chap2.txt

  
  2 Accessing and Manipulating Resolutions
  
  
  2.1 Representation-Independent Access Methods
  
  All  methods listed below take a HapResolution in any representation. If the
  other  arguments  are  compatible with the representation of the resolution,
  the  returned  value  will be in the form defined by this representation. If
  the other arguments are in a different representation, GAPs method selection
  is  used  via  TryNextMethod()  to  find  an  applicable  method (a suitable
  representation).
  
  The  idea behind this is that the results of computations have the same form
  as  the  input.  And  as  all representations are sub-representations of the
  HapResolutionRep   representation,   input  which  is  compatible  with  the
  HapResolutionRep representation is always valid.
  
  Every new representation must support the functions of this section.
  
  2.1-1 StrongestValidRepresentationForLetter
  
  > StrongestValidRepresentationForLetter( resolution, term, letter ) __method
  Returns:  filter
  
  Finds the sub-representation of HapResolutionRep for which letter is a valid
  letter   of   the   termth   module  of  resolution.  Note  that  resolution
  automatically  is  in  some  sub-representation  of HapResolutionRep.This is
  mainly meant for debugging.
  
  2.1-2 StrongestValidRepresentationForWord
  
  > StrongestValidRepresentationForWord( resolution, term, word ) ______method
  Returns:  filter
  
  Finds  the  sub-representation of HapResolutionRep for which word is a valid
  word  of the termth module of resolution. Note that resolution automatically
  is  in some sub-representation of HapResolutionRep. This is mainly meant for
  debugging.
  
  2.1-3 PositionInGroupOfResolution
  
  > PositionInGroupOfResolution( resolution, g ) _______________________method
  > PositionInGroupOfResolutionNC( resolution, g ) _____________________method
  Returns:  positive integer
  
  This  returns  the position of the group element g in the enumeration of the
  group of resolution. The NC version does not check if g really is an element
  of the group of resolution.
  
  2.1-4 IsValidGroupInt
  
  > IsValidGroupInt( resolution, n ) ___________________________________method
  Returns:  boolean
  
  Returns true if the nth element of the group of resolution is known.
  
  2.1-5 GroupElementFromPosition
  
  > GroupElementFromPosition( resolution, n ) __________________________method
  Returns:  group element or fail
  
  Returns  nth  element  of the group of resolution. If the nth element is not
  known, fail is returned.
  
  2.1-6 MultiplyGroupElts
  
  > MultiplyGroupElts( resolution, x, y ) ______________________________method
  Returns:  positive  integer or group element, depending on the type of x and
            y
  
  If  x  and  y  are given in standard representation (i.e. as integers), this
  returns the position of the product of the group elements represented by the
  positive integers x and x.
  
  If x and y are given in any other representation, the returned group element
  will also be represented in this way.
  
  2.1-7 MultiplyFreeZGLetterWithGroupElt
  
  > MultiplyFreeZGLetterWithGroupElt( resolution, letter, g ) __________method
  Returns:  A letter
  
  Multiplies  the  letter  letter  with  the  group  element g and returns the
  result.  If resolution is in standard representation, g has to be an integer
  and  letter  has  to  be  a  pair  of integer. If resolution is in any other
  representation,  letter  and  g  can  be  in  a  form  compatible  with that
  representation  or  in  the  standard form (in the latter case, the returned
  value will also have standard form).
  
  2.1-8 MultiplyFreeZGWordWithGroupElt
  
  > MultiplyFreeZGWordWithGroupElt( resolution, word, g ) ______________method
  Returns:  A word
  
  Multiplies the word word with the group element g and returns the result. If
  resolution  is  in  standard representation, g has to be an integer and word
  has  to  be  a  list  of  pairs  of  integers. If resolution is in any other
  representation,   word  and  g  can  be  in  a  form  compatible  with  that
  representation  or  in  the  standard form (in the latter case, the returned
  value will also have standard form).
  
  2.1-9 BoundaryOfFreeZGLetter
  
  > BoundaryOfFreeZGLetter( resolution, term, letter ) _________________method
  Returns:  free ZG word (in the same representation as letter)
  
  Calculates  the  boundary  of  the  letter  (word of length 1) letter of the
  termth module of resolution.
  
  The  returned  value  is a word of the term-1st module and comes in the same
  representation as letter.
  
  2.1-10 BoundaryOfFreeZGWord
  
  > BoundaryOfFreeZGWord( resolution, term, word ) _____________________method
  Returns:  free ZG word (in the same representation as letter)
  
  Calculates the boundary of the word word of the termth module of resolution.
  
  The  returned  value  is a word of the term-1st module and comes in the same
  representation as word.
  
  
  2.2 Converting Between Representations
  
  Four  methods  are  provided  to  convert  letters  and  words from standard
  representation to any other representation and back again.
  
