Review on rationality problems of algebraic k-tori

TL;DR

This paper reviews the rationality issues of algebraic k-tori, linking them to invariant fields and Noether's Problem, and discusses methods to determine their rationality.

Contribution

It provides a comprehensive overview of rationality problems for algebraic k-tori, including character group analysis and numerical approaches for rationality determination.

Findings

Rationality of k-tori is equivalent to the rationality of their invariant function fields.

Character groups are useful tools in studying k-tori rationality.

Numerical methods can assist in determining the rationality of algebraic k-tori.

Abstract

Rationality problems of algebraic k-tori are closely related to rationality problems of the invariant field, also known as Noether's Problem. We describe how a function field of algebraic k-tori can be identified as an invariant field under a group action and that a k-tori is rational if and only if its function field is rational over k. We also introduce character group of k-tori and numerical approach to determine rationality of k-tori.

Equations

$$\frac{f(x,y)}{g(x,y)}=\sigma (\frac{f(x,y)}{g(x,y)})=\frac{f(y,x)}{g(y,x)}$$

$$\psi (x+y)=s,\psi (xy)=t$$

$$z=x+yi,w=x-yi$$

$$and$$

$$\psi (f)(x,y)=f(z,w)$$

$$\psi (f)(x,y)=f(z,w)=\sigma (f(z,w))=\sigma (\sum _{n,m}{a}_{n,m}{z}^n{w}^m)=\sum _{n,m}\overline{a_{n,m}}{w}^n{z}^m$$

$$A(X):=k[x_1,...,x_n]/I(X)$$

$$(x_1,x_2)\cdot (y_1,y_2)=(x_1{y}_1-x_2{y}_2,x_1{y}_2+x_2{y}_1)$$

$$\psi (x_1,x_2)=x_1+i{x}_2$$

$$P=\{(x_1,x_2,x_3,x_4)\in {\mathbb{A}}_{\mathbb{R}}^4|x_1{x}_3-x_2{x}_4=1,x_1{x}_4+x_2{x}_3=0\}$$

$$P=\{A\in M_{2\times 2}(\mathbb{R})|A{A}^t=\left(\begin{array}{cc}s& 0\\ 0& s^{-1}\end{array}\right)s\in \mathbb{R}\backslash \{0\}\}$$

$$(x_1,x_2,x_3,x_4)\cdot (y_1,y_2,y_3,y_4)=(x_1{y}_1-x_2{y}_2,x_1{y}_2+x_2{y}_1,x_3{y}_3-x_4{y}_4,x_3{y}_4+x_4{y}_3)$$

$$(x_1+x_{2}i,x_3+x_{4}i)\cdot (y_1+y_{2}i,y_3+y_{4}i)$$

$$(x_1,x_2,x_3,x_4)\to ((x_1+x_{2}i,x_3+x_{4}i),(x_1-x_{2}i,x_3-x_{4}i))=((z,\frac{1}{z}),(w,\frac{1}{w}))$$

$$A(P(\mathbb{C}))=\mathbb{C}[x_1,x_2,x_3,x_4]/(x_1{x}_3-x_2{x}_4-1,x_1{x}_4+x_2{x}_3)\cong \mathbb{C}[z,\frac{1}{z},w,\frac{1}{w}]$$

$$K(P(\mathbb{C}))\cong \mathbb{C}(z,w)$$

$$K(P(\mathbb{R}))\cong \mathbb{C}(z,w{)}^G$$

$$\mathbb{X}(T)=\{t^n|n\in \mathbb{Z}\}\cong {\mathbb{Z}}^1$$

$$\mathbb{X}(P)=\{f_{t_1,t_2}:P\to {\mathbb{C}}^{\times }|f_{t_1,t_2}(x_1,x_2,x_3,x_4)=(x_1+x_{2}i{)}^{t_1}(x_1-x_{2}i{)}^{t_2}\}$$

$$\mathbb{X}(P(\mathbb{C}))=\{f_{t_1,t_2}:P(\mathbb{C})\to {\mathbb{C}}^{\times }|f_{t_1,t_2}((z,\frac{1}{z}),(w,\frac{1}{w}))=z^{t_1}{w}^{t_2}\}\cong {\mathbb{Z}}^2$$

$$H^1(G,M)=Z^1(G,M)/B^1(G,M)$$

$${\hat{H}}^{-1}(G,M)=Z^{-1}(G,M)/B^{-1}(G,M)$$

$$Z^{-1}(G,M)=\{m\in M|\sum _{g\in G}{m}^g=0\}$$

$$B^{-1}(G,M)=\{\sum _{g\in G}{m}_g^{g-id}|m_g\in M\}$$

$$Mispermutation\Rightarrow Misstablypermutation$$

$$Misstablypermutation\Rightarrow Misinvertible\Rightarrow Misflabbyandcoflabby$$

