From 8950fb5113ace6c04ee8ebf602168814080cc3e6 Mon Sep 17 00:00:00 2001 From: Friedrich Rober Date: Fri, 16 Apr 2021 16:10:07 +0200 Subject: [PATCH] Add `ListWreathProductElement` and `WreathProductElementList` Add implementations of these operations for the following wreath products: - perm wreath product in imprimitive action - perm wreath product in product action - matrix wreath product - generic wreath product Add documentation and tests for these operations. --- doc/ref/grpprod.xml | 2 + lib/gprd.gd | 63 +++++ lib/gprd.gi | 67 ++++++ lib/gprdmat.gi | 59 +++++ lib/gprdperm.gi | 76 +++++++ .../opers/ListWreathProductElement.tst | 215 ++++++++++++++++++ 6 files changed, 482 insertions(+) create mode 100644 tst/testinstall/opers/ListWreathProductElement.tst diff --git a/doc/ref/grpprod.xml b/doc/ref/grpprod.xml index 2255c64166..a94cf8cac5 100644 --- a/doc/ref/grpprod.xml +++ b/doc/ref/grpprod.xml @@ -118,6 +118,8 @@ has to be taken.) <#Include Label="WreathProductImprimitiveAction"> <#Include Label="WreathProductProductAction"> <#Include Label="KuKGenerators"> +<#Include Label="ListWreathProductElement"> +<#Include Label="WreathProductElementList"> diff --git a/lib/gprd.gd b/lib/gprd.gd index 6fb5c5d8bb..5403952a04 100644 --- a/lib/gprd.gd +++ b/lib/gprd.gd @@ -566,3 +566,66 @@ InstallTrueMethod(IsGeneratorsOfMagmaWithInverses, DeclareRepresentation("IsWreathProductElementDefaultRep", IsWreathProductElement and IsPositionalObjectRep,[]); + +############################################################################# +## +#F ListWreathProductElement +#O ListWreathProductElementNC +## +## <#GAPDoc Label="ListWreathProductElement"> +## +## +## +## +## +## Let x be an element of a wreath product G +## where G = K \wr H and H acts +## as a finite permutation group of degree m. +## We can identify the element x with a tuple (f_1, \ldots, f_m; h), +## where f_i \in K is the i-th base component of x +## and h \in H is the top component of x. +##

+## returns a list [f_1, \ldots, f_m, h] +## containing the components of x or fail if x cannot be decomposed in the wreath product. +##

+## If ommited, the argument testDecomposition defaults to true. +## If testDecomposition is true, makes additional tests to ensure +## that the computed decomposition of x is correct, +## i.e. it checks that x is an element of the parent wreath product of G: +##

+## If K \leq \mathop{Sym}(l), this ensures that x \in \mathop{Sym}(l) \wr \mathop{Sym}(m) +## where the parent wreath product is considered in the same action as G, +## i.e. either in imprimitive action or product action. +##

+## If K \leq \mathop{GL}(n,q), this ensures that x \in \mathop{GL}(n,q) \wr \mathop{Sym}(m). +## +## +## <#/GAPDoc> +## +DeclareGlobalFunction( "ListWreathProductElement" ); +DeclareOperation( "ListWreathProductElementNC", [HasWreathProductInfo, IsObject, IsBool] ); + +############################################################################# +## +#F WreathProductElementList +#O WreathProductElementListNC +## +## <#GAPDoc Label="WreathProductElementList"> +## +## +## +## +## +## Let list be equal to [f_1, \ldots, f_m, h] and G be a wreath product +## where G = K \wr H, H acts +## as a finite permutation group of degree m, +## f_i \in K and h \in H. +##

