begin{equation}
\rfun{S^{-1}}{x, y} \coloneqq \rbr{+\slashed{\partial}_y-\mi e
\rfun{\slashed{A}}{y} - m}\rfun{\delta^{d+1}}{x-y}.
\end{equation}

\begin{align}
\sfun{\varGamma_\text{EH}}{A_\mu} &\equiv
-\mi\rbr{\ln\tilde{\mathcal{N}} + \ln \sfun{\det}{-\mi S^{-1}}}
\nonumber \\
&= -\mi \rbr{\ln\tilde{\mathcal{N}} + \Tr \sfun{\ln}{-\mi \rfun{S^{-1}}{x,
y}}}.
\end{align}

Note that
\begin{align}
\Tr \sfun{\ln}{-\mi \rfun{S^{-1}}{x, y}} &\equiv
\Tr \sfun{\ln}{-\mi \rfun{M^\intercal}{x, y}} \nonumber \\
&= \Tr \sfun{\ln}{\mi\rbr{ +\slashed{\partial}_y^\intercal -\mi e
\rfun{\slashed{A}^\intercal}{y} - m}\rfun{\delta^{d+1}}{x-y}} \nonumber \\
&= \Tr \sfun{\ln}{\mi\rbr{ -\mathscr{C}\slashed{\partial}_y\mathscr{C}^{-1}
+\mi e \mathscr{C}\rfun{\slashed{A}}{y}\mathscr{C}^{-1}
-\mathscr{C}m\mathscr{C}^{-1}} \rfun{\delta^{d+1}}{x-y}} \nonumber \\
&= \cfun{\Tr}{\mathscr{C}\sfun{\ln}{\mi\rbr{-\slashed{\partial}_y
+\mi e \rfun{\slashed{A}}{y} -m} \rfun{\delta^{d+1}}{x-y}}\mathscr{C}^{-1}}
\nonumber \\
&= \Tr \sfun{\ln}{\mi\rbr{-\slashed{\partial}_y
+\mi e \rfun{\slashed{A}}{y} -m} \rfun{\delta^{d+1}}{x-y}}.
\end{align}
where the transpose ${}^\intercal$ is taken in the spinor space. Therefore
\begin{equation}
\Tr \sfun{\ln}{-\mi \rfun{S^{-1}}{x, y}} =
\frac{1}{2} \Tr \sfun{\ln}{\rfun{S_2^{-1}}{x, y}},
\label{eq:Tr-M_2}
\end{equation}
where
\begin{align}
\rfun{S_2^{-1}}{x, y} \coloneqq \mathcal{S}_{2y}^{-1}
\rfun{\delta^{d+1}}{x-y},\quad
\mathcal{S}_{2y}^{-1} \coloneqq \rbr{\rbr{\slashed{\partial}_y-\mi e
\rfun{\slashed{A}}{y}}^2 - m^2}.
\label{eq:M_2-def}
\end{align}
One may further simplify \cref{eq:M_2-def} by noting ($y$ suppressed)
\begin{align}
\rbr{\slashed{\partial} -\mi e \slashed{A}}^2 &=
\partial^2 - e^2 A_\mu^2 -\mi e \frac{1}{2}\rbr{\sbr{\gamma^\mu, \gamma^\nu}_-
+ \sbr{\gamma^\mu, \gamma^\nu}_+}
\rbr{\partial_\mu A_\nu + A_\mu \partial_\nu + A_\nu \partial_\mu}
\nonumber \\
&= \partial^2 - e^2 A_\mu^2 - \mi e \rbr{\partial_\mu A^\mu + 2A^\mu
\partial_\mu - \mi \sigma^{\mu\nu} \partial_\mu A_\nu} \nonumber \\
&= \rbr{\partial_\mu - \mi e A_\mu}^2
- e \sigma^{\mu\nu} \partial_{[\mu} A_{\nu]} \nonumber \\
&= \rbr{\partial_\mu - \mi e A_\mu}^2 - \frac{e}{2} \sigma^{\mu\nu} F_{\mu\nu},
\end{align}
so that
\begin{align}
\mathcal{S}_{2}^{-1} \equiv \rbr{\rbr{\partial_{\mu} - \mi e A_\mu}^2 -
\frac{e}{2} \sigma^{\mu\nu} F_{\mu\nu} - m^2},
\label{eq:M_2-simp}
\end{align}
where $y$ is suppressed as well. Now we may write
\begin{align}
\sfun{\varGamma_{\text{EH}}}{A_\mu} &= -\frac{\mi}{2} \Tr \cfun{\ln}{
\mathcal{N}^{-1} \rfun{S_{2}^{-1}}{x,y}} \nonumber \\
&= -\frac{\mi}{2} \Tr\int_0^{+\infty}\frac{\dif s}{s}\cbr{
-\me^{\mi s\rbr{\rfun{S_{2}^{-1}}{x,y}+\mi 0^+}}
+\me^{\mi s\rbr{\mathcal{N}+\mi 0^+}}} \nonumber \\
&= +\frac{\mi}{2} \int_0^{+\infty} \frac{\dif s}{s}
\int\dif^{d+1}x\,\dif^{d+1}y\,\rfun{\delta^{d+1}}{x-y}\nonumber \\
&\quad\cdot\cfun{\tr}{+\me^{\mi s\rbr{\rbr{\partial_{\mu} - \mi e A_\mu}^2 -
\frac{e}{2} \sigma^{\mu\nu} F_{\mu\nu} - m^2+\mi 0^+}\rfun{\delta^{d+1}}{x-y}}
-\me^{\mi s\rbr{\mathcal{N}+\mi 0^+}}},
\end{align}
where $\tr$ takes place in the spinor space.

