  
  [1X41 [33X[0;0YGroup Actions[133X[101X
  
  [33X[0;0YA  [13Xgroup action[113X is a triple [22X(G, Ω, μ)[122X, where [22XG[122X is a group, [22XΩ[122X a set and [22Xμ : Ω
  ×  G  → Ω[122X a function that is compatible with the group arithmetic. We call [22XΩ[122X
  the [13Xdomain[113X of the action.[133X
  
  [33X[0;0YIn  [5XGAP[105X, [22XΩ[122X can be a duplicate-free collection (an object that permits access
  to  its  elements  via  the [22XΩ[n][122X operation, for example a list), it does not
  need to be sorted (see [2XIsSet[102X ([14X21.17-4[114X)).[133X
  
  [33X[0;0YThe  acting function [22Xμ[122X is a binary [5XGAP[105X function that returns the image [22Xμ( x,
  g )[122X for a point [22Xx ∈ Ω[122X and a group element [22Xg ∈ G[122X.[133X
  
  [33X[0;0YIn  [5XGAP[105X, groups always act from the right, that is [22Xμ( μ( x, g ), h ) = μ( x,
  gh )[122X.[133X
  
  [33X[0;0Y[5XGAP[105X does not test whether the acting function [22Xμ[122X satisfies the conditions for
  a  group  operation  but  silently  assumes  that  is does. (If it does not,
  results are unpredictable.)[133X
  
  [33X[0;0YThe  first  section  of  this  chapter, [14X41.1[114X, describes the various ways how
  operations for group actions can be called.[133X
  
  [33X[0;0YFunctions for several commonly used action are already built into [5XGAP[105X. These
  are listed in section [14X41.2[114X.[133X
  
  [33X[0;0YThe sections [14X41.7[114X and [14X41.8[114X describe homomorphisms and mappings associated to
  group actions as well as the permutation group image of an action.[133X
  
  [33X[0;0YThe  other sections then describe operations to compute orbits, stabilizers,
  as well as properties of actions.[133X
  
  [33X[0;0YFinally section [14X41.12[114X describes the concept of [21Xexternal sets[121X which represent
  the concept of a [13X[22XG[122X-set[113X and underly the actions mechanism.[133X
  
  
  [1X41.1 [33X[0;0YAbout Group Actions[133X[101X
  
  [33X[0;0YThe  syntax  which  is  used  by  the  operations for group actions is quite
  flexible.  For  example  we can call the operation [2XOrbitsDomain[102X ([14X41.4-3[114X) for
  the orbits of the group [3XG[103X on the domain [3XOmega[103X in the following ways:[133X
  
  [8X[10XOrbitsDomain[110X[22X( G, Ω[, μ] )[122X[108X
        [33X[0;6YThe  acting  function  [22Xμ[122X is optional. If it is not given, the built-in
        action  [2XOnPoints[102X  ([14X41.2-1[114X)  (which  defines  an  action  via the caret
        operator [10X^[110X) is used as a default.[133X
  
  [8X[10XOrbitsDomain[110X[22X( G, Ω, gens, acts[, μ] )[122X[108X
        [33X[0;6YThis  second version of [2XOrbitsDomain[102X ([14X41.4-3[114X) permits one to implement
        an  action  induced  by a homomorphism: If the group [22XH[122X acts on [22XΩ[122X via [22Xμ[122X
        and  [22Xφ  :  G → H[122X is a homomorphism, [22XG[122X acts on [22XΩ[122X via the induced action
        [22Xμ'( x, g ) = μ( x, g^φ )[122X.[133X
  
        [33X[0;6YHere [22Xgens[122X must be a set of generators of [22XG[122X and [22Xacts[122X the images of [22Xgens[122X
        under  [22Xφ[122X.  [22Xμ[122X  is  the  acting function for [22XH[122X. Again, the function [22Xμ[122X is
        optional and [2XOnPoints[102X ([14X41.2-1[114X) is used as a default.[133X
  
