Both FpGModuleGF
(Chapter 5) and FpGModuleHomomorphismGF
(Chapter 6) store a matrix whose rows are G-generators for a module vector space (the module and the homomorphism's image respectively). The internal functions listed here provide common operations for dealing with these matrices.
> HAPPRIME_ValueOptionMaxFGExpansionSize ( field, group ) | ( operation ) |
Returns: Integer
Returns the maximum matrix expansion size. This is read from the MaxFGExpansionSize
option from the GAP options stack Reference: Options Stack, computed using the MaxFGExpansionMemoryLimit
option.
> HAPPRIME_KernelOfGeneratingRowsDestructive ( gens, rowlengths, GA ) | ( operation ) |
Returns: List
Returns a list of generating vectors for the kernel of the FG-module homomorphism defined by the generating rows gens using the group and action GA.
This function computes the kernel recursively by partitioning the generating rows into
[ B 0 ] [ C D ]
doing column reduction if necessary to get the zero block at the top right. The matrices B> and C are small enough to be expanded, while the kernel of D is calculated by recursion. The argument rowlengths lists the number of non-zero blocks in each row; the rest of each row is taken to be zero. This allows the partitioning to be more efficiently performed (i.e. column reduction is not always required).
The GAP options stack Reference: Options Stack variable MaxFGExpansionSize
can be used to specify the maximum allowable expanded matrix size. This governs the size of the B and C matrices, and thus the number of recursions before the kernel of D is also computed by recursion. A high value for will allow larger expansions and so faster computation at the cost of more memory. The MaxFGExpansionMemoryLimit
option can also be used, which sets the maximum amount of memory that GAP is allowed to use (as a string containing an integer with the suffix k
, M
or G
to indicate kilobyes, megabytes or gigabytes respectively). In this case, the function looks at the free memory available to GAP and computes an appropriate value for MaxFGExpansionSize
.
> HAPPRIME_GActMatrixColumns ( g, Vt, GA ) | ( operation ) |
> HAPPRIME_GActMatrixColumnsOnRight ( g, Vt, GA ) | ( operation ) |
Returns: Matrix
Returns the matrix that results from the applying the group action u=gv (or u=vg in the case of the OnRight
version of this function) to each column vector in the matrix Vt. By acting on columns of a matrix (i.e. the transpose of the normal GAP representation), the group action is just a permutation of the rows of the matrix, which is a fast operation. The group and action are passed in GA using the ModuleGroupAndAction
(5.4-5) record.
If the input matrix Vt is in a compressed matrix representation, then the returned matrix will also be in compressed matrix representation.
> HAPPRIME_ExpandGeneratingRow ( gen, GA ) | ( operation ) |
> HAPPRIME_ExpandGeneratingRows ( gens, GA ) | ( operation ) |
> HAPPRIME_ExpandGeneratingRowOnRight ( gen, GA ) | ( operation ) |
> HAPPRIME_ExpandGeneratingRowsOnRight ( gens, GA ) | ( operation ) |
Returns: List
Returns a list of G-generators for the vector space that corresponds to the of G-generator gen (or generators gens). This space is formed by multiplying each generator by each element of G in turn, using the group and action specified in GA (see ModuleGroupAndAction
(5.4-5)). The returned list is thus |G| times larger than the input.
For a list of generators gens [v_1, v_2, ..., v_n], HAPPRIME_ExpandGeneratingRows
returns the list [g_1v_1, g_2v_1, ..., g_1v_2, g_2v_2, ..., g_|G|v_n] In other words, the form of the returned matrix is block-wise, with the expansions of each row given in turn. This function is more efficient than repeated use of HAPPRIME_ExpandGeneratingRow
since it uses the efficient HAPPRIME_GActMatrixColumns
(11.1-3) to perform the group action on the whole set of generating rows at a time.
The function HAPPRIME_ExpandGeneratingRowsOnRight
is the same as above, but the group action operates on the right instead.
> HAPPRIME_AddGeneratingRowToSemiEchelonBasisDestructive ( basis, gen, GA ) | ( operation ) |
Returns: Record with elements vectors
and basis
This function augments a vector space basis with another generator. It returns a record consisting of two elements: vectors
, a set of semi-echelon basis vectors for the vector space spanned by the sum of the input basis and all G-multiples of the generating vector gen; and heads
, a list of the head elements, in the same format as returned by SemiEchelonMat
(Reference: SemiEchelonMat). The generator gen is expanded according to the group and action specified in the GA record (see ModuleGroupAndAction
(5.4-5)).