\subsection{Constant background field in $\rbr{3+1}$-dimensions}

\Cref{eq:Tr-M_2} can be solved exactly when $F_{\mu\nu}$ is constant throughout
space-time. %Take the case \citep{resetsky2012} where $\vec{E}\parallel\vec{B}$
%and, without loss of generality, $\vec{B}\parallel\vec{z}$.
One has
\begin{equation}
F_{0i} \equiv -F_{i0} = -E_i,\qquad
F_{ij} \equiv -F_{ji} = \epsilon_{ijk}B^k,
%F_{30} \equiv -F_{03} = E_3 \eqqcolon E,\qquad
%F_{12} \equiv -F_{21} = B_3 \eqqcolon B.
\label{eq:const-EM-strength}
\end{equation}
and the spinor part of \cref{eq:M_2-simp} can be calculated
\begin{align}
-\frac{e}{2}\sigma^{\mu\nu}F_{\mu\nu} = -e
\begin{bmatrix}
\rbr{\vec B-\mi \vec E} \cdot\vec\sigma & 0 \\
0 & \rbr{\vec B+\mi \vec E} \cdot\vec\sigma
\end{bmatrix},
\end{align}
so that
\begin{align}
&\quad\tr\me^{\mi s\rbr{\rbr{\partial_{\mu} - \mi e A_\mu}^2 -
\frac{e}{2} \sigma^{\mu\nu} F_{\mu\nu} - m^2+\mi 0^+}\rfun{\delta^{d+1}}{x-y}}
\nonumber \\
&= \me^{\mi s\rbr{\rbr{\partial_{\mu} - \mi e A_\mu}^2
- m^2+\mi 0^+}\rfun{\delta^{d+1}}{x-y}}
\tr\me^{\mi s\rbr{-\frac{e}{2}\sigma^{\mu\nu}F_{\mu\nu}}},
\end{align}
in which
\begin{align}
\tr\me^{\mi s\rbr{-\frac{e}{2}\sigma^{\mu\nu}F_{\mu\nu}}} &= 2\cbr{
\rfun{\cos}{se\sqrt{\rbr{\vec B+\mi\vec E}^2}}+
\rfun{\cos}{se\sqrt{\rbr{\vec B-\mi\vec E}^2}}} \\
&= 4\rfun{\cos}{seB}\rfun{\cosh}{seE},\qquad\text{for}\quad
\vec B = B \hat x^3,\quad \vec E = E \hat x^3.
\end{align}