Full text

1. 1 Introduction
2. 2 Algebraic $k-tori$
3. 3 Character group of $k-tori$
4. 4 Flabby resolution and numerical approach

1 Introduction

Let $k$ be a field and $K$ is a finitely generated field extension of $k$. $K$ is called rational over k or k-rational if $K$ is isomorphic to $k(x_{1},...,x_{n})$ where $x_{i}$ are transcendental over $k$ and algebraically independent. There are also relaxed notions of rationality. $K$ is called stably k-rational if $K(y_{1},...,y_{m})$ is $k-rational$ for some transcendental and algebraically independent $y_{i}$. $K$ is called $k-unirational$ if $k\subset K\subset k(x_{1},...,x_{n})$ for some pure transcendental extension $k(x_{1},...,x_{n})/k$.

The Noether’s Problem is the question of rationality of the invariant field under finite group action. For example, if $K=\mathbb{Q}(x_{1},x_{2})$ and $G=\{1,\sigma\}\cong C_{2}$ and $G$ acts on $K$ as permutation of variables $x_{1},x_{2}$ (i.e. $\sigma$ fixes $\mathbb{Q}$, $\sigma(x_{1})=x_{2}$ and $\sigma(x_{2})=x_{1}$), then the invariant field $K^{G}$ is $\mathbb{Q}-rational$.

Example 1.1

$K=\mathbb{Q}(x,y)$* and $G\cong C_{2}$, acting on $K$ as permutation of variables. Let $\frac{f}{g}\in K^{G}$, $f,g$ are coprime. We have*

[TABLE]

*By observing that $gcd(f(x,y),g(x,y))=gcd(f(y,x),g(y,x))=1$, we have $f(x,y)=f(y,x)$ and $g(x,y)=g(y,x)$.

Therefore, $K^{G}=\{\frac{f(x,y)}{g(x,y)}|f,g$ are symmetric$\}$, field of fractions (quotient field) of $S=\{f\in\mathbb{Q}[x,y]|f(x,y)=f(y,x)\}$. It is easy to see that $\psi:S\to\mathbb{Q}[s,t]$ is isomorphism, where*

[TABLE]

Therefore, $S\cong\mathbb{Q}[x,y]$ and $K^{G}\cong\mathbb{Q}(x,y)$, $\mathbb{Q}-rational$.

We can also consider case of $G$ acting on both of coefficients and variables.

Example 1.2

$K=\mathbb{C}(x,y)$* and $G=Gal(\mathbb{C}/\mathbb{R})=\{1,\sigma\}\cong C_{2}$. Suppose $G$ acts on $K$ by permuting $x,y$ and as complex conjugation on coefficients.

For example, $\sigma(ix+(1-i)xy+y^{2})=-iy+(1+i)yx+x^{2}$. Then, $K^{G}\cong\mathbb{R}(x,y)$, is $\mathbb{R}-rational$.*

Proof. For $\frac{f(z,w)}{g(z,w)}\in K^{G}$, where $f,g$ are coprime, $\sigma(f)$ and $\sigma(g)$ are also coprime. From $\frac{f}{g}=\frac{\sigma(f)}{\sigma(g)}$, we have $f=\sigma(f)$ and $g=\sigma(g)$. Thus, $K^{G}$ is quotient field of $S$ where $S:=\{f(z,w)\in\mathbb{C}[z,w]|f=\sigma(f)\}$.

Define a map $\psi:S\to\mathbb{R}[x,y]$ as

[TABLE]

[TABLE]

[TABLE]

The coefficients of $\psi(f)$ are real numbers. This is because, if we let $f(z,w)=\sum_{n,m}a_{n,m}z^{n}w^{m}$, we have that

[TABLE]

$=\sum_{n,m}\overline{a_{n,m}(x+iy)^{n}(x-iy)^{m}}=\overline{\psi(f)(x,y)}$.

