FlabbyResolutionFromBase.gap

Definition of $M_G$: Let $G$ be a finite subgroup of $\mathrm{GL}(n,\mathbb{Z})$ . 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 $\sigma(u_i)=\sum_{j=1}^n a_{i,j}u_j$ 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

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

H0

H0(G)

returns the Tate cohomology group

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

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

H1

H1(G)

returns the cohomology group

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

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

Z0lattice

Z0lattice(G)

returns a

$\mathbb{Z}$ -basis of the group of Tate

$0$ -cocycles

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

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

ConjugacyClassesSubgroups2, ConjugacyClassesSubgroupsFromPerm

ConjugacyClassesSubgroups2(G)

ConjugacyClassesSubgroupsFromPerm(G)

returns the list of conjugacy classes of subgroups of a group

$G$ . We use this function because the ordering of the conjugacy classes of subgroups of

$G$ by the built-in function ConjugacyClassesSubgroups(G) is not fixed for some groups. If a group

$G$ is too big, ConjugacyClassesSubgroups2(G) may not work well.

IsFlabby

IsFlabby(G)

returns whether

$G$ -lattice

$M_G$ is flabby or not.

IsCoflabby

IsCoflabby(G)

returns whether

$G$ -lattice

$M_G$ is coflabby or not.

IsInvertible

IsInvertible(G)

returns whether

$G$ -lattice

$M_G$ is invertible or not.

FlabbyResoluton

FlabbyResolution(G)

returns a flabby resolution

$0 \rightarrow M_G \xrightarrow{\iota} P \xrightarrow{\phi} F \rightarrow 0$ of

$M_G$ as follows:

FlabbyResolution(G).actionP

returns the matrix representation of the action of

$G$ on

$P$ ;

FlabbyResolution(G).actionF

returns the matrix representation of the action of

$G$ on

$F$ ;

FlabbyResolution(G).injection

returns the matrix which corresponds to the injection

$\iota: M_G \rightarrow P$ ;

FlabbyResolution(G).surjection

returns the matrix which corresponds to the surjection

$\phi: P \rightarrow F$ .

IsInvertibleF

IsInvertibleF(G)

returns whether

$[M_G]^{fl}$ is invertible.

flfl

flfl(G)

returns the

$G$ -lattice

$E$ with

$[[M_G]^{fl}]^{fl}=[E]$ .

PossibilityOfStablyPermutationF

PossibilityOfStablyPermutationF(G)

returns a basis

$\mathcal{L}=\{l_1,\dots,l_s\}$ of the solution space of the system of linear equations which is obtained by computing some

$\mathbb{Z}$ -class invariants. Each isomorphism class of irreducible permutation

$G$ -lattices corresponds to a conjugacy class of subgroup

$H$ of

$G$ by

$H \leftrightarrow \mathbb{Z}[G/H]$ . Let

$H_1,\dots,H_r$ be conjugacy classes of subgroups of

$G$ whose ordering corresponds to the GAP function ConjugacyClassesSubgroups2(G). Let

$F$ be the flabby class of

$M_G$ . We assume that

$F$ is stably permutation, i.e. for

$x_{r+1}=\pm 1$ ,

$\begin{align*} \left(\bigoplus_{i=1}^r \mathbb{Z}[G/H_i]^{\oplus x_i}\right)\oplus F^{\oplus x_{r+1}} \ \simeq\ \bigoplus_{i=1}^r \mathbb{Z}[G/H_i]^{\oplus y_i}.%\label{eqpos} \end{align*}$ Define

$a_i=x_i-y_i$ and

$b_1=x_{r+1}$ . Then we have for

$b_1=\pm 1$ ,

$\bigoplus_{i=1}^r \mathbb{Z}[G/H_i]^{\oplus a_i}\ \simeq\ F^{\oplus(-b_1)}.$

$[M_G]^{fl}=0\Longrightarrow$ there exist

$a_1,\ldots,a_r\in\mathbb{Z}$ and

$b_1=\pm 1$ which satisfy the system of linear equations.