A Landau-like choice of four-potential \citep{landau1930} reads
\begin{align}
A_{\mu} &\coloneqq \begin{pmatrix} 0 &
-x^0 E_1 + x^3 B^2 &
-x^0 E_2 + x^1 B^3 &
-x^0 E_3 + x^2 B^1 \end{pmatrix}, %\nonumber \\
%&=
%\rbr{0, -x^0 E_i + \epsilon_{ijk} B^j x^k}.
\end{align}
applying which leads to \cref{eq:const-EM-strength}.

Consider the case where $\vec B = B \hat x^3$, $\vec E = E \hat x^3$.
The scalar part of \cref{eq:M_2-simp} simplifies to
\begin{align}
\rbr{\partial_\mu - \mi e A_\mu}^2 - m^2 &=
-\partial_0^2 + \partial_1^2 + \rbr{\partial_2 - \mi e x^1 B}^2
+\rbr{\partial_3 + \mi e x^0 E}^2 - m^2 \nonumber \\
&= U \cbr{-\partial_0^2 + \partial_1^2 - \rbr{eB}^2 \rbr{x^1}^2 -
\rbr{eE}^2 \rbr{x^0}^2} U^\dagger,
\label{eq:scalar-part-M_2}
\end{align}
where
\begin{align}
U \coloneqq \me^{+\mi\rbr{\partial_1\partial_2/eB-\partial_3\partial_0/eE}}.
\end{align}

One can first solve the sub-eigensystem
\begin{align}
\cbr{+\partial_1^2 - \rbr{eB}^2 \rbr{x^1}^2 - \lambda}
\rfun{\tilde{f}_\lambda}{x^1} = 0,
\end{align}
and the general solution is
\begin{align}
\rfun{\tilde{f}_\lambda}{x}
= c_1 \rfun{\mathrm{D}_{-\frac{\lambda}{2eB}-\frac{1}{2}}}{\sqrt{2eB}x}
+ c_2 \rfun{\mathrm{D}_{+\frac{\lambda}{2eB}-\frac{1}{2}}}{\mi\sqrt{2eB}x},
\label{eq:parabolic-cylinder}
\end{align}
where $\rfun{\mathrm{D}_\nu}{z}$ is the parabolic cylinder function. For
\begin{align}
\lambda = -(2n+1)eB,\qquad n = -\frac{1}{2}-\frac{\lambda}{2eB}
\end{align}
\cref{eq:parabolic-cylinder} reduces to a combination of the Hermite polynomials
\begin{align}
\rfun{\tilde{f}_n}{x}
= c_1 2^{-n/2}\me^{-eBx^2/2}\rfun{\mathrm{H}_n}{\sqrt{eB}x}
+ c_2 2^{+\rbr{n+1}/2}\me^{+eBx^2/2}\rfun{\mathrm{H}_{-n-1}}{\mi\sqrt{eB}x}.
\end{align}
The first branch constitutes a complete orthogonal set of basis for
$\rfun{L^2}{\mathbb{R}}$ when $n\in\mathbb{Z}^+$, and its normalised version
will be denoted as $\cbr{f_n}$.

It has been argued that the same solution applies for
\begin{align}
\cbr{-\partial_0^2 - \rbr{eE}^2 \rbr{x^0}^2 - \lambda}
\rfun{\tilde{g}_\lambda}{x^0} = 0
\end{align}
if one applies $E\to\mi B$, so that the eigenfunctions are
\begin{align}
\rfun{g_n}{x} = \rfun{f_n}{x},\qquad n = -\frac{1}{2}-\frac{\mi \lambda}{2eE}
\quad\text{or}\quad \lambda = \mi\rbr{2n+1}eE.
\end{align}
However, we know that complex differential equations are subtle!

