FlabbyResolutionBC.gap

Definition of $M_G$: Let $G$ be a finite subgroup of ${\rm GL}(n,\mathbb{Z})$ . The $G$ -lattice $M_G$ of rank $n$ is defined to be the $G$ -lattice with a $\mathbb{Z}$ -basis $\{u_1,\ldots,u_n\}$ on which $G$ acts by $\begin{align} \sigma(u_i)=\sum_{j=1}^n a_{i,j}u_j \tag{1} \end{align}$ for any $\sigma=[a_{i,j}]\in G$ .

Hminus1

Hminus1(G)

returns the Tate cohomology group

$\widehat H^{-1}(G,M_G)$ for a finite subgroup

$G \leq {\rm GL}(n,\mathbb{Z})$ .

H0

H0(G)

returns the Tate cohomology group

$\widehat H^0(G,M_G)$ for a finite subgroup

$G \leq {\rm GL}(n,\mathbb{Z})$ .

H1

H1(G)

returns the cohomology group

$H^1(G,M_G)$ for a finite subgroup

$G \leq {\rm GL}(n,\mathbb{Z})$ .

Sha1Omega

Sha1Omega(G)

returns

$Sha^1_w(G,M_G)$ .

Sha1OmegaTr

Sha1OmegaTr(G)

returns

$Sha^1_w(G,(M_G)^\circ)$ .

ShaOmega

ShaOmega(G,n)

returns

$Sha^n_w(G,M_G)$ for

$G$ -lattice

$M_G$ .
This function needs HAP package in GAP.

ShaOmegaFromGroup

ShaOmegaFromGroup(M,n,G)

returns

$Sha^n_w(G,M)$ for

$G$ -lattice

$M$ .
This function needs HAP package in GAP.

TorusInvariants

TorusInvariants(G)

returns

$TI_G=[l_1,l_2,l_3,l_4]$ where

$l_1=\begin{cases} 0& \textrm{if}\ \ [M_G]^{fl}=0,\\ 1& \textrm{if}\ \ [M_G]^{fl}\neq 0\ \textrm{but is invertible},\\ 2& \textrm{if}\ \ [M_G]^{fl}\ \textrm{is not invertible}, \end{cases}$

$l_2=H^1(G,[M_G]^{fl})\simeq Sha^1_w(G,[M_G]^{fl})$ ,

$l_3=Sha^1_w(G,(M_G)^\circ)\simeq Sha^2_w(G,([M_G]^{fl})^\circ)$ ,

$l_4=H^1(G,([M_G]^{fl})^{fl})\simeq Sha^2_w(G,[M_G]^{fl})$ via the command H1(G).

TorusInvariantsHAP

TorusInvariantsHap(G)

returns

$TI_G=[l_1,l_2,l_3,l_4]$ where

$l_1=\begin{cases} 0& \textrm{if}\ \ [M_G]^{fl}=0,\\ 1& \textrm{if}\ \ [M_G]^{fl}\neq 0\ \textrm{but is invertible},\\ 2& \textrm{if}\ \ [M_G]^{fl}\ \textrm{is not invertible}, \end{cases}$

$l_2=H^1(G,[M_G]^{fl})\simeq Sha^1_w(G,[M_G]^{fl})$ ,

$l_3=Sha^1_w(G,(M_G)^\circ)\simeq Sha^2_w(G,([M_G]^{fl})^\circ)$ ,

$l_4=Sha^2_w(G,[M_G]^{fl})$ via the command ShaOmegaFromGroup(

$[M_G]^{fl},2,G$ ).
This function needs HAP package in GAP.

ConjugacyClassesSubgroups2TorusInvariants

ConjugacyClassesSubgroups2TorusInvariants(G)

returns the records ConjugacyClassesSubgroups2 and TorusInvariants where
ConjugacyClassesSubgroups2 is the list

$[g_1,\ldots,g_m]$ of conjugacy classes of subgroups of

$G\leq {\rm GL}(n,\mathbb{Z})$ with the fixed ordering via the function ConjugacyClassesSubgroups2(G) ( [HY17, Section 4.1]) and
TorusInvariants is the list [TorusInvariants(

$g_1$ )

$,\ldots,$ TorusInvariants(

$g_m$ )] via the function TorusInvariants(G).

PossibilityOfStablyEquivalentSubdirectProducts

PossibilityOfStablyEquivalentSubdirectProducts(G,G',
ConjugacyClassesSubgroups2TorusInvariants(G),
ConjugacyClassesSubgroups2TorusInvariants(G'))

returns the list

$l$ of the subdirect products

$\widetilde{H}\leq G\times G^\prime$ of

$G$ and

$G^\prime$ up to

$({\rm GL}(n_1,\mathbb{Z})\times {\rm GL}(n_2,\mathbb{Z}))$ -conjugacy which satisfy

$TI_{\varphi_1(H)}=TI_{\varphi_2(H)}$ for any

$H\leq \widetilde{H}$ where

$\widetilde{H}\leq G\times G^\prime$ is a subdirect product of

$G$ and

$G^\prime$ which acts on

$M_G$ and

$M_{G^\prime}$ through the surjections

$\varphi_1: \widetilde{H} \rightarrow G$ and

$\varphi_2: \widetilde{H} \rightarrow G^\prime$ respectively (indeed, this function computes it for

$H$ up to conjugacy for the sake of saving time). In particular, if the length of the list

$l$ is zero, then we find that

$[M_G]^{fl}$ and

$[M_{G^\prime}]^{fl}$ are not weak stably

$k$ -equivalent.