If the input basis is not zero, it is also modified by this function, to be the new basis (i.e. the same as the vectors
element of the returned record).
> HAPPRIME_ReduceVectorDestructive ( v, basis, heads ) | ( operation ) |
Returns: Boolean
Reduces the vector v (in-place) using the semi-echelon set of vectors basis with heads heads (as returned by SemiEchelonMat
(Reference: SemiEchelonMat)). Returns true
if the vector is completely reduced to zero, or false
otherwise.
> HAPPRIME_ReduceGeneratorsOfModuleBySemiEchelon ( gens, GA ) | ( operation ) |
> HAPPRIME_ReduceGeneratorsOfModuleBySemiEchelonDestructive ( gens, GA ) | ( operation ) |
> HAPPRIME_ReduceGeneratorsOfModuleByLeavingOneOut ( gens, GA ) | ( operation ) |
> HAPPRIME_ReduceGeneratorsOnRightByLeavingOneOut ( gens, GA ) | ( operation ) |
Returns: List of vectors
Returns a subset of the module generators gens over the group with action specified in the GA record (see ModuleGroupAndAction
(5.4-5)) that will still generate the module.
The BySemiEchelon
functions gradually expand out the module generators into an F-basis, using that F-basis to reduce the other generators, until the full vector space of the module is spanned. The generators needed to span the space are returned, and should be a small set, although not minimal. The Destructive
version of this function will modify the input gens parameter. The non-destructive version makes a copy first, so leaves the input arguments unchanged, at the expense of more memory.
The ByLeavingOneOut
function is tries repeatedly leaving out generators from the list gens to find a small subset that still generates the module. If the generators are from the field GF(2), this is guaranteed to be a minimal set of generators. The OnRight
version computes a minimal subset which generates the module under group multiplication on the right.
> HAPPRIME_DisplayGeneratingRows ( gens, GA ) | ( operation ) |
Returns: nothing
Displays a set of G-generating rows a human-readable form. The elements of each generating vector are displayed, with each block marked by a separator (since the group action on a module vector will only permute elements within a block).
This function is used by Display
for both FpGModuleGF
and FpGModuleHomomorphismGF
.
NOTE: This is currently only implemented for GF(2)
gap> HAPPRIME_DisplayGeneratingRows( > ModuleGenerators(M), HAPPRIME_ModuleGroupAndAction(M)); [...1..11|........|.......1|........|........] [........|........|........|.1....11|........] [........|........|........|........|..1.1.1.] [........|.1.1..1.|........|........|........] [........|........|......11|........|........] [........|........|1......1|........|........] |
> HAPPRIME_GeneratingRowsBlockStructure ( gens, GA ) | ( operation ) |
Returns: Matrix
Returns a matrix detailing the block structure of a set of module generating rows. The group action on a generator permutes the vector in blocks of length GA.actionBlockSize
: any block that contains non-zero elements will still contain non-zero elements after the group action; any block that is all zero will remain all zero. This operation returns a matrix giving this block structure: it has a one where the block is non-zero and zero where the block is all zero.
gap> b := HAPPRIME_GeneratingRowsBlockStructure( > ModuleGenerators(M), ModuleActionBlockSize(M)); [ [ 1, 0, 1, 1, 1 ], [ 1, 0, 1, 1, 1 ], [ 0, 1, 1, 1, 1 ], [ 0, 0, 1, 1, 1 ] ] |
> HAPPRIME_DisplayGeneratingRowsBlocks ( gens, actionBlockSize ) | ( operation ) |
Returns: nothing
Displays a set of G-generating rows a compact human-readable form. Each generating rows can be divided into blocks of length actionBlockSize. The generating rows are displayed in a per-block form: a *
where the block is non-zero and .
where the block is all zero.