The eigenfunctions of \cref{eq:scalar-part-M_2} can now be written down
\begin{align}
&\quad\cbr{\rbr{\partial_\mu - \mi e A_\mu}^2 - m^2}
\cbr{U\rfun{f_n}{x^1}\rfun{g_k}{x^0}} \nonumber \\
&= U \cbr{-\partial_0^2 + \partial_1^2 - \rbr{eB}^2 \rbr{x^1}^2 -
\rbr{eE}^2 \rbr{x^0}^2} \rfun{f_n}{x^1} \rfun{g_k}{x^0} \nonumber \\
&= -\mi\rbr{2n+1}\rbr{2k+1}e^2BE \cbr{U \rfun{f_n}{x^1} \rfun{g_k}{x^0}},
\end{align}
and they are
\begin{align}
\rfun{\phi_{nk}}{x} \coloneqq U\rfun{f_n}{x^1}\rfun{g_k}{x^0}.
\end{align}

\begin{align}
\rfun{\delta^4}{x-y} &=
\sum_{n,k} \rfun{\phi_{nk}^\dagger}{x} \rfun{\phi_{nk}}{y} \nonumber \\
&= \sum_{n,k}
\cbr{\rfun{U}{x}\rfun{f_n}{x^1}\rfun{g_k}{x^0}}^\dagger
\cbr{\rfun{U}{y}\rfun{f_n}{y^1}\rfun{g_k}{y^0}}.
\end{align}

\cite{heisenberg1936}

%Further using the trick
%\begin{equation}
%\rfun{\ln}{A+\mi 0^+} =
%-\int_0^{+\infty}\frac{\dif s}{s}\,\me^{\mi s \rbr{A+\mi 0^+}},
%\end{equation}
%one obtains
%\begin{align}
%\sfun{\varGamma}{A_\mu} &= \frac{\mi}{2}\rbr{-\Tr \ln M_2-\ln \mathcal{N}_2}
%\nonumber \\
%&= \frac{\mi}{2}\rbr{\Tr\int_0^{+\infty}\frac{\dif s}{s}\,
%\me^{\mi s \rbr{M_2 + \mi 0^+}}-\ln \mathcal{N}_2} \nonumber \\
%&=
%\end{align}

%\section*{Acknowledgements}
%\addcontentsline{toc}{section}{\protect\numberline{}Acknowledgements}%

\appendix

\section{Notions and conventions}

The metric convention is mostly positive, i.e.\
$\eta_{\mu\nu} \coloneqq \rfun{\diag}{-, +, +, \ldots}$

\section{Fresnel functional integral}

\citep[ch.~10]{mosel2004}

\section{Algebra}
\begin{align}
\sbr{a\partial_1 \partial_2, x^1}_- = a\partial_2
\end{align}
central.

Baker--Campbell--Hausdorff formula

\begin{align}
\me^{+a\partial_1\partial_2} x^1 \me^{-a\partial_1\partial_2}
= x^1 + a \partial_2.
\end{align}

\bibliographystyle{plainnat}
\bibliography{./eft}

%\printbibliography

\end{document}


\chapter{Fixed Wind sUAS Development}
\label{chap:Fixed_Wing}

\section{Range \& Endurance}
The fixed wing sUAS we used for this project was the Aerosonde UAV. The parameters as given for the Aerosonde UAV are in the Figure below.

\subsection{Range}
The range of an aircraft is technically defined as the total distance measured with respect to the ground traversed by the aircraft on one full tank of fuel. It is used to estimate engine performance in terms of fuel consumption. The maximum total range is the maximum distance an aircraft can fly between takeoff and landing, as limited by fuel capacity in powered aircraft, or cross-country speed and environmental conditions in unpowered aircraft. In this project, the maximum range of the given aircraft is calculated with respect to the propeller data obtained from the UIUC database for the given propeller (APC 16X8).