FlabbyResolutionLowRank

FlabbyResolutionLowRank(G).actionF

returns the matrix representation of the action of

$G$ on

$F$ where

$F$ is a suitable flabby class of

$M_G$ (

$F]=[M_G]^{fl}$ ) with low rank by using backtracking techniques (see [HY17, Chapter 5], see also [HHY Algorithm 4.1 (3)]).

Each isomorphism class of irreducible permutation $\widetilde{H}$ -lattices corresponds to a conjugacy class of subgroup $H$ of $\widetilde{H}$ by $H \leftrightarrow \mathbb{Z}[\widetilde{H}/H]$ . Let $H_1=\{1\},\ldots,H_r=\widetilde{H}$ be all conjugacy classes of subgroups of $\widetilde{H}$ whose ordering corresponds to the GAP function ConjugacyClassesSubgroups2( $\widetilde{H}$ ) (see [HY17, Section 4.1, page 42]).

We suppose that $[F]=[F^\prime]$ as $\widetilde{H}$ -lattices. Then we have $\begin{align} \left(\bigoplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus x_i}\right)\oplus F^{\oplus b_1} \ \simeq\ \left(\bigoplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus y_i}\right)\oplus F^{\prime\,\oplus b_1}\tag{2} \end{align}$ where $b_1=1$ . We write the equation $(2)$ as $\begin{align} \bigoplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i}\ \simeq\ (F-F^\prime)^{\oplus (-b_1)}\tag{3} \end{align}$ formally where $a_i=x_i-y_i\in\mathbb{Z}$ . Then we may consider " $\,F-F^\prime\,$ " formally in the sene of $(2)$ . By computing some ${\rm GL}(n,\mathbb{Z})$ -conjugacy class invariants, we will give a necessary condition for $[F]=[F^\prime]$ .

Let $\{c_1,\ldots,c_r\}$ be a set of complete representatives of the conjugacy classes of $\widetilde{H}$ . Let $A_i(c_j)$ be the matrix representation of the factor coset action of $c_j\in \widetilde{H}$ on $\mathbb{Z}[\widetilde{H}/H_i]$ and $B(c_j)$ be the matrix representation of the action of $c_j\in \widetilde{H}$ on $F-F^\prime$ .

By $(3)$ , for each $c_j\in \widetilde{H}$ , we have $\begin{align} \sum_{i=1}^r a_i\, {\rm tr}\, A_i(c_j)+ b_1\, {\rm tr}\, B(c_j)=0\tag{4} \end{align}$ where ${\rm tr}\,A$ is the trace of the matrix $A$ . Similarly, we consider the rank of $H^0=\widehat Z^0$ . For each $H_j$ , we get $\begin{align} \sum_{i=1}^r a_i\, {\rm rank}\ \widehat Z^0(H_j,\mathbb{Z}[\widetilde{H}/H_i]) +b_1\, {\rm rank}\ \widehat Z^0(H_j,F-F^\prime)=0.\tag{5} \end{align}$ Finally, we compute $\widehat H^0$ . Let $Sy_p(A)$ be a $p$ -Sylow subgroup of an abelian group $A$ . $Sy_p(A)$ can be written as a direct product of cyclic groups uniquely. Let $n_{p,e}(Sy_p(A))$ be the number of direct summands of cyclic groups of order $p^e$ . For each $H_j,p,e$ , we get $\begin{align} \sum_{i=1}^r a_i\, n_{p,e}(Sy_p(\widehat H^0(H_j,\mathbb{Z}[\widetilde{H}/H_i]))) +b_1\, n_{p,e}(Sy_p(\widehat H^0(H_j,F-F^\prime)))=0.\tag{6} \end{align}$ By the equalities $(4)$ , $(5)$ and $(6)$ , we may get a system of linear equations in $a_1,\dots,a_r,b_1$ over $\mathbb{Z}$ . Namely, we have that $[F]=[F^\prime]$ as $\widetilde{H}$ -lattices $\Longrightarrow$ there exist $a_1,\ldots,a_r\in\mathbb{Z}$ and $b_1=\pm 1$ which satisfy $(3)$ $\Longrightarrow$ this system of linear equations has an integer solution in $a_1,\ldots,a_r$ with $b_1=\pm 1$ .

In particular, if this system of linear equations has no integer solutions, then we conclude that $[F]\neq [F^\prime]$ as $\widetilde{H}$ -lattices.