This function is used by DisplayBlocks
(5.4-10) (for FpGModuleGF
) and DisplayBlocks
(6.5-4) (for FpGModuleHomomorphismGF
).
gap> HAPPRIME_DisplayGeneratingRowsBlocks( > ModuleGenerators(M), HAPPRIME_ModuleGroupAndAction(M)); [*.*..] [...*.] [....*] [.*...] [..*..] [..*..] |
> HAPPRIME_IndependentGeneratingRows ( blocks ) | ( operation ) |
Returns: List of lists
Given a block structure as returned by HAPPRIME_GeneratingRowsBlockStructure
(11.1-9), this decomposes a set of generating rows into sets of independent rows. These are returned as a list of row indices, where each set of rows share no blocks with any other set.
gap> DisplayBlocks(M); Module over the group ring of Group( [ f1, f2, f3 ] ) in characteristic 2 with 6 generators in FG^5. [**...] [.*...] [.**..] [.**..] [...*.] [....*] Generators are in minimal echelon form. gap> gens := ModuleGenerators(M);; gap> G := ModuleGroup(M);; gap> blocks := HAPPRIME_GeneratingRowsBlockStructure(gens, G); [ [ 1, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 1, 1, 0, 0 ], [ 0, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ] gap> HAPPRIME_IndependentGeneratingRows(blocks); [ [ 1, 2, 3, 4 ], [ 5 ], [ 6 ] ] |
> HAPPRIME_GactFGvector ( g, v, MT ) | ( operation ) |
Returns: Vector
Returns the vector that is the result of the action u=gv of the group element g on a module vector v (according to the group multiplication table MT. This operation is the quickest current method for a single vector. To perform the same action on a set of vectors, it is faster write the vectors as columns of a matrix and use HAPPRIME_GActMatrixColumns
(11.1-3) instead.
> HAPPRIME_CoefficientsOfGeneratingRows ( gens, GA, v ) | ( operation ) |
> HAPPRIME_CoefficientsOfGeneratingRows ( gens, GA, coll ) | ( operation ) |
> HAPPRIME_CoefficientsOfGeneratingRowsDestructive ( gens, GA, v ) | ( operation ) |
> HAPPRIME_CoefficientsOfGeneratingRowsDestructive ( gens, GA, coll ) | ( operation ) |
> HAPPRIME_CoefficientsOfGeneratingRowsGF ( gens, GA, v ) | ( operation ) |
> HAPPRIME_CoefficientsOfGeneratingRowsGF ( gens, GA, coll ) | ( operation ) |
> HAPPRIME_CoefficientsOfGeneratingRowsGFDestructive ( gens, GA, v ) | ( operation ) |
> HAPPRIME_CoefficientsOfGeneratingRowsGFDestructive ( gens, GA, coll ) | ( operation ) |
> HAPPRIME_CoefficientsOfGeneratingRowsGFDestructive2 ( gens, GA, v ) | ( operation ) |
> HAPPRIME_CoefficientsOfGeneratingRowsGFDestructive2 ( gens, GA, coll ) | ( operation ) |
Returns: Vector, or list of vectors
For a single vector v, this function returns a vector x giving the G-coefficients from gens needed to generate v, i.e. the solution to the equation x*A=v, where A is the expansion of gens. If there is no solution, fail
is returned. If a list of vectors, coll, then a vector is returned that lists the solution for each vector (any of which may be fail
). The standard forms of this function use standard linear algebra to solve for the coefficients. The Destructive
version will corrupt both gens and v. The GF
versions use the block structure of the generating rows to expand only the blocks that are needed to find the solution before using linear algebra. If the generators are in echelon form, this can save memory, but is slower.
The GFDestructive2
functions also assume an echelon form for the generators, but use back-substitution to find a set of coefficients. This can save a lot of memory but is again slower.
> HAPPRIME_GenerateFromGeneratingRowsCoefficients ( gens, GA, c ) | ( operation ) |
> HAPPRIME_GenerateFromGeneratingRowsCoefficients ( gens, GA, coll ) | ( operation ) |
> HAPPRIME_GenerateFromGeneratingRowsCoefficientsGF ( gens, GA, c ) | ( operation ) |
> HAPPRIME_GenerateFromGeneratingRowsCoefficientsGF ( gens, GA, coll ) | ( operation ) |
Returns: Vector, or list of vectors
For a vector c, returns (as a vector), the module element generated by multiplying c by the expansion of the generators gens. For a list of coefficient vectors coll, this returns a list of generating vectors.