Therefore, $\psi(f)=\overline{\psi(f)}$, $\psi(f)\in\mathbb{R}[x,y]$. It is easy to see that $\psi$ is actually isomorphism, $S\cong\mathbb{R}[x,y]$, and $K^{G}\cong\mathbb{R}(x,y)$.

Another perspective to view this change of variables is identifying the field with rational function field of algebraic $k-tori$. (see Example 2.5 and Example 2.6)

2 Algebraic $k-tori$

Let $k$ be a field. Then $\mathbb{A}_{k}^{n}$ is n-dimension affine space over the field $k$, simply $k^{n}$ with usual vector space structure on it. A subset $X$ of $\mathbb{A}_{k}^{n}$ is an algebraic k-variety (k-variety in short) if it is a set of zeros of a system of equations with $n$ variables $x_{1},...x_{n}$ over $k$. The ideal of polynomials that vanish on every points of $X$ will be denoted by $I(X)$. The coordinate ring of a variety $X$ is defined to be the quotient

[TABLE]

Projective varieties can be similarly defined as the set of zeros of a system of homogeneous equations. Projective $n-space$ $\mathbb{P}_{k}^{n}$ is defined as set of lines passing the origin in $\mathbb{A}_{k}^{n+1}$.

If $X,Y$ are varieties, a map $f:X\to Y$ is called regular if it can be presented as fraction of polynomials $p/q$, where $q$ does not vanishes in $X$. A map $f:X\to Y$ is called rational if it is regular on Zariski open dense set. (Formally, a regular map is defined as an equivalence class of pairs $<U,f_{U}>$ where $U$ is Zariski open subset of $U$. See [1]) Let $X$ be a variety, $K(X)$ is the rational function field, or function field in short, the set of rational maps $f:X\to\mathbb{A}_{k}$. For example, if $X$ is an affine variety over algebraically closed field $k$, $K(X)$ is quotient field of $A(X)$.

Example 2.1

*Let $X=\{(x,y)\in\mathbb{A}_{\mathbb{C}}^{2}|xy=1\}$ be a variety over $\mathbb{C}$.

Then, $A(X)=\mathbb{C}[x,y]/(xy-1)\cong\mathbb{C}[x,\frac{1}{x}]$ and $K(X)\cong\mathbb{C}(x)$.*

Two varieties $X,Y$ are isomorphic (resp. birationally isomorphic) if there is a bijective regular map (resp. rational map) $f:X\to Y$ and its inverse is also regular (resp. rational).

A variety $X$ in $\mathbb{A}_{k}^{n}$ is an algebraic group if it has a group structure on it, where the group operation and inversions are regular maps. (i.e. $\ast:X\times X\to X$ and ${}^{-1}:X\to X$ are regular)

Algebraic $k-tori$, or algebraic $k-torus$, is a special type of algebraic group over $k$. We call an algebraic group as $k-tori$ when it is isomorphic to some power of multiplicative group over $\overline{k}$, the algebraic closure of $k$.

Definition 2.1** (Multiplicative Group)**

Let $k$ be a field, the multiplicative group $\mathbb{G}_{m}(k)$ is algebraic group in $\mathbb{A}_{k}^{2}$, defined as $\{(x,y)\in\mathbb{A}_{k}^{2}|xy=1\}$, with operation $\cdot:\mathbb{G}_{m}(k)\times\mathbb{G}_{m}(k)\to\mathbb{G}_{m}(k)$ of $(x,\frac{1}{x})\cdot(y,\frac{1}{y})=(xy,\frac{1}{xy})$

Example 2.2

$\mathbb{G}_{m}(\mathbb{R})$* is the curve $xy=1$ on the real affine plane. It is isomorphic to $\mathbb{R}^{\times}$ as a group. $((x,y)\to x$ is group isomorphism.$)$*

As field changes, same system of equations can define different varieties. For instance, the equation $xy=1$ in previous example defines $\mathbb{G}_{m}(\mathbb{C})$ in $\mathbb{A}_{\mathbb{C}}^{2}$, which is different from $\mathbb{G}_{m}(\mathbb{R})$. If $E$ is a field and $F$ is its algebraic closure, an irreducible variety $V$ over $F$ entails the ring of equations, $I$. If $I$ happens to be in $E[\textbf{x}]$ (ring of polynomials over $E$), we can define $V(E)$, a variety over $E$ defined by equations in $I$. This can be viewed as restriction of scalar. Extension of scalar can be defined similarly.

