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There is a special type for the basic algebras which are the
modular group algebras of p-groups for p a prime. If G
is a finite p and k is a field of characteristic p, then
the commands BasicAlgebra(G, k) and BasicAlgebra(G) automatically create a basic algebra of
type AlgBasGrpP. The type is optimized for the
computation of cohomology rings. Included for this type are
restriction and inflation maps. Most of the functions for modules
and complexes are the same as for general basic algebras.
Subsections
The group which defines the algebra A.
The internal PC group of the algebra A.
The map from Group(A) to PCGroup(A) for an algebra A.
Converts a GModule M over a p-group to a module over the basic algebra of
that group.
Returns the standard module of the algebra A as a module over Group(A)
and as a module over PCGroup(A).
Converts a module M for the basic algebra of a p-group into a module over
the p-group.
Returns the data needed to compute the projective resolution of an A-module
for an algebra A.
The data is given as a record with the fields:
- (a)
- The matrices of the PCGenerators of the p-group on the standard
indecomposable projective module for the algebra (field name
PCgenMats).
- (b)
- The matrices of the minimal generators of the p-group on the
standard indecomposable projective module for the algebra (field name
MingenMats).
- (c)
- The algebra A (field name Algebra).
CompactProjectiveResolution(M, n) : ModAlgBas, RngIntElt -> Rec
Computes the projective resolution of the module M out to n steps.
The function returns a record with the fields:
- (a)
- The list of the ranks of the projective modules in
the resolution (field name BettiNumbers).
- (b)
- The record of the boundary maps (field name ResolutionRecord).
- (c)
- The module M (field name Module).
- (d)
- The augmentation map (field name AugmentationMap).
- (e)
- The type of the resolution, whether projective or injective
(field name Typ).
The projective resolution as a complex of modules over the basic algebra
of the group algebra, computed from the compact projective resolution PR.
The projective resolution of the module M computed as a complex out to
n steps. The function also returns the augmentation map.
The projective resolution computed from a compact projective resolution PR
as a complex. The function also returns the augmentation map.
Creates the data on the chain maps for all generators of the cohomology
of the simple module k in degrees within the limits of the compact
projective resolution PR of the simple module. The function returns
a record having the following information.
- (a)
- The record of the chain maps of the generators of cohomology
(field name ChainMapRecord).
- (b)
- The sequence of sizes of the chain map record
(field name ChainSizes).
- (c)
- The degrees of the chain maps (field name
ChainDegrees).
- (d)
- The list of cocycles representing the generators
(field name Cocycles).
- (e)
- The record of the products of the generators (field name
ProductRecord).
- (f)
- The locations of the products of the generators (field name
ProductLocations).
Much of the information is for use in the computation
of the cohomology ring.
Creates a chain map from the projective resolution P to
itself for the element number n in degree d of cohomology.
Creates a chain map in compact form from the compact projective resolution
PR to itself for the element number n in degree d of cohomology.
CohomologyRing(PR, AC) : Rec, Rec -> Rec
The cohomology ring of the unique simple module k for the basic algebra
of the group algebra of a p-group. The input can be given either as the
module k and the number of steps n or as the compact projective
resolution PR of k together with AC, the calculation of the chain
map generators of the cohomology. In the former case the
compact resolution and
the chain map of the generators are computed in the process. The ring
is returned as a record having the following fields:
- (a)
- The polynomial ring or free graded-commutative k-algebra R
generated by the cohomology generators (field name PolRing).
- (b)
- The ideal of relations in R satisfied by the cohomology
generators (field name RelationsIdeal).
- (c)
- The list of relations that have been computed (field name
ComputedRelations).
- (d)
- The chain maps giving the tops of the monomial in the
cohomology generators (field name MonomialData).
- (e)
- The number of computed steps in the resolution
(field name NumberOfSteps).
A minimal set of relations generating the relations ideal of a cohomology
ring R.
Assuming that A is the basic algebra of a p-group G and that B is
the basic algebra of a subgroup of G, the function
returns the change of basis matrix that make the standard free module
for A into a directs sum of standard free modules for B. It also returns
the inverse of the matrix and a set of coset representatives of the
PCGroup(B) in PCGroup(A).
Takes the compact projective resolution PR for the trivial module
of G and the resolution data RD for
the basic algebra of a subgroup H and returns the restriction of the
resolution to a complex of modules over the basic algebra for H.
Computes the chain map from the resolution P2 of the simple module for the
basic algebra of a subgroup H of a group G to the restriction to
H of the resolution P1 of the simple module for the basic algebra of G.
The inputs P1 and P2 must be in compact form.
