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FlabbyResolutionFromBase.gap

Definition of MG
Let G be a finite subgroup of GL(n,Z). The G-lattice MG of rank n is defined to be the G-lattice with a Z-basis {u1,,un} on which G acts by σ(ui)=nj=1ai,juj for any σ=[ai,j]G.

Hminus1

Hminus1(G)
returns the Tate cohomology group ˆH1(G,MG) for a finite subgroup GGL(n,Z).

H0

H0(G)
returns the Tate cohomology group ˆH0(G,MG) for a finite subgroup GGL(n,Z).

H1

H1(G)
returns the cohomology group H1(G,MG) for a finite subgroup GGL(n,Z).

Z0lattice

Z0lattice(G)
returns a Z-basis of the group of Tate 0-cocycles ˆZ0(G,MG) for a finite subgroup GGL(n,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 MG is flabby or not.

IsCoflabby

IsCoflabby(G)
returns whether G-lattice MG is coflabby or not.

IsInvertible

IsInvertible(G)
returns whether G-lattice MG is invertible or not.

FlabbyResoluton

FlabbyResolution(G)
returns a flabby resolution 0MGιPϕF0 of MG 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 ι:MGP;
FlabbyResolution(G).surjection
returns the matrix which corresponds to the surjection ϕ:PF.

IsInvertibleF

IsInvertibleF(G)
returns whether [MG]fl is invertible.

flfl

flfl(G)
returns the G-lattice E with [[MG]fl]fl=[E].

PossibilityOfStablyPermutationF

PossibilityOfStablyPermutationF(G)
returns a basis L={l1,,ls} of the solution space of the system of linear equations which is obtained by computing some Z-class invariants. Each isomorphism class of irreducible permutation G-lattices corresponds to a conjugacy class of subgroup H of G by HZ[G/H]. Let H1,,Hr be conjugacy classes of subgroups of G whose ordering corresponds to the GAP function ConjugacyClassesSubgroups2(G). Let F be the flabby class of MG. We assume that F is stably permutation, i.e. for xr+1=±1, (ri=1Z[G/Hi]xi)Fxr+1  ri=1Z[G/Hi]yi. Define ai=xiyi and b1=xr+1. Then we have for b1=±1, ri=1Z[G/Hi]ai  F(b1). [MG]fl=0 there exist a1,,arZ and b1=±1 which satisfy the system of linear equations.

PossibilityOfStablyPermutationM

PossibilityOfStablyPermutationM(G)
returns the same as PossibilityOfStablyPermutationF(G) but with respect to MG 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 G1P=PG2 where G1 (resp. G2) is the matrix representation group of the action of G on (ri=1Z[G/Hi]ai)Fb1 (resp. (ri=1Z[G/Hi]ai)Fb1) with the isomorphism (ri=1Z[G/Hi]ai)Fb1(ri=1Z[G/Hi]ai)Fb1 for lists L1=[a1,,ar,b1] and L2=[a1,,ar,b1], 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 MG instead of F.

StablyPermutationFCheckP

StablyPermutationFCheckP(G,L1,L2)
returns a basis P={P1,,Pm} of the solution space of G1P=PG2 where G1 (resp. G2) is the matrix representation group of the action of G on (ri=1Z[G/Hi]ai)Fb1 (resp. (ri=1Z[G/Hi]ai)Fb1) for lists L1=[a1,,ar,b1] and L2=[a1,,ar,b1], 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 MG instead of F.

StablyPermutationFCheckMat

StablyPermutationFCheckMat(G,L1,L2,P)
returns true if G1P=PG2 and det P=±1 where G1 (resp. G2) is the matrix representation group of the action of G on (ri=1Z[G/Hi]ai)Fb1 (resp. (ri=1Z[G/Hi]ai)Fb1) for lists L1=[a1,,ar,b1] and L2=[a1,,ar,b1]. If not, this returns false.

StablyPermutationMCheckMat

StablyPermutationMCheckMat(G,L1,L2,P)
returns the same as StablyPermutationFCheckMat(G,L1,L2,P) but with respect to MG instead of F.

StablyPermutationFCheckGen

StablyPermutationFCheckP(G,L1,L2)
returns the list [M1,M2] where M1=[g1,,gt] (resp. M2=[g1,,gt]) is a list of the generators of G1 (resp. G2) which is the matrix representation group of the action of G on (ri=1Z[G/Hi]ai)Fb1 (resp. (ri=1Z[G/Hi]ai)Fb1) for lists L1=[a1,,ar,b1] and L2=[a1,,ar,b1].

StablyPermutationMCheckGen

StablyPermutationMCheckGen(G,L1,L2)
returns the same as StablyPermutationFCheckGen(G,L1,L2) but with respect to MG instead of F.

DirectSumMatrixGroup

DirectSumMatrixGroup(l)
returns the direct sum of the groups G1,,Gn for the list l=[G1,,Gn].

DirectProductMatrixGroup

DirectProductMatrixGroup(l)
returns the direct product of the groups G1,,Gn for the list l=[G1,,Gn].

Norm1TorusJ

Norm1TorusJ(d,m)
returns the Chevalley module JG/H for the m-th transitive subgroup G=dTmSd 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]fln:=[[M]fln1]fl inductively, [M]fl=0 is provided if we may find some n with [M]fln=0 (this method is slightly improved to the flfl algorithm, see above).

Hcandidates

Hcandidates(G)
retruns subgroups H of G which satisfy σGHσ={1} where Hσ=σ1Hσ (hence H contains no normal subgroup of G except for {1}).

Norm1TorusJTransitiveGroup

Norm1TorusJTransitiveGroup(d,m)
returns the Chevalley module JG/H for the m-th transitive subgroup G=dTmSd 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 JG/H for a group G and a subgroup HG.

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 mi=P instead of the original P 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].)

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 A5 and PSL2(F8) extensions, arXiv:2309.16187.
[PARI2] The PARI Group, PARI/GP version 2.13.3, Univ. Bordeaux, 2021, http://pari.math.u-bordeaux.fr/.