Definition 2.2** (Algebraic $k$-tori)**

Let $k$ be a field with algebraic closure $\overline{k}$. If $T$ is an algebraic group over $k$, it is $k-torus$ if and only if

$T(\overline{k})\cong(\mathbb{G}_{m}(\overline{k}))^{r}$**

for some $r$. The $r$ is called dimension of $T$.

Example 2.3

$T=\mathbb{G}_{m}(\mathbb{R})$* is one dimensional $\mathbb{R}-tori$. This is because $T(\mathbb{C})=\mathbb{G}_{m}(\mathbb{C})$.*

From now, let $k^{\times}=\mathbb{G}_{m}(k)$ be the one dimensional torus over $k$. There are two one-dimensional $\mathbb{R}$-tori, one can be recognized as $\mathbb{R}^{\times}$, the other one can be recognized as $SO(2)$ as a group.

Example 2.4

The norm one torus $N$ is a real algebraic group in $\mathbb{A}_{\mathbb{R}}^{2}$, defined by equation $x_{1}^{2}+x_{2}^{2}=1$ $(i.e.$ $N=\{(x_{1},x_{2})\in\mathbb{A}_{\mathbb{R}}^{2}|x_{1}^{2}+x_{2}^{2}=1\})$, and operation $\cdot:N\times N\to N$ such that

[TABLE]

*Indeed, $N$ is isomorphic to $SO(2)$ as a group.

Also, $N(\mathbb{C})=\{(x_{1},x_{2})\in\mathbb{A}_{\mathbb{C}}^{2}|x_{1}^{2}+x_{2}^{2}=1\}$ is isomorphic to $C^{\times}$ as algebraic group. The map $\psi:N(\mathbb{C})\to\mathbb{C}^{\times}$*

[TABLE]

is isomorphism. Therefore, $N$ is one dimensional real torus.

If $T$ is a $k-torus$, $T$ is called split over K if it satisfies $T(K)\cong(K^{\times})^{s}$ for some extension $K/k$ and some $s$. For instance, $\mathbb{R}^{\times}$ is split over $\mathbb{R}$, $N$ is not.

It is easy to find split torus such as $(\mathbb{R}^{\times})^{2}$ or $(\mathbb{R}^{\times})^{3}$, being another torus. Also, for any integer $r$, $N^{r}$ is $r$-dimensional $\mathbb{R}-tori$. Meanwhile, there are also some non-trivial(not a product of low-dimensional torus) torus.

Example 2.5

Let $P$ be a real algebraic group in $\mathbb{A}_{\mathbb{R}}^{4}$, defined as

[TABLE]

Alternatively,

[TABLE]

and operation $\cdot:P\times P\to P$ such that

[TABLE]

Which is compatible with complex multiplication of

[TABLE]

Moreover, $P(\mathbb{C})$ is isomorphic to $(\mathbb{C}^{\times})^{2}$, by sending

[TABLE]

Therefore, $P$ is 2-dimensional $\mathbb{R}-tori$.

By tracking the function fields of $P(\mathbb{R})$ and $P(\mathbb{C})$, we have the same trick of change of variables as in Example 1.2.

Example 2.6

In the previous example, the coordinate ring of $P(\mathbb{C})$ is

[TABLE]

where $z=x_{1}+x_{2}i$ and $w=x_{1}-x_{2}i$. The function field of $P(\mathbb{C})$ is

[TABLE]

Let $G=Gal(\mathbb{C}/\mathbb{R})$ acts on $K(P(\mathbb{C}))$ as in Example 1.2. Observe that the coordinate ring of $P(\mathbb{R})$ is $A(P(\mathbb{R}))=A(P(\mathbb{C}))^{G}$ and the function field of $P(\mathbb{R})$ is $K(P(\mathbb{R}))=K(P(\mathbb{C}))^{G}\cong\mathbb{C}(z,w)^{G}$ (note that $G$ actions on $K(P(\mathbb{C}))$ and $\mathbb{C}(z,w)$ are equivalent through the isomorphism). In short, we have that

[TABLE]

Therefore, when $G=Gal(\mathbb{C}/\mathbb{R})$ action on $C(z,w)$ is given, we can convert the rationality problem to the rationality problem of $K(P(\mathbb{R}))$, the function field of $P(\mathbb{R})$. In this sense, the following definition and theorem are natural.