  2.2-1 ConvertStandardLetter
  
  > ConvertStandardLetter( resolution, term, letter ) __________________method
  > ConvertStandardLetterNC( resolution, term, letter ) ________________method
  Returns:  letter in the representation of resolution
  
  Converts  the letter letter in standard representation to the representation
  of  resolution.  The  NC  version  does not check whether letter really is a
  letter in standard representation.
  
  2.2-2 ConvertStandardWord
  
  > ConvertStandardWord( resolution, term, word ) ______________________method
  > ConvertStandardWordNC( resolution, term, word ) ____________________method
  Returns:  word in the representation of resolution
  
  Converts  the  word word in standard representation to the representation of
  resolution.  The  NC  version does not check whether word is a valid word in
  standard representation.
  
  2.2-3 ConvertLetterToStandardRep
  
  > ConvertLetterToStandardRep( resolution, term, letter ) _____________method
  > ConvertLetterToStandardRepNC( resolution, term, letter ) ___________method
  Returns:  letter in standard representation
  
  Converts  the  letter  letter  in  the  representation  of resolution to the
  standard  representation.  The NC version does not check whether letter is a
  valid letter of resolution.
  
  2.2-4 ConvertWordToStandardRep
  
  > ConvertWordToStandardRep( resolution, term, word ) _________________method
  > ConvertWordToStandardRepNC( resolution, term, word ) _______________method
  Returns:  word in standard representation
  
  Converts  the  word word in the representation of resolution to the standard
  representation.  The  NC version does not check whether word is a valid word
  of resolution.
  
  
  2.3 Special Methods for HapResolutionRep
  
  Some   methods   explicitely  require  the  input  to  be  in  the  standard
  representation  (HapResolutionRep).  Two  of  these test if a word/letter is
  really  in standard representation, the other ones are non-check versions of
  the universal methods.
  
  2.3-1 IsFreeZGLetter
  
  > IsFreeZGLetter( resolution, term, letter ) _________________________method
  Returns:  boolean
  
  Checks  if  letter  is  an  valid  letter  (word  of  length  1) in standard
  representation of the termth module of resolution.
  
  2.3-2 IsFreeZGWord
  
  > IsFreeZGWord( resolution, term, word ) _____________________________method
  Returns:  boolean
  
  Check if word is a valid word in large standard representation of the termth
  module in resolution.
  
  2.3-3 MultiplyGroupEltsNC
  
  > MultiplyGroupEltsNC( resolution, x, y ) ____________________________method
  Returns:  positive integer
  
  Given positive integers x and y, this returns the position of the product of
  the  group  elements  represented  by  the  positive  integers x and x. This
  assumes  that all input is in standard representation and does not check the
  input.
  
  2.3-4 MultiplyFreeZGLetterWithGroupEltNC
  
  > MultiplyFreeZGLetterWithGroupEltNC( resolution, letter, g ) ________method
  Returns:  A letter in standard representation
  
  Multiplies  the  letter  letter  with  the  group element represented by the
  positive  integer  g  and  returns the result. The input is assumed to be in
  HapResolutionRep and is not checked.
  
  2.3-5 MultiplyFreeZGWordWithGroupEltNC
  
  > MultiplyFreeZGWordWithGroupEltNC( resolution, word, g ) ____________method
  Returns:  A letter in standard representation
  
  Multiplies  the word word with the group element represented by the positive
  integer   g  and  returns  the  result.  The  input  is  assumed  to  be  in
  HapResolutionRep and is not checked.
  
  2.3-6 BoundaryOfFreeZGLetterNC
  
  > BoundaryOfFreeZGLetterNC( resolution, term, letter ) _______________method
  Returns:  free ZG word in standard representation
  
  Calculates  the  boundary  of  the  letter  (word of length 1) letter of the
  termth  module  of  resolution.  The  input  is  assumed  to  be in standard
  representation and not checked.
  
  2.3-7 BoundaryOfFreeZGWordNC
  
  > BoundaryOfFreeZGWordNC( resolution, term, word ) ___________________method
  Returns:  free ZG word in standard representation
  
  Calculates the boundary of the word word of the termth module of resolution.
  The input is assumed to be in standard representation and not checked.
  
  
  2.4 The HapLargeGroupResolutionRep Representation
  
  The   large   group  representation  has  one  additional  component  called
  groupring.  Elements of the modules in a resolution are represented by lists
  of  group ring elements. The length of the list corresponds to the dimension
  of the free module.
  
  All  methods for the generic representation do also work for the large group
  representation.  Some of them are wrappers for special methods which do only
  work  for  this representation. The following list only contains the methods
  which are not already present in the generic representation.
  
  Note  that  the  input or the output of these functions does not comply with
  the standard representation.
  
  2.4-1 GroupRingOfResolution
  
  > GroupRingOfResolution( resolution ) ________________________________method
  Returns:  group ring
  
  This  returns  the group ring of resolution. Note that by the way that group
  rings     are     handled     in    GAP,    this    is    not    equal    to
  GroupRing(R,GroupOfResolution(resolution))  where  R  is  the  ring  of  the
  resolution.
  