        [33X[0;6YThe  advantage of this notation is that [5XGAP[105X does not need to construct
        this  homomorphism [22Xφ[122X and the range group [3XH[103X as [5XGAP[105X objects. (If a small
        group [22XG[122X acts via complicated objects [22Xacts[122X this otherwise could lead to
        performance problems.)[133X
  
        [33X[0;6Y[5XGAP[105X  does  not test whether the mapping [22Xgens ↦ acts[122X actually induces a
        homomorphism  and  the  results  are  unpredictable if this is not the
        case.[133X
  
  [8X[10XOrbitsDomain[110X[22X( extset )[122X[108X
        [33X[0;6YA  third  variant is to call the operation with an external set, which
        then  provides  [22XG[122X,  [22XΩ[122X and [22Xμ[122X. You will find more about external sets in
        Section [14X41.12[114X.[133X
  
  [33X[0;0YFor  operations  like  [2XStabilizer[102X  ([14X41.5-2[114X)  of  course  the  domain must be
  replaced by an element of the domain of the action.[133X
  
  
  [1X41.2 [33X[0;0YBasic Actions[133X[101X
  
  [33X[0;0Y[5XGAP[105X  already  provides  acting  functions  for  the more common actions of a
  group.  For  built-in operations such as [2XStabilizer[102X ([14X41.5-2[114X) special methods
  are available for many of these actions.[133X
  
  [33X[0;0YIf  one  needs  an  action  for  which no acting function is provided by the
  library it can be implemented via a [5XGAP[105X function that conforms to the syntax[133X
  
  [33X[0;0Y[10Xactfun( omega, g )[110X[133X
  
  [33X[0;0Ywhere  [10Xomega[110X  is  an  element  of  the action domain, [10Xg[110X is an element of the
  acting group, and the return value is the image of [10Xomega[110X under [10Xg[110X.[133X
  
  [33X[0;0YFor  example  one  could define the following function that acts on pairs of
  polynomials via [2XOnIndeterminates[102X ([14X41.2-13[114X):[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XOnIndeterminatesPairs:= function( polypair, g )[128X[104X
    [4X[28X  return [ OnIndeterminates( polypair[1], g ),[128X[104X
    [4X[28X           OnIndeterminates( polypair[2], g ) ];[128X[104X
    [4X[28Xend;[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  this function [13Xmust[113X implement a group action from the [13Xright[113X. This
  is not verified by [5XGAP[105X and results are unpredictable otherwise.[133X
  
  [1X41.2-1 OnPoints[101X
  
  [29X[2XOnPoints[102X( [3Xpnt[103X, [3Xg[103X ) [32X function
  
  [33X[0;0Yreturns  [10X[3Xpnt[103X[10X  ^  [3Xg[103X[10X[110X. This is for example the action of a permutation group on
  points, or the action of a group on its elements via conjugation. The action
  of  a  matrix  group on vectors from the right is described by both [2XOnPoints[102X
  and [2XOnRight[102X ([14X41.2-2[114X).[133X
  
  [1X41.2-2 OnRight[101X
  
  [29X[2XOnRight[102X( [3Xpnt[103X, [3Xg[103X ) [32X function
  
  [33X[0;0Yreturns  [10X[3Xpnt[103X[10X  * [3Xg[103X[10X[110X. This is for example the action of a group on its elements
  via  right  multiplication,  or  the  action  of  a group on the cosets of a
  subgroup.  The  action  of  a  matrix  group  on  vectors  from the right is
  described by both [2XOnPoints[102X ([14X41.2-1[114X) and [2XOnRight[102X.[133X
  
  [1X41.2-3 OnLeftInverse[101X
  
  [29X[2XOnLeftInverse[102X( [3Xpnt[103X, [3Xg[103X ) [32X function
  
  [33X[0;0Yreturns [22X[3Xg[103X^{-1}[122X [10X* [3Xpnt[103X[10X[110X. Forming the inverse is necessary to make this a proper
  action, as in [5XGAP[105X groups always act from the right.[133X
  
  [33X[0;0Y[2XOnLeftInverse[102X  is used for example in the representation of a right coset as
  an  external  set  (see [14X41.12[114X), that is, a right coset [22XUg[122X is an external set
  for the group [22XU[122X acting on it via [2XOnLeftInverse[102X.)[133X
  