PossibilityOfStablyPermutationM

PossibilityOfStablyPermutationM(G)

returns the same as PossibilityOfStablyPermutationF(G) but with respect to

$M_G$ instead of

$F$ .

Nlist

Nlist(l)

returns the negative part of the list

$l$ .

Plist

Plist(l)

returns the positive part of the list

$l$ .

StablyPermutationFCheck

StablyPermutationFCheck(G,L1,L2)

returns the matrix

$P$ which satisfies

$G_1P=PG_2$ where

$G_1$ (resp.

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

$G$ on

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

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

$\left(\bigoplus_{i=1}^r \mathbb{Z}[G/H_i]^{\oplus a_i}\right)\oplus F^{\oplus b_1} \simeq \left(\bigoplus_{i=1}^r \mathbb{Z}[G/H_i]^{\oplus a_i^{\prime}}\right)\oplus F^{\oplus b_1^{\prime}}$ for lists

$L_1=[a_1,\ldots,a_r,b_1]$ and

$L_2=[a_1^{\prime},\ldots,a_r^{\prime},b_1^{\prime}]$ , if

$P$ exists. If such

$P$ does not exist, this returns false.

StablyPermutationMCheck

StablyPermutationMCheck(G,L1,L2)

returns the same as StablyPermutationFCheck(G,L1,L2) but with respect to

$M_G$ instead of

$F$ .

StablyPermutationFCheckP

StablyPermutationFCheckP(G,L1,L2)

returns 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

$G$ on

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

$(\oplus_{i=1}^r \mathbb{Z}[G/H_i]^{\oplus a_i^{\prime}})\oplus F^{\oplus b_1^{\prime}}$ ) for lists

$L_1=[a_1,\ldots,a_r,b_1]$ and

$L_2=[a_1^{\prime},\ldots,a_r^{\prime},b_1^{\prime}]$ , if

$P$ exists. If such

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

StablyPermutationMCheckP

StablyPermutationMCheckP(G,L1,L2)

returns the same as StablyPermutationFCheckP(G,L1,L2) but with respect to

$M_G$ instead of

$F$ .

StablyPermutationFCheckMat

StablyPermutationFCheckMat(G,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

$G$ on

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

$(\oplus_{i=1}^r \mathbb{Z}[G/H_i]^{\oplus a_i^{\prime}})\oplus F^{\oplus b_1^{\prime}}$ ) for lists

$L_1=[a_1,\ldots,a_r,b_1]$ and

$L_2=[a_1^{\prime},\ldots,a_r^{\prime},b_1^{\prime}]$ . If not, this returns false.

StablyPermutationMCheckMat

StablyPermutationMCheckMat(G,L1,L2,P)

returns the same as StablyPermutationFCheckMat(G,L1,L2,P) but with respect to

$M_G$ instead of

$F$ .

StablyPermutationFCheckGen

StablyPermutationFCheckP(G,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

$G$ on

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

$(\oplus_{i=1}^r \mathbb{Z}[G/H_i]^{\oplus a_i^{\prime}})\oplus F^{\oplus b_1^{\prime}}$ ) for lists

$L_1=[a_1,\ldots,a_r,b_1]$ and

$L_2=[a_1^{\prime},\ldots,a_r^{\prime},b_1^{\prime}]$ .

StablyPermutationMCheckGen

StablyPermutationMCheckGen(G,L1,L2)

returns the same as StablyPermutationFCheckGen(G,L1,L2) but with respect to

$M_G$ instead of

$F$ .

DirectSumMatrixGroup

DirectSumMatrixGroup(l)

returns the direct sum of the groups

$G_1,\ldots,G_n$ for the list

$l=[G_1,\ldots,G_n]$ .

DirectProductMatrixGroup

DirectProductMatrixGroup(l)

returns the direct product of the groups

$G_1,\ldots,G_n$ for the list

$l=[G_1,\ldots,G_n]$ .