Definition 2.3** (Rationality of $k-variety$)**

*We say that a variety $X$ over $k$ is rational if, equivalently,

(1) $X$ is birationally isomorphic to $\mathbb{P}_{k}^{n}$ for some $n$.

(2) $K(X)\cong k(x_{1},..,x_{n})$

If $K/k$ is Galois extension, a $k-tori$ $T$ is $K-rational$ if it is rational as a $K$-variety $T(K)$. If $k$ is algebraically closed, there is unique $n$-dimension tori $T_{n}=(k^{\times})^{n}$. Since the function field of $T_{n}$ is $k(x_{1},...,x_{n})$, thus $T_{n}$ is $k$-rational.

Theorem 2.1

*The following two problems are equivalent.

(1) The rationality problem of $n$ dimensional $k-tori$ $T$

*(2) The rationality problem of invariant field $K^{G}$

where $G=Gal(\overline{k}/k)$ and $K=k(x_{1},...,x_{n})$.

There is a connection between the $G$ action on $K$ and $k-tori$ $T$, connecting the two rationality problems given in the previous theorem. To be specific, the character group of $T$ determines both the $G$ action and $T$ uniquely.

3 Character group of $k-tori$

Definition 3.1** (Character group of $k-tori$)**

Let $T$ be $k-tori$. Then $\mathbb{X}(T)$, the character group of $T$ is the set of algebraic group homomorphisms(a regular map preserving the group structure) from $T$ to $\overline{k}^{\times}$, denoted by $Hom(T,\mathbb{G}_{m})$ or $Hom(T,\overline{k}^{\times})$.

The character group $\mathbb{X}(T)$ of $T$ has a group structure defined by component-wise multiplication. Also, if $T$ is split over $L$ for finite Galois extension of base field $k$, $G=Gal(L/k)$ acts on $\mathbb{X}(T)$. Moreover, it is known that $\mathbb{X}(T)$ is torsion-free $\mathbb{Z}$-module(i.e. isomorphic to $\mathbb{Z}^{n}$ for some $n$). Therefore, $\mathbb{X}(T)$ is a $G-lattice$ (a free $\mathbb{Z}-module$ with $G$-action).

Example 3.1

If $T=\mathbb{C}^{\times}$ is multiplicative group of $\mathbb{C}$, then $\mathbb{X}(T)$ is set of regular functions $f:\mathbb{C}^{\times}\to\mathbb{C}^{\times}$ such that $f(xy)=f(x)f(y)$ for $x,y\in\mathbb{C}^{\times}$. Since $f$ is a rational function, it is a meromorphic function over $\mathbb{C}$. Also, we have $f(\mathbb{C}^{\times})\subset\mathbb{C}^{\times}$, which implies 0 is the only point where $f$ can have zeros or poles. Therefore, $f(t)=t^{n}$ for some $n\in\mathbb{Z}$. If we write a function $t\to t^{n}$ as $t^{n}$, we have

[TABLE]

as a group. $G=Gal(\mathbb{C}/\mathbb{C})=\{id\}$ acts trivially on $\mathbb{X}(T)$.

In general, if $k$ is algebraically closed, the character group of $(k^{\times})^{n}=\mathbb{G}_{m}^{n}$ is

$\mathbb{X}(\mathbb{G}_{m}^{n})=\{f_{t_{1},...t_{n}}:\mathbb{G}_{m}^{n}\to\mathbb{G}_{m}|f_{t_{1},...t_{n}}(x_{1},...x_{n})=\prod_{i}x_{i}^{t_{i}},t_{i}\in\mathbb{Z}\}$

$=\prod_{i=1}^{n}\{f_{t}:\mathbb{G}_{m}\to\mathbb{G}_{m}|f_{t}(x_{i})=x_{i}^{t},t\in\mathbb{Z}\}\cong\mathbb{Z}^{n}$

Example 3.2

Let $P$ be the 2-dimension $\mathbb{R}-tori$ in Example 2.5. Then, the character group of $P$ is

[TABLE]

Let $z=x_{1}+x_{2}i$, $w=x_{1}-x_{2}i$, then we have the natural extension of $\mathbb{X}(P)$ to $\mathbb{X}(P(\mathbb{C}))$

[TABLE]

Observe that the complex conjugation $\sigma\in G$, exchanges $z$ and $w$, thus acting on $\mathbb{Z}^{2}$ as $2\times 2$ matrix $\begin{bmatrix}0&1\\ 1&0\end{bmatrix}$.