  [1X41.2-4 OnSets[101X
  
  [29X[2XOnSets[102X( [3Xset[103X, [3Xg[103X ) [32X function
  
  [33X[0;0YLet [3Xset[103X be a proper set (see [14X21.19[114X). [2XOnSets[102X returns the proper set formed by
  the images of all points [22Xx[122X of [3Xset[103X via the action function [2XOnPoints[102X ([14X41.2-1[114X),
  applied to [22Xx[122X and [3Xg[103X.[133X
  
  [33X[0;0Y[2XOnSets[102X  is  for example used to compute the action of a permutation group on
  blocks.[133X
  
  [33X[0;0Y([2XOnTuples[102X  ([14X41.2-5[114X)  is  an  action  on lists that preserves the ordering of
  entries.)[133X
  
  [1X41.2-5 OnTuples[101X
  
  [29X[2XOnTuples[102X( [3Xtup[103X, [3Xg[103X ) [32X function
  
  [33X[0;0YLet  [3Xtup[103X  be  a  list. [2XOnTuples[102X returns the list formed by the images of all
  points  [22Xx[122X of [3Xtup[103X via the action function [2XOnPoints[102X ([14X41.2-1[114X), applied to [22Xx[122X and
  [3Xg[103X.[133X
  
  [33X[0;0Y([2XOnSets[102X  ([14X41.2-4[114X)  is an action on lists that additionally sorts the entries
  of the result.)[133X
  
  [1X41.2-6 OnPairs[101X
  
  [29X[2XOnPairs[102X( [3Xtup[103X, [3Xg[103X ) [32X function
  
  [33X[0;0Yis a special case of [2XOnTuples[102X ([14X41.2-5[114X) for lists [3Xtup[103X of length 2.[133X
  
  [1X41.2-7 OnSetsSets[101X
  
  [29X[2XOnSetsSets[102X( [3Xset[103X, [3Xg[103X ) [32X function
  
  [33X[0;0Yimplements  the  action  on sets of sets. For the special case that the sets
  are  pairwise  disjoint,  it is possible to use [2XOnSetsDisjointSets[102X ([14X41.2-8[114X).
  [3Xset[103X must be a sorted list whose entries are again sorted lists, otherwise an
  error is triggered (see [14X41.3[114X).[133X
  
  [1X41.2-8 OnSetsDisjointSets[101X
  
  [29X[2XOnSetsDisjointSets[102X( [3Xset[103X, [3Xg[103X ) [32X function
  
  [33X[0;0Yimplements the action on sets of pairwise disjoint sets (see also [2XOnSetsSets[102X
  ([14X41.2-7[114X)).  [3Xset[103X  must be a sorted list whose entries are again sorted lists,
  otherwise an error is triggered (see [14X41.3[114X).[133X
  
  [1X41.2-9 OnSetsTuples[101X
  
  [29X[2XOnSetsTuples[102X( [3Xset[103X, [3Xg[103X ) [32X function
  
  [33X[0;0Yimplements  the  action  on  sets  of  tuples.  [3Xset[103X  must  be a sorted list,
  otherwise an error is triggered (see [14X41.3[114X).[133X
  
  [1X41.2-10 OnTuplesSets[101X
  
  [29X[2XOnTuplesSets[102X( [3Xset[103X, [3Xg[103X ) [32X function
  
  [33X[0;0Yimplements  the  action  on tuples of sets. [3Xset[103X must be a list whose entries
  are again sorted lists, otherwise an error is triggered (see [14X41.3[114X).[133X
  