Norm1TorusJ

Norm1TorusJ(d,m)

returns the Chevalley module

$J_{G/H}$ for the

$m$ -th transitive subgroup

$G=dTm\leq S_d$ of degree

$d$ where

$H$ is the stabilizer of one of the letters in

$G$ .

FlabbyResolutionLowRankFromGroup

FlabbyResolutionLowRankFromGroup(M,G)

returns a suitable flabby resolution with low rank for

$G$ -lattice

$M$ by using backtracking techniques. Repeating the algorithm, by defining

$[M]^{fl^n}:=[[M]^{fl^{n-1}}]^{fl}$ inductively,

$[M]^{fl}=0$ is provided if we may find some

$n$ with

$[M]^{fl^n}=0$ (this method is slightly improved to the flfl algorithm, see above).

Hcandidates

Hcandidates(G)

retruns subgroups

$H$ of

$G$ which satisfy

$\bigcap_{\sigma\in G} H^\sigma=\{1\}$ where

$H^\sigma=\sigma^{-1}H\sigma$ (hence

$H$ contains no normal subgroup of

$G$ except for

$\{1\}$ ).

Norm1TorusJTransitiveGroup

Norm1TorusJTransitiveGroup(d,m)

returns the Chevalley module

$J_{G/H}$ for the

$m$ -th transitive subgroup

$G = {}_dT_m \leq S_d$ of degree

$d$ where

$H$ is the stabilizer of one of the letters in

$G$ . (The input and output of this function is the same as the function Norm1TorusJ(d,m) but this function is more efficient.)

Norm1TorusJCoset

Norm1TorusJCoset(G,H)

retruns the Chevalley module

$J_{G/H}$ for a group

$G$ and a subgroup

$H\leq G$ .

StablyPermutationMCheckPPari

StablyPermutationMCheckPPari(G,L1,L2)

returns the same as StablyPermutationMCheckP(G,L1,L2) but using efficient PARI/GP functions (e.g. matker, matsnf) [PARI2]. (This function applies union-find algorithm and it also requires PARI/GP [PARI2].)

StablyPermutationFCheckPPari

StablyPermutationFCheckPPari(G,L1,L2)

returns the same as StablyPermutationFCheckP(G,L1,L2) but using efficient PARI/GP functions (e.g. matker, matsnf) [PARI2]. (This function applies union-find algorithm and it also requires PARI/GP [PARI2].)

StablyPermutationFCheckPFromBasePari

StablyPermutationFCheckPFromBasePari(G,mi,L1,L2)

returns the same as StablyPermutationFCheckPPari(G,L1,L2) but with respect to

$m_i=\mathcal{P}^\circ$ instead of the original

$\mathcal{P}^\circ$ as in Hoshi and Yamasaki [HY17, Equation (4) in Section 5.1]. (See [HY17, Section 5.7, Method III]. This function applies union-find algorithm and it also requires PARI/GP [PARI2].)

FlabbyResolutionNorm1TorusJ

FlabbyResolutionNorm1TorusJ(d,m).actionF

returns the matrix representation of the action of

$G$ on a flabby class

$F=[J_{G/H}]^{fl}$ for the

$m$ -th transitive subgroup

$G=dTm \leq S_n$ of degree

$n$ where

$H$ is the stabilizer of one of the letters in

$G$ . (This function is similar to FlabbyResolution(Norm1TorusJ(d,m)).actionF but it may speed up and save memory resources.)

References

[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.

[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.

[HY] Akinari Hoshi and Aiichi Yamasaki, Rationality problem for norm one tori for

$A_5$ and

${\rm PSL}_2(\mathbb{F}_8)$ extensions, arXiv:2309.16187.

[PARI2] The PARI Group, PARI/GP version 2.13.3, Univ. Bordeaux, 2021, http://pari.math.u-bordeaux.fr/.