The standard versions of this function use standard linear algebra. The GF
versions only performs the expansion of necessary generating rows, and only expands by one group element at a time, so will only need at most twice the amount of memory as that to store gens, which is a large saving over expanding the generators by every group element at the same time, as in a naive implementation. It may also be faster.
> HAPPRIME_RemoveZeroBlocks ( gens, GA ) | ( operation ) |
Returns: Vector
Removes from a set of generating vectors gens (with ModuleGroupAndAction
(5.4-5) GA) any blocks that are zero in every generating vector. Removal is done in-place, i.e. the input argument gens will be modified to remove the zero blocks. Zero blocks are unaffected by any row or expansion operation, and can be removed to save time or memory in those operations. The function returns the original block structure as a vector, and this can be used in the function HAPPRIME_AddZeroBlocks
(11.1-16) to reinstate the zero blocks later, if required. See the documentation for that function for more detail of the block structure vector.
> HAPPRIME_AddZeroBlocks ( gens, blockStructure, GA ) | ( operation ) |
Returns: List of vectors
Adds zero blocks to a set of generating vectors gens to make it have the block structure given in blockStructure (for a given ModuleGroupAndAction
(5.4-5) GA). The generators gens are modified in place, and also returned.
The blockStructure parameter is a vector of which is the length of the required output vector and has zeros where zero blocks should be, and is non-zero elsewhere. Typically, an earlier call to HAPPRIME_RemoveZeroBlocks
(11.1-15) will have been used to remove the zero blocks, and this function and such a blockStructure vector is returned by this function. HAPPRIME_AddZeroBlocks
can be used to reinstate these zero blocks.
FG-modules in HAPprime use the datatype FpGModuleGF
(Chapter 5). Internally, this uses many of the functions listed in Section 11.1, and further internal functions are listed below.
> HAPPRIME_DirectSumForm ( current, new ) | ( operation ) |
Returns: String
Returns a string containing the form of the generator matrix if the direct sum is formed between a FpGModuleGF
with the form current and a FpGModuleGF
with the form new. The direct sum is formed by placing the two module generating matrices in diagonal form. Given the form of the two generating matrices, this allows the form of the direct sum to be stated. See ModuleGeneratorsForm
(5.5-5) for information about form strings.
> HAPPRIME_PrintModuleDescription ( M, func ) | ( operation ) |
Returns: nothing
Used by PrintObj
(Reference: PrintObj), ViewObj
(Reference: ViewObj), Display
(Reference: Display) and DisplayBlocks
(5.4-10), this helper function prints a description of the module M. The parameter func can be one of the strings "print"
, "view"
, "display"
or "displayblocks"
, corresponding to the print different functions that might be called.
> HAPPRIME_ModuleGeneratorCoefficients ( M, elm ) | ( operation ) |
> HAPPRIME_ModuleGeneratorCoefficientsDestructive ( M, elm ) | ( operation ) |
> HAPPRIME_ModuleGeneratorCoefficients ( M, coll ) | ( operation ) |
> HAPPRIME_ModuleGeneratorCoefficientsDestructive ( M, coll ) | ( operation ) |
Returns: Vector
Returns the coefficients needed to make the module element elm as a linear and G-combination of the module generators of the FpGModuleGF
M. The coefficients are returned in standard vector form, or if there is no solution then fail
is returned. If a list of elements is given, then a list of coefficients (or fails
) is returned. The Destructive
form of this function might change the elements of of M or elm. The non-Destructive
version makes copies to ensure that they are not changed.
See also HAPPRIME_ModuleElementFromGeneratorCoefficients
(11.2-4).
> HAPPRIME_ModuleElementFromGeneratorCoefficients ( M, c ) | ( operation ) |
> HAPPRIME_ModuleElementFromGeneratorCoefficients ( M, coll ) | ( operation ) |
Returns: Vector
Returns an element from the module M, constructed as a linear and G-sum of the module generators as specified in c. If a list of coefficient vectors is given, a list of corresponding module elements is returned.