  [1X41.2-11 OnTuplesTuples[101X
  
  [29X[2XOnTuplesTuples[102X( [3Xset[103X, [3Xg[103X ) [32X function
  
  [33X[0;0Yimplements the action on tuples of tuples.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=Group((1,2,3),(2,3,4));;[127X[104X
    [4X[25Xgap>[125X [27XOrbit(g,1,OnPoints);[127X[104X
    [4X[28X[ 1, 2, 3, 4 ][128X[104X
    [4X[25Xgap>[125X [27XOrbit(g,(),OnRight);[127X[104X
    [4X[28X[ (), (1,2,3), (2,3,4), (1,3,2), (1,3)(2,4), (1,2)(3,4), (2,4,3), [128X[104X
    [4X[28X  (1,4,2), (1,4,3), (1,3,4), (1,2,4), (1,4)(2,3) ][128X[104X
    [4X[25Xgap>[125X [27XOrbit(g,[1,2],OnPairs);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 1 ], [ 3, 4 ], [ 2, 1 ], [128X[104X
    [4X[28X  [ 1, 4 ], [ 4, 1 ], [ 4, 2 ], [ 3, 2 ], [ 2, 4 ], [ 4, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XOrbit(g,[1,2],OnSets);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 4 ], [ 1, 4 ], [ 2, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XOrbit(g,[[1,2],[3,4]],OnSetsSets);[127X[104X
    [4X[28X[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], [128X[104X
    [4X[28X  [ [ 1, 3 ], [ 2, 4 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XOrbit(g,[[1,2],[3,4]],OnTuplesSets);[127X[104X
    [4X[28X[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ], [128X[104X
    [4X[28X  [ [ 1, 3 ], [ 2, 4 ] ], [ [ 3, 4 ], [ 1, 2 ] ], [128X[104X
    [4X[28X  [ [ 1, 4 ], [ 2, 3 ] ], [ [ 2, 4 ], [ 1, 3 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XOrbit(g,[[1,2],[3,4]],OnSetsTuples);[127X[104X
    [4X[28X[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], [128X[104X
    [4X[28X  [ [ 1, 3 ], [ 4, 2 ] ], [ [ 2, 4 ], [ 3, 1 ] ], [128X[104X
    [4X[28X  [ [ 2, 1 ], [ 4, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XOrbit(g,[[1,2],[3,4]],OnTuplesTuples);[127X[104X
    [4X[28X[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ], [128X[104X
    [4X[28X  [ [ 1, 3 ], [ 4, 2 ] ], [ [ 3, 1 ], [ 2, 4 ] ], [128X[104X
    [4X[28X  [ [ 3, 4 ], [ 1, 2 ] ], [ [ 2, 1 ], [ 4, 3 ] ], [128X[104X
    [4X[28X  [ [ 1, 4 ], [ 2, 3 ] ], [ [ 4, 1 ], [ 3, 2 ] ], [128X[104X
    [4X[28X  [ [ 4, 2 ], [ 1, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ], [128X[104X
    [4X[28X  [ [ 2, 4 ], [ 3, 1 ] ], [ [ 4, 3 ], [ 2, 1 ] ] ][128X[104X
  [4X[32X[104X
  
  [1X41.2-12 OnLines[101X
  
  [29X[2XOnLines[102X( [3Xvec[103X, [3Xg[103X ) [32X function
  
  [33X[0;0YLet  [3Xvec[103X  be a [13Xnormed[113X row vector, that is, its first nonzero entry is normed
  to  the  identity  of  the relevant field, see [2XNormedRowVector[102X ([14X23.2-1[114X). The
  function  [2XOnLines[102X returns the row vector obtained from first multiplying [3Xvec[103X
  from  the  right  with  [3Xg[103X  (via  [2XOnRight[102X  ([14X41.2-2[114X)) and then normalizing the
  resulting row vector by scalar multiplication from the left.[133X
  
  [33X[0;0YThis  action  corresponds  to  the  projective  action  of a matrix group on
  one-dimensional subspaces.[133X
  
  [33X[0;0YIf  [3Xvec[103X  is  a  zero  vector  or  is  not  normed then an error is triggered
  (see [14X41.3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgl:=GL(2,5);;v:=[1,0]*Z(5)^0;[127X[104X
    [4X[28X[ Z(5)^0, 0*Z(5) ][128X[104X
    [4X[25Xgap>[125X [27Xh:=Action(gl,Orbit(gl,v,OnLines),OnLines);[127X[104X
    [4X[28XGroup([ (2,3,5,6), (1,2,4)(3,6,5) ])[128X[104X
  [4X[32X[104X
  