In order to calculate the range of the aircraft with respect to the different powertrain systems, the following specifications are adopted:

\begin{enumerate}
	\item \textbf{Specific Fuel Consumption (SFC) (for propeller)} : Specific fuel consumption is one of the important metrics in determining the performance of a propeller-driven aircraft. SFC is defined as the weight of the fuel consumed by the reciprocating engine per unit power per unit time.

	\item \textbf{Thrust specific fuel consumption (TSFC) (for jet engine)} : The thrust specific fuel consumption of a jet engine is defined as the fuel efficiency of an engine design with respect to thrust output. TSFC is the mass of fuel burned by an engine in one hour divided by the thrust that the engine produces.
\end{enumerate}

The maximum range for the APC 16X8 propeller is calculated using the given formula:

\begin{equation}
R = \int_{W_0}^{W_1} \frac{\eta}{c_p} \frac{C_L}{C_D} \frac{dW}{W} = \frac{\eta}{c_p} \frac{C_L}{C_D} ln(\frac{W_0}{W_1})
\end{equation}

\begin{equation}
	R_{max} = \frac{\eta}{c_p} \frac{C_L}{C_D} ln(\frac{W_0}{W_1}) = \frac{\eta/c_p}{2\sqrt{KC_{D_0}}}ln(\frac{W_0}{W_1})
\end{equation}

\begin{figure}
	\centering
	\includegraphics[width=0.8\textwidth]{Images/Fixed_Wing_parameters}
	\caption{Given Parameters}
	\label{fig:Parameters}
\end{figure}

\textbf{Note :} To cover the longest distance, common sense says that we must use the minimum fuel consumption per unit distance (e.g., km or mile).

\subsection{Endurance}

The endurance of an aircraft is defined as the total time that an aircraft stays in the air on a tank of fuel. Like the range characteristics, to achieve maximum endurance, it is advised to use the minimum thrust per unit time. For a steady level flight, the lift is equal to weight and the thrust is equal to the drag (L = W, T = D).

To calculate the maximum endurance for a propeller-driven aircraft, the following parameters are required :

\begin{enumerate}
	\item Maximum weight loss (Wf = W0 – W1)
	\item Maximum $ C_l^\frac{3}{2}/C_d $
\end{enumerate}

The maximum endurance for a propeller-driven aircraft is calculated using the given formula :

\begin{equation}
	E = \int_{W_0}^{W_1} \frac{\eta}{c_p V}\frac{C_L}{C_D}\frac{dW}{W}
\end{equation}

\begin{equation}
	 = \int_{W_0}^{W_1} \frac{\eta}{c_p}\sqrt{\frac{\rho S}{2}}\frac{C_L^{1.5}}{C_D}\frac{dW}{W^{1.5}}
\end{equation}

\begin{equation}
	E = \frac{\eta}{c_p}\sqrt{2\rho S}(\frac{C_L^{1.5}}{C_D})(\frac{1}{\sqrt{W_1}} - \frac{1}{\sqrt{W_0}})
\end{equation}

Based on the above-mentioned procedures, the results of the multi-rotor drone (quadcopter) are as follows:

Max. Range = 390.04 kms

Max. Endurance = 5.88 hours

\section{Fixed Wing Aircraft Dynamics}
We are using the Linearized Models in the \cite{Randal2012} for Developing the Flight Dynamics for the Fixed Wing Aircraft. The Aircraft Dynamics is primarily divided into two different parts, The Longitudinal Dynamics and the Lateral Dynamics. The Linearized Equations are demonstrated in the State Space Models as given in the further subsections. The Longitudinal Dynamics has the elevator and the thrust as the inputs. The elevator controls the pitching and the thrust controls the Speed. The Lateral Dynamics has the aileron and the rudder as the inputs.

\subsection{Longitudinal Flight Dynamics}
The Longitudinal Dynamics has 6 states, u, w, q, $\theta$, h. The Longitudinal Statespace can be seen as below. The Longitudinal The Dynamics Equations for all these states are linearized using Taylor's sequence at the Equilibrium Locations. We calculate the Equilibrium values using the non-linear Dynamic Equations.