  2.4-2 MultiplyGroupElts_LargeGroupRep
  
  > MultiplyGroupElts_LargeGroupRep( resolution, x, y ) ________________method
  > MultiplyGroupEltsNC_LargeGroupRep( resolution, x, y ) ______________method
  Returns:  group element
  
  Returns  the product of x and y. The NC version does not check whether x and
  y are actually elements of the group of resolution.
  
  2.4-3 IsFreeZGLetterNoTermCheck_LargeGroupRep
  
  > IsFreeZGLetterNoTermCheck_LargeGroupRep( resolution, letter ) ______method
  Returns:  boolean
  
  Returns  true  if  letter  has  the  form of a letter (a module element with
  exactly  one  non-zero  entry  which has exactly one non-zero coefficient) a
  module  of  resolution  in  the HapLargeGroupResolution representation. Note
  that  it  is  not  tested  if  letter  actually  is  a letter in any term of
  resolution
  
  2.4-4 IsFreeZGWordNoTermCheck_LargeGroupRep
  
  > IsFreeZGWordNoTermCheck_LargeGroupRep( resolution, word ) __________method
  Returns:  boolean
  
  Returns true if word has the form of a word of a module of resolution in the
  HapLargeGroupResolution  representation.  Note that it is not tested if word
  actually is a word in any term of resolution.
  
  2.4-5 IsFreeZGLetter_LargeGroupRep
  
  > IsFreeZGLetter_LargeGroupRep( resolution, term, letter ) ___________method
  Returns:  boolean
  
  Returns  true  if and only if letter is a letter (a word of length 1) of the
  termth  module  of resolution in the hapLargeGroupResolution representation.
  I.e.  it tests if letter is a module element with exactly one non-zero entry
  which has exactly one non-zero coefficient.
  
  2.4-6 IsFreeZGWord_LargeGroupRep
  
  > IsFreeZGWord_LargeGroupRep( resolution, term, word ) _______________method
  Returns:  boolean
  
  Tests if word is an element of the termth module of resoultion.
  
  2.4-7 MultiplyFreeZGLetterWithGroupElt_LargeGroupRep
  
  > MultiplyFreeZGLetterWithGroupElt_LargeGroupRep( resolution, letter, g ) method
  > MultiplyFreeZGLetterWithGroupEltNC_LargeGroupRep( resolution, letter, g ) method
  Returns:  free ZG letter in large group representation
  
  Multiplies  the  letter  letter  with  the  group  element g and returns the
  result.  The  NC version does not check whether g is an element of the group
  of resolution and letter can be a letter.
  
  2.4-8 MultiplyFreeZGWordWithGroupElt_LargeGroupRep
  
  > MultiplyFreeZGWordWithGroupElt_LargeGroupRep( resolution, word, g ) method
  > MultiplyFreeZGWordWithGroupEltNC_LargeGroupRep( resolution, word, g ) method
  Returns:  free ZG word in large group representation
  
  Multiplies  the  word  word with the group element g and returns the result.
  The  NC  version  does  not  check  whether  g is an element of the group of
  resolution and word can be a word.
  
  2.4-9 GeneratorsOfModuleOfResolution_LargeGroupRep
  
  > GeneratorsOfModuleOfResolution_LargeGroupRep( resolution, term ) ___method
  Returns:  list of letters/words in large group representation
  
  Returns  a  set  of  generators  for  the  termth  module of resolution. The
  returned value is a list of vectors of group ring elements.
  
  2.4-10 BoundaryOfGenerator_LargeGroupRep
  
  > BoundaryOfGenerator_LargeGroupRep( resolution, term, n ) ___________method
  > BoundaryOfGeneratorNC_LargeGroupRep( resolution, term, n ) _________method
  Returns:  free ZG word in the large group representation
  
  Returns the boundary of the nth generator of the termth module of resolution
  as  a  word  in  the  n-1st  module  (in large group representation). The NC
  version  does  not  check  whether there is a termth module and if it has at
  least n generators.
  
  2.4-11 BoundaryOfFreeZGLetterNC_LargeGroupRep
  
  > BoundaryOfFreeZGLetterNC_LargeGroupRep( resolution, term, letter ) _method
  > BoundaryOfFreeZGLetter_LargeGroupRep( resolution, term, letter ) ___method
  Returns:  free ZG word in large group representation
  
  Calculates  the  boundary  of  the  letter  letter  of  the termth module of
  resolution  in  large  group  representation.  The NC version does not check
  whether letter actually is a letter in the termth module.
  
  2.4-12 BoundaryOfFreeZGWord_LargeGroupRep
  
  > BoundaryOfFreeZGWord_LargeGroupRep( resolution, term, word ) _______method
  Returns:  free ZG word in large group representation
  
  Calculates  the  boundary  of  the  element  word  of  the  termth module of
  resolution  in  large  group  representation.  The NC version does not check
  whether word actually is a word in the termth module.