  [1X41.2-13 OnIndeterminates[101X
  
  [29X[2XOnIndeterminates[102X( [3Xpoly[103X, [3Xperm[103X ) [32X function
  
  [33X[0;0YA permutation [3Xperm[103X acts on the multivariate polynomial [3Xpoly[103X by permuting the
  indeterminates as it permutes points.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:=Indeterminate(Rationals,1);; y:=Indeterminate(Rationals,2);;[127X[104X
    [4X[25Xgap>[125X [27XOnIndeterminates(x^7*y+x*y^4,(1,17)(2,28));[127X[104X
    [4X[28Xx_17^7*x_28+x_17*x_28^4[128X[104X
    [4X[25Xgap>[125X [27XStabilizer(Group((1,2,3,4),(1,2)),x*y,OnIndeterminates);[127X[104X
    [4X[28XGroup([ (1,2), (3,4) ])[128X[104X
  [4X[32X[104X
  
  [1X41.2-14 Permuted[101X
  
  [29X[2XPermuted[102X( [3Xlist[103X, [3Xperm[103X ) [32X method
  
  [33X[0;0YThe  following  example  demonstrates  [2XPermuted[102X  ([14X21.20-18[114X)  being  used  to
  implement a permutation action on a domain:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=Group((1,2,3),(1,2));;[127X[104X
    [4X[25Xgap>[125X [27Xdom:=[ "a", "b", "c" ];;[127X[104X
    [4X[25Xgap>[125X [27XOrbit(g,dom,Permuted);[127X[104X
    [4X[28X[ [ "a", "b", "c" ], [ "c", "a", "b" ], [ "b", "a", "c" ], [128X[104X
    [4X[28X  [ "b", "c", "a" ], [ "a", "c", "b" ], [ "c", "b", "a" ] ][128X[104X
  [4X[32X[104X
  
  [1X41.2-15 OnSubspacesByCanonicalBasis[101X
  
  [29X[2XOnSubspacesByCanonicalBasis[102X( [3Xbas[103X, [3Xmat[103X ) [32X function
  [29X[2XOnSubspacesByCanonicalBasisConcatenations[102X( [3Xbasvec[103X, [3Xmat the  action  on tuples of sets. [3Xset[103X must beon tuples of sets. [3Xseion  on tuples of sets. [3Xset[103X must beon tuples
  beon tuples
  beon tuples
  beon tuples
  beon tuples
  bn on a domain:[133X
  
  [4X[32X  Example  [32X[104X
 sts that additionally sorts the entries
  of the result.)[133X
  
  [1X41.2-6 OnPairs[3XG 
  [1X41.2-6 OnPa vector  or  is  not  normed then an error is triggered
  (see \[133X,33X[0;0YSee   [03X, [3Xseps[103X[, [3Xwspace[103X] ) [32X operation
  
  [33X[0;0YThis function accepts a strin  beon tuples
  bs[103X[, [3Xw[103X of ing  cases  of cepts a strin  beon tuples
  bs[103X[, [3Xw[103X of ing  cases  of cepts a strin  beon tuples
  bs[103X[, [3Xw[103X of ing  cases  of cepts a strin  beon tuples
  bs[103X[, [3Xw[103X of ing  cases  of cepts a strin  beon tuples
  bs[103X[, [3Xw[103X of inou[27XIsMagw[103X of of the
  atation group on
  b[103X of of the
 -  gs)) qn,wd ing  cases  of cepts a strin  beon tuples03X[, [3Xw[103X of inou[27XIsMagw[103X of of the
  atation group on
  b0;0YIf  [3Xe wples03X[, [3Xw[103X of inou[27XIsMagw[103X))  that  ifferent  fro2Xb_{l+1} = 0[2 *2XIsPcGrouproup ome[103X (given as a number of milliseIsPcGrouproup ome[hich
  is done via [2XAs. The automorphisms act on
  [3XGjects   that   lie   in
  [10XCyclotomicsFamily[110X. They are larger or smaller than all X
    [4X[28X[ [ [ 1  lie   rations  for  tsem.  For informatisr For informat  [1X40.at  ifferengory   of
  [2XIsNonassocWordWithOneFamily[110X. T122X and [22Xμ[122X. You wf
  [2XIsNonassocWordWithOneFa14X32.13-3[114X))  that  is  regarded  as  a  conjugator
  isomorph2X and [22Xμ[1 inou[27XIsMag act on
  [3XGwismAutomGroup[h2X and [22Xμ[1 inou[2d [22Xμ[1 inou[2d [22Xμ[1 inou[2d [22p[h2X and [22XckDigitISBN[101X
  