PossibilityOfStablyEquivalentFSubdirectProduct

PossibilityOfStablyEquivalentFSubdirectProduct(H~)

returns a basis

$\mathcal{L}=\{l_1,\dots,l_s\}$ of the solution space

$\{[a_1,\ldots,a_r,b_1]\mid a_i, b_1\in\mathbb{Z}\}$ of the system of linear equations which is obtained by the equalities

$(4)$ ,

$(5)$ and

$(6)$ and gives all possibilities that establish the equation

$(3)$ for a subdirect product

$\widetilde{H}\leq G\times G^\prime$ of

$G$ and

$G^\prime$ .

PossibilityOfStablyEquivalentMSubdirectProduct

PossibilityOfStablyEquivalentMSubdirectProduct(H~)

returns the same as PossibilityOfStablyEquivalentFSubdirectProduct(H~) but with respect to

$M_G$ and

$M_{G^\prime}$ instead of

$F$ and

$F^\prime$ .

PossibilityOfStablyEquivalentFSubdirectProduct with "H2" option

PossibilityOfStablyEquivalentFSubdirectProduct(H~:H2)

returns the same as PossibilityOfStablyEquivalentFSubdirectProduct(H~) but using also the additional equality

$\begin{align} \sum_{i=1}^r a_i\, n_{p,e}(Sy_p(H^2(\widetilde{H},\mathbb{Z}[\widetilde{H}/H_i])))+b_1\, n_{p,e}(Sy_p(H^2(\widetilde{H},F-F^\prime)))=0\tag{7} \end{align}$ and the equalities

$(4)$ ,

$(5)$ and

$(6)$ .

PossibilityOfStablyEquivalentMSubdirectProduct with "H2" option

PossibilityOfStablyEquivalentMSubdirectProduct(H~:H2)

returns the same as PossibilityOfStablyEquivalentFSubdirectProduct(H~:H2) but with respect to

$M_G$ and

$M_{G^\prime}$ instead of

$F$ and

$F^\prime$ .

In general, we will provide a method in order to confirm the isomorphism $\begin{align} \left(\bigoplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i}\right)\oplus F^{\oplus b_1} \simeq \left(\bigoplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i^{\prime}}\right)\oplus F^{\prime\,\oplus b_1^{\prime}} \tag{8} \end{align}$ with $a_i,a_i^{\prime}\geq 0$ , $b_1,b_1^\prime\geq 1$ , although it is needed by trial and error.

Let $G_1$ (resp. $G_2$ ) be the matrix representation group of the action of $\widetilde{H}$ on the left-hand side $(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i})$ $\oplus$ $F^{\oplus b_1}$ (resp. the right-hand side $(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i^{\prime}})$ $\oplus$ $F^{\oplus b_1^{\prime}}$ ) of the isomorphism $(8)$ . Let $\mathcal{P}=\{P_1,\dots,P_m\}$ be a basis of the solution space of $G_1P=PG_2$ where $m={\rm rank}_\mathbb{Z}$ ${\rm Hom}(G_1,G_2)$ $=$ ${\rm rank}_\mathbb{Z}$ ${\rm Hom}_{\widetilde{H}}(M_{G_1},M_{G_2})$ . Our aim is to find the matrix $P$ which satisfies $G_1 P=P G_2$ by using computer effectively. If we can get a matrix $P$ with det $P=\pm 1$ , then $G_1$ and $G_2$ are ${\rm GL}(n,\mathbb{Z})$ -conjugate where $n$ is the rank of both sides of $(8)$ and hence the isomorphism $(8)$ established. This implies that the flabby class $[F^{\oplus b_1}]=[F^{\prime\,\oplus b_1^{\prime}}]$ as $\widetilde{H}$ -lattices.

StablyEquivalentFCheckPSubdirectProduct

StablyEquivalentFCheckPSubdirectProduct(H~,l1,l2)

returns a basis

$\mathcal{P}=\{P_1,\dots,P_m\}$ of the solution space of

$G_1P=PG_2$ where

$m={\rm rank}_\mathbb{Z}$

${\rm Hom}(G_1,G_2)$ and

$G_1$ (resp.

$G_2$ ) is the matrix representation group of the action of

$\widetilde{H}$ on

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i})\oplus F^{\oplus b_1}$ (resp.

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i^{\prime}})\oplus F^{\prime\,\oplus b_1^{\prime}}$ ) with the isomorphism

$(8)$ for a subdirect product

$\widetilde{H}\leq G\times G^\prime$ of

$G$ and

$G^\prime$ , and lists

$l_1=[a_1,\ldots,a_r,b_1]$ ,

$l_2=[a_1^{\prime},\ldots,a_r^{\prime},b_1^{\prime}]$ , if

$P$ exists. If such

$P$ does not exist, this returns [ ].

StablyEquivalentMCheckPSubdirectProduct

StablyEquivalentMCheckPSubdirectProduct(H~,l1,l2)

returns the same as StablyEquivalentFCheckPSubdirectProduct(H~,l1,l2) but with respect to

$M_G$ and

$M_{G^\prime}$ instead of

$F$ and

$F^\prime$ .