Computes the sequence of images of the generators of the cohomology ring
of G restricted to a subgroup H. The input is the projective resolutions
and cohomology generators for the basic algebra of G (PR1 and AC1)
and for the basic algebra of the subgroup (PR2 and AC2), as well as
the cohomology relations for the subgroup, REL2.
Returns the images of the generators of the cohomology ring of a
quotient group Q in the cohomology ring of a group G. The input
θis the quotient map G -> Q. Other input is the
projective resolutions and cohomology generators for the basic
algebra of G (PR1 and AC1) and for the quotient group Q
(PR2 and AC2) as well as the cohomology relations for G, REL1.
We create the cohomology ring of a group G of order 64 and find a
cyclic subgroup Z of the center of G. We compute the restriction
of the cohomology of G to the cohomology of Z and also the inflation
of the cohomology of G/Z to the cohomology ring of G.
> SetSeed(1);
> G := SmallGroup(64,7);
> Z := sub<G| Random(Center(G))>;
> G;
GrpPC : G of order 64 = 2^6
PC-Relations:
G.1^2 = G.4,
G.2^2 = G.5,
G.3^2 = G.5,
G.4^2 = G.6,
G.2^G.1 = G.2 * G.3,
G.3^G.1 = G.3 * G.5,
G.3^G.2 = G.3 * G.5
> #Z, [G!Z.i: i in [1 .. Ngens(Z)]];
4 [ G.4 ]
So we see that Z has order 4 and is generated by the element G.4.
Now construct the quotient and the basic Algebras.
> Q, mu := quo<G|Z>;
> A := BasicAlgebra(G);
> B := BasicAlgebra(Q);
> C := BasicAlgebra(Z);
Next we want the simple modules and the cohomology rings. We compute
the cohomology out to 17 steps which should be more than enough to
get the generators and relations.
> k := SimpleModule(A,1);
> kk := SimpleModule(B,1);
> kkk := SimpleModule(C,1);
> time R := CohomologyRing(k,17);
Time: 2.060
> time S := CohomologyRing(kk,17);
Time: 0.140
> time T := CohomologyRing(kkk,17);
Time: 0.060
The structure of the cohomology rings can be from the following outputs.
> R`RelationsIdeal,S`RelationsIdeal,T`RelationsIdeal;
First the cohomology ring for $G$.
Ideal of Graded Polynomial ring of rank 6 over GF(2)
Lexicographical Order
Variables: $.1, $.2, $.3, $.4, $.5, $.6
Variable weights: 1 1 2 2 3 4
Basis:
[
$.1^2,
$.1*$.2,
$.2^3,
$.1*$.3,
$.2*$.5,
$.3^2,
$.1*$.5 + $.2^2*$.3,
$.3*$.5,
$.5^2
]
Now the cohomology ring for Q.
Ideal of Graded Polynomial ring of rank 4 over GF(2)
Lexicographical Order
Variables: $.1, $.2, $.3, $.4
Variable weights: 1 1 3 4
Basis:
[
$.1*$.2,
$.1^3,
$.1*$.3,
$.2^2*$.4 + $.3^2
]
And finally the cohomology ring for Z.
Ideal of Graded Polynomial ring of rank 2 over GF(2)
Lexicographical Order
Variables: $.1, $.2
Variable weights: 1 2
Basis:
[
$.1^2
]
Next we require the inputs for the restriction and inflation maps.
> Pr1 := k`CompactProjectiveResolution;
> Pr2 := kk`CompactProjectiveResolution;
> Pr3 := kkk`CompactProjectiveResolution;
> Ac1 := k`AllCompactChainMaps;
> Ac2 := kk`AllCompactChainMaps;
> Ac3 := kkk`AllCompactChainMaps;
Now the inflation map from Q to G sends the generators of the
cohomology of Q to the given list of elements in the cohomology
ring of G.
> inf := InflationMap(Pr2,Pr1,Ac2,Ac1,R,mu);
> inf;
[
$.2,
$.1,
$.5,
$.6
]
The restriction map from the cohomology ring of G to the cohomology
ring of Z sends the generators of R to the corresponding elements
in the computed sequence.
> res := RestrictionOfGenerators(Pr1,Pr3,Ac1,Ac3,T);
> res;
[
0,
0,
0,
$.2,
0,
0
]
Finally, a set of minimal relations is determined for the cohomology ring R.
> MinimalRelations(R);
[
$.1^2,
$.1*$.2,
$.2^3,
$.1*$.3,
$.2*$.5,
$.3^2,
$.1*$.5 + $.2^2*$.3,
$.3*$.5,
$.5^2
]
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