It is known that when a $G=Gal(K/k)$ action (as $\mathbb{Z}$-linear function) on $\mathbb{Z}^{n}$ is given, there exists unique $n$-dimensional $k-tori$ which has the given $G-lattice$ as its character group. Furthermore, there are conditions of $G-lattice$ corresponding to the rationality conditions of $k-tori$ and of invariant fields.

4 Flabby resolution and numerical approach

This section contains many results in [2]. Let $G$ be a group and $M$ be a $G-lattice$ ($M\cong\mathbb{Z}^{n}$ as group and has $G$-linear action on it). $M$ is called a permutation G-lattice if $M\cong\bigoplus_{1\leq i\leq m}\mathbb{Z}[G/H_{i}]$ for some subgroups $H_{1},...,H_{m}$ of $G$ (equivalently, there exists a $\mathbb{Z}$-basis of $M$ such that $G$ acts on $M$ as permutation of the basis). $M$ is called stably permutation G-lattice if $M\bigoplus P\cong Q$ for some permutation $G-lattices$ $P$ and $Q$. $M$ is called invertible if it is a direct summand of a permutation $G$-lattice, i.e. $P\cong M\bigoplus M^{\prime}$ for some permutation $G$-lattice $P$ and $M^{\prime}$.

Definition 4.1** (1st Group Cohomology)**

Let $G$ be a group and $M$ be a $G$-lattice. For $g\in G$ and $m\in M$, let $g.m=m^{g}$ be $g$ acting on $m$. The first group cohomology $H^{1}(G,M)$ is a group defined as

[TABLE]

where $Z^{1}(G,M)=\{f:G\to M|f(gh)=f(g)^{h}f(h)\}$ and $B^{1}(G,M)=\{f:G\to M|f(g)=m_{f}^{g}m_{f}^{-1}\;for\;some\;m_{f}\in M\}$

$H^{1}(G,M)=0$ simply implies that if $f:G\to M$ satisfies $f(gh)=f(g)^{h}f(h)$, then there exists $m\in M$ such that $f(g)=m^{g}m^{-1}$. $M$ is called coflabby if $H^{1}(G,M)=0$.

Definition 4.2** (-1st Tate Cohomology)**

Let $G$ be finite group of order n and $M$ be a $G$-lattice. The -1st group cohomology $\hat{H}^{-1}(G,M)$ is a group defined as

[TABLE]

where

[TABLE]

,

[TABLE]

Similarly, $M$ is called flabby if $\hat{H}^{-1}(G,M)=0$. It is clear that a $k-tori$ is rational if and only if $\mathbb{X}(T)$ is permutation $G$-lattice. Thus, the rationality problems of $k-tori$ and invariant fields can be reduced into problem of finding permutation $G$-lattice(equivalent to find finite subgroup of $GL(n,\mathbb{Z})$. However, this problem is not solved yet, even though there are many results in weakened problems.

Let $C(G)$ be the category of all $G$-lattices and $S(G)$ be the category of all permutation $G$-lattices. Define equivalence relation   on $C(G)$ by $M_{1}~{}M_{2}$ if and only if there exist $P_{1},P_{2}\in S(G)$ such that $M_{1}\bigoplus P_{1}\cong M_{2}\bigoplus P_{2}$. Let $[M]$ be equivalence class containing $M$ under this relation.

Theorem 4.1

(Endo and Miyata [3, Lemma 1.1], Colliot-Thélène and Sansuc [4, Lemma 3])* For any $G$-lattice $M$, there is a short exact sequence of $G$-lattices $0\to M\to P\to F\to 0$ where $P$ is permutation and $F$ is flabby.*

In the previous theorem, $[F]$ is called the flabby class of $M$, denoted by $[M]^{fl}$.

Theorem 4.2

(Akinari and Aiichi [2, 17pp])* If $M$ is stably permutation, then $[M]^{fl}$. If $M$ is invertible, $[M]^{fl}$ is invertible.*

It is not difficult to see that

[TABLE]

Furthermore, it is true that

[TABLE]

In [2], they gave the complete list of stably permutation lattices for dimension 4 and 5 by computing $[M]^{fl}$ for finite subgroup of $GL(n,\mathbb{Z})$, which is equivalent to classifying stably rational tori. Thus, the rationality problems for low dimensional $k-tori$ can be resolved by finding conditions which can determine a stably permutation $M$ is permutation or not.