StablyEquivalentFCheckMatSubdirectProduct

StablyEquivalentFCheckMatSubdirectProduct(H~,l1,l2,P)

returns true if

$G_1P=PG_2$ and det

$P=\pm 1$ where

$G_1$ (resp.

$G_2$ ) is the matrix representation group of the action of

$\widetilde{H}$ on

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i})\oplus F^{\oplus b_1}$ (resp.

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i^{\prime}})\oplus F^{\prime\,\oplus b_1^{\prime}}$ ) with the isomorphism

$(8)$ for a subdirect product

$\widetilde{H}\leq G\times G^\prime$ of

$G$ and

$G^\prime$ , and lists

$l_1=[a_1,\ldots,a_r,b_1]$ ,

$l_2=[a_1^{\prime},\ldots,a_r^{\prime},b_1^{\prime}]$ . If not, this returns false.

StablyEquivalentMCheckMatSubdirectProduct

StablyEquivalentMCheckMatSubdirectProduct(H~,l1,l2,P)

returns the same as StablyEquivalentFCheckMatSubdirectProduct(H~,l1,l2,P) but with respect to

$M_G$ and

$M_{G^\prime}$ instead of

$F$ and

$F^\prime$ .

StablyEquivalentFCheckGenSubdirectProduct

StablyEquivalentFCheckGenSubdirectProduct(H~,l1,l2)

returns the list

$[\mathcal{M}_1,\mathcal{M}_2]$ where

$\mathcal{M}_1=[g_1,\ldots,g_t]$ (resp.

$\mathcal{M}_2=[g_1^{\prime},\ldots,g_t^{\prime}]$ ) is a list of the generators of

$G_1$ (resp.

$G_2$ ) which is the matrix representation group of the action of

$\widetilde{H}$ on

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i})\oplus F^{\oplus b_1}$ (resp.

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i^{\prime}})\oplus F^{\prime\,\oplus b_1^{\prime}}$ ) with the isomorphism

$(8)$ for a subdirect product

$\widetilde{H}\leq G\times G^\prime$ of

$G$ and

$G^\prime$ , and lists

$l_1=[a_1,\ldots,a_r,b_1]$ ,

$l_2=[a_1^{\prime},\ldots,a_r^{\prime},b_1^{\prime}]$ .

StablyEquivalentMCheckGenSubdirectProduct

StablyEquivalentMCheckGenSubdirectProduct(H~,l1,l2)

returns the same as StablyEquivalentMCheckGenSubdirectProduct(H~,l1,l2) but with respect to

$M_G$ and

$M_{G^\prime}$ instead of

$F$ and

$F^\prime$ .

By applying the function StablyEquivalentFCheckPSubdirectProduct, we get a basis $\mathcal{P}=\{P_1,\dots,P_m\}$ of the solution space of $G_1P=PG_2$ with det $P_i=\pm 1$ for some $1\leq i\leq m$ where $G_1$ (resp. $G_2$ ) is the matrix representation group of the action of $\widetilde{H}$ on the left-hand side $(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i})\oplus F^{\oplus b_1}$ (resp. the right-hand side $(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i^{\prime}})\oplus F^{\prime\,\oplus b_1^{\prime}}$ ) of the isomorphism $(8)$ and $m={\rm rank}_\mathbb{Z}$ ${\rm Hom}(G_1,G_2)$ .

However, in general, we have that det $P_i\neq\pm 1$ for any $1\leq i\leq m$ . In the general case, we should seek a matrix $P$ with det $P=\pm 1$ which is given as a linear combination $P=\sum_{i=1}^m c_i P_i$ . This task is important for us and not easy in general even if we use a computer.

We made the following GAP algorithms which may find a matrix $P=\sum_{i=1}^m c_i P_i$ with $G_1P=PG_2$ and det $P=\pm 1$ .

StablyEquivalentFCheckPSubdirectProduct( $\widetilde{H},l_1,l_2$ ) although it works in more general situations.

SearchPRowBlocks

SearchPRowBlocks(P)

returns the records bpBlocks and rowBlocks where bpBlocks (resp. rowBlocks) is the decomposition of the list

$l=[1,\ldots,m]$ (resp.

$l=[1,\ldots,n]$ ) with

$m={\rm rank}_\mathbb{Z}$

${\rm Hom}(G_1,G_2)$ (resp.

$n={\rm size}$

$G_1$ ) according to the direct sum decomposition of

$M_{G_1}$ for a basis

$\mathcal{P}=\{P_1,\dots,P_m\}$ of the solution space of

$G_1P=PG_2$ where

$G_1$ (resp.

$G_2$ ) is the matrix representation group of the action of

$\widetilde{H}$ on the left-hand side

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i})\oplus F^{\oplus b_1}$ (resp. the right-hand side

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i^{\prime}})\oplus F^{\prime\,\oplus b_1^{\prime}}$ ) of the isomorphism

$(8)$ .

We write $B$ [ $t$ ] $=$ SearchPRowBlocks( $\mathcal{P}$ ).bpBlocks[ $t$ ] and $R$ [ $t$ ] $=$ SearchPRowBlocks( $\mathcal{P}$ ).rowBlocks[ $t$ ].