See also HAPPRIME_ModuleGeneratorCoefficients
(11.2-3)
> HAPPRIME_MinimalGeneratorsVectorSpaceGeneratingRowsDestructive ( vgens, GA ) | ( operation ) |
> HAPPRIME_MinimalGeneratorsVectorSpaceGeneratingRowsOnRightDestructive ( vgens, GA ) | ( operation ) |
Returns: FpGModuleGF
Returns a module with minimal generators that is equal to the FG-module with vector space basis vgens and ModuleGroupAndAction
(5.4-5) as specified in GA. The solution is computed by the module radical method, which is fast at the expense of memory. This function will corrupt the matrix gens.
This is a helper function for MinimalGeneratorsModuleRadical
(5.5-9) that is also used by ExtendResolutionPrimePowerGroupRadical
(HAPprime: ExtendResolutionPrimePowerGroupRadical) (which knows that its module is already in vector-space form).
> HAPPRIME_IsGroupAndAction ( obj ) | ( operation ) |
Returns: Boolean
Returns true
if obj appears to be a groupAndAction
record (see ModuleGroupAndAction
(5.4-5)), or false
otherwise.
For details of the main resolution functions in HAPprime, see Chapter 2 of this datatypes reference manual, and HAPprime: Resolutions in the HAPprime user guide. This section describes the internal helper functions used by the higher-level functions.
> HAPPRIME_WordToVector ( w, dim, orderG ) | ( method ) |
Returns: HAP word (list of lists)
Returns the boundary map vector that corresponds to the HAP word vector w with module ambient dimension dim and group order orderG (assumed to be the actionBlockSize
). A HAP word vector has the following format: [ [block, elm], [block, elm], ... ]
where block
is a block number and elm
is a group element index (see example below).
See also HAPPRIME_VectorToWord
(11.3-2)
gap> G := CyclicGroup(4);; gap> v := HAPPRIME_WordToVector([ [1,2],[2,3] ], 2, Order(G)); <a GF2 vector of length 8> gap> HAPPRIME_DisplayGeneratingRows([v], CanonicalGroupAndAction(G)); [.1..|..1.] gap> HAPPRIME_VectorToWord(v, Order(G)); [ [ 1, 2 ], [ 2, 3 ] ] |
> HAPPRIME_VectorToWord ( vec, orderG ) | ( function ) |
Returns: Vector
The HAP word format vector that corresponds to the boundary vector vec with actionBlockSize
assumed to be orderG.
See HAPPRIME_WordToVector
(11.3-1) for a few more details and an example.
> HAPPRIME_BoundaryMatrices ( R ) | ( attribute ) |
Returns: List of matrices
If R is a resolution which stores its boundaries as a list of matrices (e.g. one created by HAPprime, this list is returned. Otherwise, fail
is returned. Note that the first matrix in this list corresponds to the zeroth degree: for resolutions of modules, this is the generators of the module; for resolutions of groups, this is the empty matrix. The second matrix corresponds to the first degree, and so on.
> HAPPRIME_AddNextResolutionBoundaryMapMatNC ( R, BndMat ) | ( operation ) |
Returns: HapResolution
Returns the resolution R extended by one term, where that term is given by the boundary map matrix BndMat. If BndMat is not already in compressed matrix form, it will be converted into this form, and if the boundaries in R are not already in matrix form, they are all converted into this form.
> HAPPRIME_CreateResolutionWithBoundaryMapMatsNC ( G, BndMats ) | ( operation ) |
Returns: HapResolution
Returns a HAP resolution object for group G where the module homomorphisms are given by the boundary matrices in the list BndMats. This list is indexed with the boundary matrix for degree zero as the first element. If the resolution is the resolution of a module, the module's minimal generators are this first boundary matrix, otherwise (for the resolution of a group), this should be set to be the empty matrix []
.
Ths internal functions in this section implement isomorphisms between polynomial rings via maps from one set of indeterminates to another.
> HAPPRIME_SwitchPolynomialIndeterminates ( R, S, poly ) | ( function ) |
Returns: Polynomial or List of Polynomials
Changes the indeterminates in poly, which should be a polynomial or a list of polynomials, substituting the indeterminates of the polynomial ring S one-for-one for those in R (from which all polynomials in poly must come). The returned object is either a polynomial or a list of polynomials in the new indeterminates, depending on the input object.
See HAPPRIME_MapPolynomialIndeterminates
(11.4-2) for a function that can work with more general indeterminate maps.