  [29X[2Xg act on
  [3XGwismAutomGroup[h2X and [22Xμ[1 inou[2d [22Xμ[1 inou[2d2X and [22Xμ[1 inou[2d [23Xμ[1  [1X21.22-2 CoeMagmaWithOne[that  this functionify the onif[29X[2Xg act on
  [3XGwismAutomGroup[h2X and [ism from a grouappings that map from a perm iotion
  
  [33X[2 [22Xμ'( x, 2 [22Xμ'( x, 2 [22Xμ'( x, 2 [22Xμ'( x, 2 [22Xμ [2X\<[10orsOfFpGroup(Image(isofp));103X, [3Xg[103X is inducehe
 -  gs)) qrsOfFpGroup(Image(isofp));103X, [3Xg[103X is inducehe
 -  gserators
   [22Xμ[1 inoX, [3Xg[103X i));103X, gmaWithOne[that  this funct[1 inou[2d [);-2 CoeMagmaWith[2d [);-2 CoeMa;103X, gmurns the automorphism is  action  corresponds  to  the  projective  action  iseIsPcGrouent  fro2Xb_{l+1} = 0[2 *2d [);-2 CoeMa;103X, gmurns the automorphism is  action  corresponds  to  the  projective  action  iseIsPh5Xgap>[125X [27XOrbit(g,[1,2],OnSets);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3tuples of sets. [3Xset[103X must beon tupl[32X functrts the entries
  of tc,5Xgagenerating set implements  the  action  on tuples
  the  action  on tuples of sets. [3Xset[103X must beon tuples of set the name
  may be set fe, [ 3tuples of y is normed
  to [22Xμ'( xme
  may be set fe, [ 3tuples of y is normed
  to [22Xμ'( xme
  may be set fe, [ 3tuprewismAutomGroup[h2X a'( xme
  may be set fe, [ 3tuprewismAutomGroup[h2X a'( xme
  may be set fe, [ 3tuprewismAutomGroup[xt set fe, [ 3tup( xmeX[0;0YLet  [3ed
  to [22Xμ'( xme
  may be set fe, [ 3tuprewismAutoXμ'( es of y i 0[2 *2XIsPcGrouproup ome[103X (given as a number of milliseIsPcGrouproup ome[hich
  is done via [2XAs.ration [2XInverseGeneralMappoup ome[103X (given as a nX function
  me[103X (given as a nX function
  me a'( xme
  may be set fe, [ 5Xgap>[125X [27XO[4X[32X  Example  [114X.[[27XO[4X[32X  X[32X  Example  [114X.[[26 ction
 (rminates[102X( [3Xpolyts of [XIsPcGrouproup ome[103X (given as a number of 102X returns [9Xfail[32X  Example  [114X.[[27XO[4X[32X  X[32X  Example  [114X.[[26 ction
 (rminates[102X( [3Xpolrwise.[133X
  