SearchPFilterRowBlocks

SearchPFilterRowBlocks(P,B[t],R[t],j)

returns the lists

$\{M_s\}$ where

$M_s$ is the

$n_t \times n$ matrix with all invariant factors

$1$ which is of the form

$M_s=\sum_{i\,\in\,B{\rm [}t{\rm ]}} c_i P_i^\prime$

$(c_i\in\{0,1\})$ at most

$j$ non-zero

$c_i$ 's and

$P_i^\prime$ is the submatrix of

$P_i$ consists of

$R$ [

$t$ ] rows with

$n_t={\rm length}($

$R$ [

$t$ ]

$)$ for a basis

$\mathcal{P}=\{P_1,\dots,P_m\}$ of the solution space of

$G_1P=PG_2$ where

$G_1$ (resp.

$G_2$ ) is the matrix representation group of the action of

$\widetilde{H}$ on the left-hand side

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i})\oplus F^{\oplus b_1}$ (resp. the right-hand side

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i^{\prime}})\oplus F^{\prime\,\oplus b_1^{\prime}}$ ) of the isomorphism

$(8)$ ,

$B$ [

$t$ ]

$=$ SearchPRowBlocks(

$\mathcal{P}$ ).bpBlocks[

$t$ ],

$R$ [

$t$ ]

$=$ SearchPRowBlocks(

$\mathcal{P}$ ).rowBlocks[

$t$ ]) and

$j\geq 1$ .

SearchPFilterRowBlocks(P,B[t],R[t],j,C)

returns the same as SearchPFilterRowBlocks(P,B[t],R[t],j) but with respect to

$c_i\in C$ instead of

$c_i\in\{0,1\}$ for the list

$C$ of integers.

SearchPFilterRowBlocksRandomMT

SearchPFilterRowBlocksRandomMT(P,B[t],R[t],u)

returns the same as SearchPFilterRowBlocks(P,B[t],R[t],j) but with respect to random

$u$

$c_i$ 's via Mersenne Twister instead of at most

$j$ non-zero

$c_i$ 's for integer

$u\geq 1$ .

SearchPFilterRowBlocksRandomMT(P,B[t],R[t],u,C)

returns the same as SearchPFilterRowBlocksRandomMT(P,B[t],R[t],u) but with respect to

$c_i\in C$ instead of

$c_i\in\{0,1\}$ for the list

$C$ of integers.

SearchPMergeRowBlock

SearchPMergeRowBlock(m1,m2)

returns all concatenations of the matrices

$M_s$ and

$M_t$ vertically with all invariant factors

$1$ (resp. a concatenation of the matrices

$M_s$ and

$M_t$ vertically with determinant

$\pm 1$ ) for

$m_1=\{M_s\}$ and

$m_2=\{M_t\}$ where

$M_s$ are

$n_1\times n$ matrices and

$M_t$ are

$n_2\times n$ matrices with

$n_1+n_2$ <

$n$ (resp.

$n_1+n_2=n$ ).

When there exists $t\in\mathbb{Z}$ such that $R$ [ $t$ ] $=\{j\}$ , we can use:

SearchPLinear

SearchPLinear(M,P1)

returns the list

$\{{\rm det}(M+P_i)\}_{i\,\in\,B{\rm [}t{\rm ]}}$ of integers for an

$n\times n$ matrix

$M$ which is obtained by inserting the zero row into the

$j$ -th row of

$(n-1)\times n$ matrix

$M_s=\sum_{i\,\not\in\,B{\rm [}t{\rm ]}} c_i P_i^\prime$ with all invariant factors

$1$ and

$\mathcal{P}_1=\{P_i\}_{i\,\in\, B{\rm [}t{\rm ]}}$ where

$B$ [

$t$ ]

$=$ SearchPRowBlocks(

$\mathcal{P}$ ).bpBlocks[

$t$ ],

$P_i^\prime$ is the submatrix of

$P_i$ deleting the

$j$ -th row, and

$\mathcal{P}=\{P_1,\ldots,P_m\}$ is obtained by StablyEquivalentFCheckPSubdirectProduct(

$\widetilde{H}$ ,

$l_1$ ,

$l_2$ ) under the assumption that there exists

$t\in\mathbb{Z}$ such that

$R$ [

$t$ ]

$=\{j\}$ .