> HAPPRIME_MapPolynomialIndeterminates ( old, new, poly ) | ( function ) |
Returns: Polynomial or List of Polynomials
Changes the indeterminates in poly, which can be a polynomial or a list of polynomials, substituting the polynomials in old for those in new. The returned object is either a polynomial or a list of polynomials in the new indeterminates, depending on the input object. The change of variable arguments, old and new, do not have to be simply indeterminates: they can be can be lists of polynomials which are equivalent in the two different sets of indeterminates. If a polynomial cannot be converted (i.e. if it cannot be generated from the polynomials in old) then fail
is returned for that polynomial.
> HAPPRIME_CombineIndeterminateMaps ( coeff, M, N ) | ( function ) |
Returns: List
Returns the indeterminate map that results from applying map M followed by map N. An indeterminate map is a list containing two lists, the first of which is a list of polynomials in the original indeterminates, the second the equivalent polynomials in the new ring indeterminates.
Ths internal functions in this section provide a transparent way of using Singular's Gröbner basis functions, if they are available. See the singular package documentation singular: singular: the GAP interface to Singular for further details.
> HAPPRIME_SingularGroebnerBasis ( pols, O ) | ( function ) |
> HAPPRIME_SingularReducedGroebnerBasis ( pols, O ) | ( function ) |
Returns: List
Returns the Gröbner basis (or reduced Gröbner basis) with respect to the ordering O for the ideal generated by the polynomials pols. This function uses the Gröbner basis implementation from singular, for preference, if available (GroebnerBasis
(singular: GroebnerBasis)), and if so it also manuipulates the result to fix a bug in singular where the returned polynomials are not necessarily returned with a value external representation (see Reference: The Defining Attributes of Rational Functions).
If the option obeyGBASIS
is true
, then this function will use whichever algorithm is specified by the GBASIS
global variable (see SINGULARGBASIS
(singular: SINGULARGBASIS)).
For the main functions dealing with presentations of graded algebras, and details of the datatype see Chapter 4. This section details the internal functions used by the higher-level functions.
> HAPPRIME_HilbertSeries ( P ) | ( attribute ) |
Returns: List
The Hilbert-Poincaré series for a graded ring is a polynomial H_P(t) whose coefficients are the dimensions of each degree of the ring. It can be written as
H_P(t) = Q(t) / (1-t^a)(1-t^b)...(1-t^n)
This function returns the list of coefficients for the polynomial Q(t) in the Hilbert Series for the graded algebra with presentation P.
This function simply calls the Singular function hilb
via the singular package and returns the result.
> HAPPRIME_SwitchGradedAlgebraRing ( A, R ) | ( attribute ) |
Returns: GradedAlgebraPresentation
Returns a new presentation for the graded algebra A which uses the ring R instead of the one in A.
> HAPPRIME_GradedAlgebraPresentationAvoidingIndeterminates ( A, avoid ) | ( attribute ) |
Returns: GradedAlgebraPresentation
Returns a new presentation for the graded algebra A which avoids the indeterminates listed in avoid.
> HAPPRIME_LastLHSBicomplexSize | ( global variable ) |
Stores the last bicomplex size last used by LHSSpectralSequenceLastSheet
(4.4-10). Used for testing and generating results for paper.
> HAPPRIME_LHSSpectralSequence ( G, N, n ) | ( operation ) |
Returns: GradedAlgebraPresentation
or list
This function is called by both LHSSpectralSequence
(4.4-10) and LHSSpectralSequenceLastSheet
(4.4-10) and does the actual work of computing the Lyndon-Hoschild-Serre spectral sequence for the group extension N -> G -> G/N. See the documentation for those functions for the main details, including options that are passed through to this function.
If n is the string "Einf"
then only the limiting sheet is returned (and the other sheets are discarded during computation, which can save time and memory). Otherwise, n sheets are returned, or enough until convergence is proved, if that is smaller.
> HAPPRIME_CohomologyRingWithoutResolution ( G ) | ( function ) |
Returns: GradedAlgebraPresentation
or fail
If the mod-p cohomology ring for G can be computed without building a resolution for G then this ring is returned, otherwise this function returns fail
.