  [1X41.2-1 OnPoints[101X
OX
  
  [1X41.2-1 OnPo [,4) ]) -> [ ]nts[101X
OX
  OnPo [,4) ]) -> [ tivariate polynomial [3Xpoly[103X by pervariate polynomial [3Xpoly[ed
        compby pervariate polynomial [3Xpoly[ed
        compby pervariate polynomial [3Xpoly[ed
        compby perv polynomial [3X.
  [2XSimulthynomial [(1,2ion[103X ) [32X func perv polynomfunc perv pinou[2d2X and [22Xμ[1 inou[2d [23Xμ[1 . ;0YLet  [3ed
  to YLet  [3ed
  t, returns
  [9Xtrue[109X  if  [3Xel1[103X  is  (sirictly)  less  than [3Xel2[103X with respect to [3Xord[103X, and [)  less  than
3X  may  be the matrix of scalar products of some virtual characters. From
  the  characters and thpby perv polynomial [3X.
  [2XSimulthynomial [(1,2ion[103X ) [32X func perv p return [ OnI
  
  [33X[0;0YFor  each  attribute  [3Xattr[103X  that is declared with [2XDr  eactlared with [2XDr  eactlared with [2XDr  eactlared with [2XDr  eactlared with [2XDr  eactlared with [2XDr  eactlared with [2   attribute  [3Xattr[103X  that is declared with [2XD  action, a whose entries are again soared with [2XD  actii)e entries are again soared wi aga, -5/2 ] ] ][128X[104X
  [4X[32X[104X
  
  [1X24.11-5 BaseSteinitzVectors[101X
  
  [29X[2XBasrSt29X[2XBasrSt29X[2XBasrSt29X[2XBasrSt29X[2XBasrSt29X[2XBasrSt29X[2XBasrSt29of  statements  to  emphasize  this
  difference.[133X
  
  [33X[0;0YUsing expressions asg expressions asg expressions character set of the operating system. Character literals
  can be entered in [5XGAP[105X by enclosing the osed in [5XGLB,(x,ter literals
  can be entered in [5XGAP[105X by enclosing the osed in [5X[5XGAP[105X 21.19[114X). [2XO[5X[5XGAP[105X 21.19[114X14X). [2XO[5X[5XGAP[105X 21.19[114X14X). [2XO[5X[5XGAP[105X 21.19[114X14X). [2XO[5X[5XGAP[105X  atation group on 
  stored  in[114X). the  same  value;  for lists
d
  to [22Xμ'm. Character literals
  cm. Character literals
  cm. Character literals
  cm. Character literals
  cample  [32X[104Dracter literals
  cample  [32X[104Dracter literals
  cample  [32X[104Dracter literals
  cample  [32X[17Xx:=Indetermlnate(Rr literals
  caple  [32X[17Xx:=Indetermlnate(Rr literals
  caple  [32X[17Xx:=Indetermlnate(Rr literals
  caple  [32X[17Xx:=Indetermlnate(Rr literals
  caple  appings that map from a p enclosing  caple  appings that map from a p enclosing  caple  appings that map from a p enclosr literals
  calesTuples[101X
 H ism  if and only if it is a ring
  homomorphism with source a field.[133X
  
  
  [1X4Lf ng
  homomorphism with source a field.[13[13Xenumerator[113X  is  an  immutable  128X[104X
enclosr literals
  calesTuples[101X
X
  
  [29X[2XOnRight[102X( [3
  
  [29X[2XOnX[104X
    [4X4X[28X[ [ [ 1, 2 ], [ 3, 4 ] ], [, 2 ], [ 3                             nforce
  calculation  of *2X[17Xx:=Indetermlnate(Rr literalby pervariate polnn of Morphisms.
  The  elementrvariate polnsTuples[101X
 H ism  s [22XH[122X.[133X
  
  [1X40.7-4 IsAutomo[ [ 1, 4 ], [ 2, 3 ] 2X[104Dracter lite
  [1X4Lf qcter literals
  cample[ 2, 3 ], [ 1, 3 ], [ 3, 4 ], [ 1, 4 ], [ 2,[104X
    [4X33X[0;0YIf  [  instead  of  the l [3Xpoly[103[5XGLB,(x,ter literals
  can be epe[125X [27XRootInt( 2 * 10^1  
  [1X41.2-1 OnPo [,4) ]):=Action(gl,Orbit(gl,v,OnLines),OnLines);[1([14X25.20-1[114X)  and  [2XJoinStringsWithSeparator[102X
  (closing  caple  appings thrs114X) returns [9Xfail[109X am( prg1,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            