When there exist $t_1, t_2\in\mathbb{Z}$ such that $R$ [ $t_1$ ] $=\{j_1\}$ , $R$ [ $t_2$ ] $=\{j_2\}$ , we can use:

SearchPBilinear

SearchPBilinear(M,P1,P2)

returns the matrix

$[{\rm det}(M+P_{i_1}+P_{i_2})]_{i_1\,\in\,B{\rm [}t_1{\rm ]},i_2\,\in\,B{\rm [}t_2{\rm ]}}$ for an

$n\times n$ matrix

$M$ which is obtained by inserting the two zero rows into the

$j_1$ -th row and the

$j_2$ -th row of

$(n-2)\times n$ matrix

$M_s=\sum_{i\,\not\in\,B{\rm [}t_1{\rm ]} \,\cup\,B{\rm [}t_2{\rm ]}} c_i P_i^\prime$ with all invariant factors

$1$ and

$\mathcal{P}_1=\{P_{i_1}\}_{i_1 \in B{\rm [}t_1{\rm ]}}$ ,

$\mathcal{P}_2=\{P_{i_2}\}_{i_2 \in B{\rm [}t_2{\rm ]}}$ , where

$B$ [

$t_1$ ]

$=$ SearchPRowBlocks(

$\mathcal{P}$ ).bpBlocks[

$t_1$ ],

$B$ [

$t_2$ ]

$=$ SearchPRowBlocks(

$\mathcal{P}$ ).bpBlocks[

$t_2$ ],

$P_i^\prime$ is the submatrix of

$P_i$ deleting the

$j_1$ -th and the

$j_2$ -th rows, and

$\mathcal{P}=\{P_1,\ldots,P_m\}$ is obtained by StablyEquivalentFCheckPSubdirectProduct(

$\widetilde{H}$ ,

$l_1$ ,

$l_2$ ) under the assumption that there exist

$t_1, t_2\in\mathbb{Z}$ such that

$R$ [

$t_1$ ]

$=\{j_1\}$ and

$R$ [

$t_2$ ]

$=\{j_2\}$ .

When there exists $t\in\mathbb{Z}$ such that $R$ [ $t$ ] $=\{j_1,j_2\}$ , we can use:

SearchPQuadratic

SearchPQuadratic(M,P1)

returns the matrix

$[\frac{1}{2}({\rm det}(M+P_{i_1}+P_{i_2})-{\rm det}(M+P_{i_1})-{\rm det}(M+P_{i_2}))]_{i_1,i_2\,\in\,B{\rm [}t{\rm ]}}$ for an

$n\times n$ matrix

$M$ which is obtained by inserting the two zero rows into the

$j_1$ -th row and the

$j_2$ -th row of

$(n-2)\times n$ matrix

$M_s=\sum_{i\,\not\in\,B{\rm [}t{\rm ]}} c_i P_i^\prime$ with all invariant factors

$1$ and

$\mathcal{P}_1=\{P_i\}_{i \in B{\rm [}t{\rm ]}}$ , where

$B$ [

$t$ ]

$=$ SearchPRowBlocks(

$\mathcal{P}$ ).bpBlocks[

$t$ ],

$P_i^\prime$ is the submatrix of

$P_i$ deleting the

$j_1$ -th and

$j_2$ -th rows and

$\mathcal{P}=\{P_1,\ldots,P_m\}$ is obtained by StablyEquivalentFCheckPSubdirectProduct(

$\widetilde{H}$ ,

$l_1$ ,

$l_2$ ) under the assumption that there exists

$t\in\mathbb{Z}$ such that

$R$ [

$t$ ]

$=\{j_1,j_2\}$ .

When $R$ [ $1$ ] $=\{1,\ldots,m\}$ , we can use:

SearchP1

SearchP1(P)

returns a matrix

$P=\sum_{i=1}^m c_i P_i$ with

$c_i\in\{0,1\}$ ,

$G_1P=PG_2$ and det

$P=\pm 1$ where

$G_1$ (resp.

$G_2$ ) is the matrix representation group of the action of

$\widetilde{H}$ on the left-hand side

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i})\oplus F^{\oplus b_1}$ (resp. the right-hand side

$(\oplus_{i=1}^r \mathbb{Z}[\widetilde{H}/H_i]^{\oplus a_i^{\prime}})\oplus F^{\prime\,\oplus b_1^{\prime}}$ ) of the isomorphism

$(8)$ for

$\mathcal{P}=\{P_1,\ldots,P_m\}$ which is obtained by StablyEquivalentFCheckPSubdirectProduct(

$\widetilde{H}$ ,

$l_1$ ,

$l_2$ ) under the assumption that

$R$ [

$1$ ]

$=\{1,\ldots,m\}$ .

SearchP1(P,C)

returns the same as SearchP1(P) but with respect to

$c_i\in C$ instead of

$c_i\in\{0,1\}$ for the list

$C$ of integers.

Endomorphismring

Endomorphismring(G)

returns a

$\mathbb{Z}$ -basis of

${\rm End}_{\mathbb{Z}[G]}(M_G)$ for a finite subgroup

$G$ of

${\rm GL}(n,\mathbb{Z})$ .

IsCodimJacobsonEnd1

IsCodimJacobsonEnd1(G,p)

returns true (resp. false) if

${\rm dim}_{\mathbb{Z}/p\mathbb{Z}} (E/pE)\big/J(E/pE)=1$ (resp.