Current cases where the resolution for G is not needed are
if G is the group of order two then H^*(G, F) = F[x] where x has degree one
if G is cyclic of order greater than two then H^*(G, F) = F[x,y]/x^2 where x has degree one and y has degree two
if G can be expressed as the direct sum of other groups then the cohomology rings for those groups are found (by recursive calls to ModPCohomologyRingPresentation
(HAPprime: ModPCohomologyRingPresentation (for group))) and combined using TensorProduct
(4.4-1)
> HAPPRIME_Polynomial2Algebra ( A[, ringA], poly ) | ( function ) |
Returns: Algebra element
Converts a polynomial poly in ringA (the mod-p cohomology ring presentation for the cohomology algebra A) into the equivalent element in A. If ringA is not provided, it is recovered using the attribute PresentationOfGradedStructureConstantAlgebra
(3.3-1). The ringA argument is provided to allow cases where the ring and polynomial use different (but isomorphic) indeterminants from those provided by PresentationOfGradedStructureConstantAlgebra
(3.3-1) (as is the case in HAPPRIME_LHSSpectralSequence
(11.6-5)).
> HAPPRIME_Algebra2Polynomial ( A, alg ) | ( function ) |
Returns: Polynomial
Converts an algebra element alg in the mod-p cohomology algebra A (given by ModPCohomologyRing
(HAP: ModPCohomologyRing)) into the equivalent polynomial from the corresponding ring presentation (given by PresentationOfGradedStructureConstantAlgebra
(3.3-1)).
The kernel of a derivation is efficiently computed by writing the ring R as an S-module where S is the subring of squares (or some larger ring in the kernel). The internal function in this section is used to describe this S-module and allow conversion between elements of R and elements of the S-module.
> HAPPRIME_SModule ( R, exps ) | ( function ) |
Returns: Record
For a polynomial ring R, k[x_1, x_2, ..., x_n], and list of exponents exps, [e_1, e_2, ..., e_n], returns a record which represents R as an S-module, where S is the subring of R given by k[x_1^e_1, x_2^e_2, ..., x_n^e_n]
The record has the following components:
Rring
the original ring R
Sring
a ring isomorphic to the subring of powers S
Spows
the exponents exps
generators
the list of generators of R as an S-module
FromPoly
a function which takes a polynomial in R and returns the corresponding element in the S-module (as a vector)
ToPoly
a function which takes an element in the S-module (as a vector) and returns the corresponding polynomial in R
Internal helper functions for testing HAPprime.
> HAPPRIME_VersionWithSVN ( ) | ( function ) |
Returns: String
Returns a string giving a current version number for the HAPprime installation assuming that it is checked out from a subversion repository. This fetches the current version number from PackageInfo.g
and appends the return of the svnversion
program to this, returning the resulting composite string.
gap> HAPPRIME_VersionWithSVN(); "0.2.1.302:319M" |
> HAPPRIME_Random2Group ( [orderG] ) | ( operation ) |
> HAPPRIME_Random2GroupAndAction ( [orderG] ) | ( operation ) |
Returns: Group or groupAndAction
record
Returns a random 2-group, or a groupAndAction
record (see ModuleGroupAndAction
(5.4-5)) with the canonical action. The order may be specified as an argument, or if not then a group is chosen randomly (from a uniform distribution) over all of the possible groups with order from 2 to 128.
gap> HAPPRIME_Random2Group(); <pc group of size 8 with 3 generators> gap> HAPPRIME_Random2Group(); <pc group of size 32 with 5 generators> |
> HAPPRIME_TestResolutionPrimePowerGroup ( [ntests] ) | ( operation ) |
Returns: Boolean
Returns true
if ResolutionPrimePowerGroupGF
(HAPprime: ResolutionPrimePowerGroupGF (for group)) and ResolutionPrimePowerGroupRadical
(HAPprime: ResolutionPrimePowerGroupRadical (for group)) appear to be working correctly, or false
otherwise. This repeatedly creates resolutions of length 6 for random 2-groups (up to order 128) using both of the HAPprime resolution algorithms, and compares them both with the original HAP ResolutionPrimePowerGroup
(HAP: ResolutionPrimePowerGroup) and checks that they are equal. The optional argument ntests specifies how many resolutions to try: the default is 25.
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