Given a nonlinear system described by the differential equations
$$
\dot{x}=f(x, u),
$$
where $f: \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}, x$ is the state of the system, and $u$ is the input, the system is said to be in equilibrium at the state $x^{*}$ and input $u^{*}$ if
$$
f\left(x^{*}, u^{*}\right)=0 .
$$

When an Aircraft is in constant-altitude, wings-level steady flight, a subset of its states are in equilibrium. In particular, the altitude h; the body frame velocities u, v, w; the Euler angles $\phi$, $\theta$, $\psi$; and the angular rates p, q, and r are all constant, Zero in this case.

The final state-Space Equation is as given below \\\

\begin{aligned}
	&\left(\begin{array}{c}
		\dot{\bar{u}} \\
		\dot{\bar{w}} \\
		\dot{\bar{q}} \\
		\dot{\bar{\theta}} \\
		\dot{\bar{h}}
	\end{array}\right)=\left(\begin{array}{ccccc}
		X_{u} & X_{w} & X_{q} & -g \cos \theta^{*} & 0 \\
		Z_{u} & Z_{w} & Z_{q} & -g \sin \theta^{*} & 0 \\
		M_{u} & M_{w} & M_{q} & 0 & 0 \\
		0 & 0 & 1 & 0 & 0 \\
		\sin \theta^{*}-\cos \theta^{*} & 0 & u^{*} \cos \theta^{*}+w^{*} \sin \theta^{*} 0
	\end{array}\right)\left(\begin{array}{c}
		\bar{u} \\
		\bar{w} \\
		\bar{q} \\
		\bar{\theta} \\
		\bar{h}
	\end{array}\right)\\
	&+\left(\begin{array}{cc}
		X_{\delta_{e}} & X_{\delta_{t}} \\
		Z_{\delta_{e}} & 0 \\
		M_{\delta_{e}} & 0 \\
		0 & 0 \\
		0 & 0
	\end{array}\right)\left(\begin{array}{l}
		\bar{\delta}_{e} \\
		\bar{\delta}_{t}
	\end{array}\right)
\end{aligned} \\\\

The Coefficients of the Matrix are as given below: \\\
\begin{lstlisting}

Cxo = -P.CDo*cos(alpha) + P.CLo*sin(alpha)
Cxa = -P.CDa*cos(alpha) + P.CLa*sin(alpha)
Cxq = -P.CDq*cos(alpha) + P.CLq*sin(alpha)
Cxdelta_e = -P.CDdelta_e*cos(alpha) + P.CLdelta_e*sin(alpha)

Xu  = (2*qs*Cxo - rho*Sprop*Ue*C_T)/m
Xw  = Cxa*qs/m
Xq  = Cxq*qs1*c_bar/(2*m)
Xdelta_e = Cxdelta_e*qs1_2/m
Xdelta_t = (rho*Sprop*C_T*k^2)/m

Czo = -P.CLo*cos(alpha) - P.CDo*sin(alpha)
Cza = -P.CLa*cos(alpha) - P.CDa*sin(alpha)
Czq = -P.CLq*cos(alpha) - P.CDq*sin(alpha)
Czdelta_e = -P.CLdelta_e*cos(alpha) - P.CDdelta_e*sin(alpha)

Zu  = 2*qs*Czo/m
Zw  = Cza*qs/m
Zq  = Ue + Czq*qs1*c_bar/(2*m)
Zdelta_e = Czdelta_e*qs1_2/m

Mu  = 2*qs*c_bar*Cmo/Iyy
Mw  = Cma * qs * c_bar/Iyy
Mq  = Cmq*qs1*c_bar*c_bar/(2*Iyy)
Mdelta_e = Cmdelta_e*qs1_2*c_bar/Iyy
\end{lstlisting}\\\