$\neq 1$ ) where

$E={\rm End}_{\mathbb{Z}[G]}(M_G)$ for a finite subgroup

$G$ of

${\rm GL}(n,\mathbb{Z})$ and prime number

$p$ . If this returns true, then

$M_G \otimes_\mathbb{Z} \mathbb{Z}_p$ is an indecomposable

$\mathbb{Z}_p[G]$ -lattice. In particular,

$M_G$ is an indecomposable

$G$ -lattice (see [HY, Lemma 6.11]).

IdempotentsModp

IdempotentsModp(B,p)

returns all idempotents of

$R/pR$ for a

$\mathbb{Z}$ -basis

$B$ of a subring

$R$ of

$n\times n$ matrices

$M(n,\mathbb{Z})$ over

$\mathbb{Z}$ and prime number

$p$ . If this returns only the zero and the identity matrices when

$R={\rm End}_{\mathbb{Z}[G]}(M_G)$ , then

$M_G \otimes_\mathbb{Z} \mathbb{Z}_p$ is an indecomposable

$\mathbb{Z}_p[G]$ -lattice. In particular,

$M_G$ is an indecomposable

$G$ -lattice (see [HY, Lemma 6.10]).

ConjugacyClassesSubgroups2WSEC

ConjugacyClassesSubgroups2WSEC(G)

returns the records ConjugacyClassesSubgroups2 and WSEC where
ConjugacyClassesSubgroups2 is the list

$[g_1,\ldots,g_m]$ of conjugacy classes of subgroups of

$G\leq {\rm GL}(n,\mathbb{Z})$

$(n=3,4)$ with the fixed ordering via the function ConjugacyClassesSubgroups2(

$G$ ) (see [HY17, Section 4.1]) and
WSEC is the list

$[w_1,\ldots,w_m]$ where

$g_i$ is in the

$w_i$ -th weak stably

$k$ -equivalent class

${\rm WSEC}_{w_i}$ in dimension

$n$ .

MaximalInvariantNormalSubgroup

MaximalInvariantNormalSubgroup(G,ConjugacyClassesSubgroups2WSEC(G))

returns the maximal normal subgroup

$N$ of

$G$ which satisfies that

$\pi(H_1)=\pi(H_2)$ implies

$\psi(H_1)=\psi(H_2)$ for any

$H_1,H_2\leq G$ where

$\pi:G\rightarrow G/N$ is the natural homomorphism,

$\psi : H_i\mapsto w_i$ , and

$H_i$ is in the

$w_i$ -th weak stably

$k$ -equivalent class

${\rm WSEC}_{w_i}$ in dimension

$n$ .

PossibilityOfStablyEquivalentSubdirectProducts with "WSEC" option

PossibilityOfStablyEquivalentSubdirectProducts(G,G',
ConjugacyClassesSubgroups2WSEC(G),
ConjugacyClassesSubgroups2WSEC(G'),["WSEC"])

returns the list

$l$ of the subdirect products

$\widetilde{H}\leq G\times G^\prime$ of

$G$ and

$G^\prime$ up to

$({\rm GL}(n_1,\mathbb{Z})\times {\rm GL}(n_2,\mathbb{Z}))$ -conjugacy which satisfy

$w_1=w_2$ for any

$H\leq \widetilde{H}$ where

$\varphi_i(H)$ is in the

$w_i$ -th weak stably

$k$ -equivalent class

${\rm WSEC}_{w_i}$ in dimension

$n$

$(n=3,4)$ and

$\widetilde{H}\leq G\times G^\prime$ is a subdirect product of

$G$ and

$G^\prime$ which acts on

$M_G$ and

$M_{G^\prime}$ through the surjections

$\varphi_1: \widetilde{H} \rightarrow G$ and

$\varphi_2: \widetilde{H} \rightarrow G^\prime$ respectively (indeed, this function computes it for

$H$ up to conjugacy for the sake of saving time).

IsomorphismFromSubdirectProduct

IsomorphismFromSubdirectProduct(H~)

returns the isomorphism

$\sigma: G/N\rightarrow G^\prime/N^\prime$ which satisfies

$\sigma(\varphi_1(h)N)=\varphi_2(h)N^\prime$ for any

$h \in \widetilde{H}$ where

$N=\varphi_1({\rm Ker}(\varphi_2))$ and

$N^\prime=\varphi_2({\rm Ker}(\varphi_1))$ for a subdirect product

$\widetilde{H}\leq G\times G^\prime$ of

$G$ and

$G^\prime$ with surjections

$\varphi_1: \widetilde{H} \rightarrow G$ and

$\varphi_2: \widetilde{H} \rightarrow G^\prime$ .

AutGSubdirectProductsWSECInvariant

AutGSubdirectProductsWSECInvariant(G)

returns subdirect products

$\widetilde{H}_m$

$=$

$\{(g,g^{\sigma_m})\mid g\in G, g^{\sigma_m}\in G^{\sigma_m}\}$

$(1\leq m\leq s)$ of

$G$ and

$G^{\sigma_m}$ where

$\{\sigma_1,\ldots,\sigma_s\}$ is a complete set of representatives of the double coset

$X\backslash Z/X$ ,

$\begin{align*} {\rm Inn}(G)\leq X\leq Y\leq Z\leq {\rm Aut}(G), \end{align*}$

$\begin{align*} X&={\rm Aut}_{{\rm GL}(n,\mathbb{Z})}(G)=\{\sigma\in{\rm Aut}(G)\mid G\ {\rm and}\ G^\sigma\ {\rm are}\ {\rm conjugate}\ {\rm in} {\rm GL}(n,\mathbb{Z})\}\simeq N_{{\rm GL}(n,\mathbb{Z})}(G)/Z_{{\rm GL}(n,\mathbb{Z})}(G),\\ Y&=\{\sigma\in{\rm Aut}(G)\mid [M_G]^{fl}=[M_{G^\sigma}]^{fl}\ {\rm as}\ \widetilde{H}\textrm{-lattices}\ {\rm where}\ \widetilde{H}=\{(g,g^\sigma)\mid g\in G\}\simeq G\},\\ Z&=\{\sigma\in{\rm Aut}(G)\mid [M_H]^{fl}\sim [M_{H^\sigma}]^{fl}\ {\rm for}\ {\rm any}\ H\leq G\}, \end{align*}$

${\rm Inn}(G)$ is the group of inner automorphisms on

$G$ ,

${\rm Aut}(G)$ is the group of automorphisms on

$G$ ,

$N_{{\rm GL}(n,\mathbb{Z})}(G)$ is the normalizer of

$G$ in

${\rm GL}(n,\mathbb{Z})$ and

$Z_{{\rm GL}(n,\mathbb{Z})}(G)$ is the centralizer of

$G$ in

${\rm GL}(n,\mathbb{Z})$ .

AutGSubdirectProductsWSECInvariantGen

AutGSubdirectProductsWSECInvariantGen(G)

returns the same as AutGSubdirectProductsWSECInvariant(

$G$ ) but with respect to

$\{\sigma_1,\ldots,\sigma_t\}$ where

$\sigma_1,\ldots,\sigma_t\in Z$ are some minimal number of generators of the double cosets of

$X\backslash Z/X$ , i.e. minimal number of elements

$\sigma_1,\ldots,\sigma_t\in Z$ which satisfy

$\langle \sigma_1,\ldots,\sigma_t,x\mid x\in X\rangle=Z$ , instead of a complete set of representatives of the double coset

$X\backslash Z/X$ . If this returns [], then we get

$X=Y=Z$ .

AutGLnZ

AutGLnZ(G)

returns

$\begin{align*} X={\rm Aut}_{{\rm GL}(n,\mathbb{Z})}(G)=\{\sigma\in{\rm Aut}(G)\mid G\ {\rm and}\ G^\sigma\ {\rm are}\ {\rm conjugate}\ {\rm in}\ {\rm GL}(n,\mathbb{Z})\}\simeq N_{{\rm GL}(n,\mathbb{Z})}(G)/Z_{{\rm GL}(n,\mathbb{Z})}(G). \end{align*}$

N3WSECMembersTable

N3WSECMembersTable[r][i]

returns an integer

$j$ which satisfies that

$N_{3,j}$ is the

$i$ -th group in the weak stably

$k$ -equivalent class

${\rm WSEC}_r$ .

N4WSECMembersTable

N4WSECMembersTable[r][i]

is the same as N3WSECMembersTable[r][i] but using

$N_{4,j}$ instead of

$N_{3,j}$ .

I4WSECMembersTable

I4WSECMembersTable[r][i]

is the same as N3WSECMembersTable[r][i] but using

$I_{4,j}$ instead of

$N_{3,j}$ .

AutGWSECINvariantSmallDegreeTest

AutGWSECINvariantSmallDegreeTest(G)

returns the list

$l=[l_1,\ldots,l_s]$

$(l_1\leq \cdots\leq l_s$ ) of integers with the minimal

$l_s,\ldots,l_1$ which satisfies

$Z=Z^\prime$ where

$\begin{align*} Z&=\{\sigma\in{\rm Aut}(G)\mid [M_H]^{fl}\sim [M_{H^\sigma}]^{fl} \mbox{ for any } H\leq G\},\\ Z^\prime&=\{\sigma\in{\rm Aut}(G)\mid [M_H]^{fl}\sim [M_{H^\sigma}]^{fl} \mbox{ for any } H\leq G \mbox{ with } [G:H]\in l\} \end{align*}$ for

$G\leq {\rm GL}(n,\mathbb{Z})$

$(n=3,4)$ .

References

[HHY20] Sumito Hasegawa, Akinari Hoshi and Aiichi Yamasaki, Rationality problem for norm one tori in small dimensions, Math. Comp. 89 (2020) 923-940. AMS Extended version: arXiv:1811.02145.

[HY17] Akinari Hoshi and Aiichi Yamasaki, Rationality problem for algebraic tori, Mem. Amer. Math. Soc. 248 (2017) no. 1176, v+215 pp. AMS Preprint version: arXiv:1210.4525.

[HY] Akinari Hoshi and Aiichi Yamasaki, Birational classification for algebraic tori, arXiv:2112.02280.