\subsection{Lateral Flight Dynamics}
For the lateral dynamics, the variables of interest are the roll angle $\phi$, the
roll rate p, the heading angle $\psi$, and the yaw rate r. The control surfaces
used to influence the lateral dynamics are the ailerons $\delta_{a}$, and the rudder
$\delta_{r}$ . The ailerons are primarily used to influence the roll rate p, while the
rudder is primarily used to control the yaw $\psi$ of the aircraft.\\\

\begin{aligned}
	\left(\begin{array}{c}
		\dot{\bar{v}} \\
		\dot{p} \\
		\dot{\bar{r}} \\
		\dot{\bar{\phi}} \\
		\dot{\bar{\psi}}
	\end{array}\right)=&\left(\begin{array}{ccccr}
		Y_{v} & Y_{p} & Y_{r} & g \cos \theta^{*} \cos \phi^{*} & 0 \\
		L_{v} & L_{p} & L_{r} & 0 & 0 \\
		N_{v} & N_{p} & N_{r} & 0 & 0 \\
		0 & 1 & \cos \phi^{*} \tan \theta^{*} & q^{*} \cos \phi^{*} \tan \theta^{*}-r^{*} \sin \phi^{*} \tan \theta^{*} & 0 \\
		0 & 0 & \cos \phi^{*} \sec \theta^{*} & p^{*} \cos \phi^{*} \sec \theta^{*}-r^{*} \sin \phi^{*} \sec \theta^{*} & 0
	\end{array}\right) \\
	& \times\left(\begin{array}{c}
		\bar{v} \\
		\bar{p} \\
		\bar{r} \\
		\bar{\phi} \\
		\bar{\psi}
	\end{array}\right)+\left(\begin{array}{cc}
		Y_{\delta_{a}} & Y_{\delta_{r}} \\
		L_{\delta_{a}} & L_{\delta_{r}} \\
		N_{\delta_{a}} & N_{\delta_{r}} \\
		0 & 0 \\
		0 & 0
	\end{array}\right)\left(\begin{array}{c}
		\bar{\delta}_{a} \\
		\bar{\delta}_{r}
	\end{array}\right)
\end{aligned} \\\\

The Coefficients of the Matrix are as given below: \\\

\begin{lstlisting}

	Yv = 2*Cyo*qs/m + qs*Cyb/m
	Yp = Cyp*qs1*b/(2*m)
	Yr = -Ue + (Cyr*qs1*b/(2*m))
	Ydelta_a = Cydelta_a*qs1_2/m
	Ydelta_r = Cydelta_r*qs1_2/m

	Lv = Cpb*qs*b
	Lp = Cpp*qs1*b^2/2
	Lr = Cpr*qs*b^2/2
	Ldelta_a = Cpdelta_a * qs1_2 * b
	Ldelta_r = Cpdelta_r * qs1_2 * b

	Nv = Crb * qs * b
	Np = Crp * qs1 * b^2 / 2
	Nr = Crr * qs1 * b^2 / 2
	Ndelta_a = Crdelta_a * qs1_2 * b
	Ndelta_r = Crdelta_r * qs1_2 * b
\end{lstlisting} \\\

\section{Simulink Model}
The Given State-Space setup was implemented in Matlab with a Level2 S Function Block. The Block takes 4 inputs in the following order : $\delta_{e}$, $\delta_{a}$, $\delta_{r}$, $\delta_{t}$. The Block further provides 12 outputs, giving the state for both the longitudinal and lateral Systems. The complete Simulink Model can be seen in the figure \ref{fig:fixed_wing_simulink}.\\

The Simulink model performs all the three manuevers in the order provided in the Queation. The input for the system is controlled from the refernece position subsystem, which gives out 5 outputs, height h, $\psi$, $\phi$, V, U. The error are calculated using feedback loops and are further fed to PIDs. A Radian to Degree converter is used in $\phi$ and $\psi$  as the outputs are in Radians. The reference position subsystem is time-based with a digital clock as an input and time based if-else block. We calculated the time for each manuever and used the digital clock to change the input for the positions.

\begin{figure}
	\centering
	\includegraphics[width=1\textwidth]{Images/fixed_wing_simulink}
	\caption{Simulink Model}
	\label{fig:fixed_wing_simulink}
\end{figure}