+## returns the element x \in G +## identified by the tuple (f_1, \ldots, f_m; h). +## +## +## <#/GAPDoc> +## +DeclareGlobalFunction( "WreathProductElementList" ); +DeclareOperation( "WreathProductElementListNC", [HasWreathProductInfo, IsList] ); diff --git a/lib/gprd.gi b/lib/gprd.gi index 398c4fc8ff..06ab2d7a0d 100644 --- a/lib/gprd.gi +++ b/lib/gprd.gi @@ -896,6 +896,73 @@ local I,n,fam,typ,gens,hgens,id,i,e,info,W,p,dom; end); +############################################################################# +## +#M ListWreathProductElement(, [, ]) +## +InstallGlobalFunction( ListWreathProductElement, +function(G, x, testDecomposition...) + local info; + if Length(testDecomposition) = 0 then + testDecomposition := true; + elif Length(testDecomposition) = 1 then + testDecomposition := testDecomposition[1]; + elif Length(testDecomposition) > 1 then + ErrorNoReturn("too many arguments"); + fi; + if not HasWreathProductInfo(G) then + ErrorNoReturn("usage: must be a wreath product"); + fi; + return ListWreathProductElementNC(G, x, testDecomposition); +end); + +InstallMethod( ListWreathProductElementNC, "generic wreath product", true, + [ HasWreathProductInfo, IsWreathProductElement, IsBool ], 0, +function(G, x, testDecomposition) + local info, list, i; + info := WreathProductInfo(G); + list := EmptyPlist(info!.degI + 1); + for i in [1 .. info!.degI + 1] do + list[i] := StructuralCopy(x![i]); + od; + return list; +end); + +############################################################################# +## +#M WreathProductElementList(, ) +## +InstallGlobalFunction( WreathProductElementList, +function(G, list) + local info, i; + + if not HasWreathProductInfo(G) then + ErrorNoReturn("usage: must be a wreath product"); + fi; + info := WreathProductInfo(G); + if Length(list) <> info.degI + 1 then + ErrorNoReturn("usage: must have ", + "length 1 + "); + fi; + for i in [1 .. info.degI] do + if not list[i] in info.groups[1] then + ErrorNoReturn("usage: must contain ", + "elements of "); + fi; + od; + if not list[info.degI + 1] in info.groups[2] then + ErrorNoReturn("usage: must be ", + "an element of "); + fi; + return WreathProductElementListNC(G, list); +end); + +InstallMethod( WreathProductElementListNC, "generic wreath product", true, + [ HasWreathProductInfo, IsList ], 0, +function(G, list) + return Objectify(FamilyObj(One(G))!.defaultType, StructuralCopy(list)); +end); + ############################################################################# ## #M PrintObj() diff --git a/lib/gprdmat.gi b/lib/gprdmat.gi index 0f6ea2eabd..e9797831cc 100644 --- a/lib/gprdmat.gi +++ b/lib/gprdmat.gi @@ -440,6 +440,65 @@ local info, degI, dimA, zero, projFunc; return info.projection; end); +############################################################################# +## +#M ListWreathProductElementNC( , ) +## +InstallMethod( ListWreathProductElementNC, "matrix wreath product", true, + [ IsMatrixGroup and HasWreathProductInfo, IsObject, IsBool], 0, +function(G, x, testDecomposition) + local info, degI, dimA, h, list, i, j, k, zeroMat; + + info := WreathProductInfo(G); + degI := info.degI; + dimA := info.dimA; + + # The top group element + h := x ^ Projection(G); + if h = fail then + return fail; + fi; + list := EmptyPlist(degI + 1); + list[degI + 1] := h; + if testDecomposition then + # ZeroMatrix does not accept IsPlistRep + if IsPlistRep(x) then + zeroMat := NullMat(dimA, dimA, info.field); + else + zeroMat := ZeroMatrix(dimA, dimA, x); + fi; + fi; + for i in [1 .. degI] do + j := i ^ h; + list[i] := ExtractSubMatrix(x, [dimA * (i - 1) + 1 .. dimA * i], [dimA * (j - 1) + 1 .. dimA * j]); + if testDecomposition then + for k in [1 .. degI] do + if k = j then + continue; + fi; + if ExtractSubMatrix(x, [dimA * (i - 1) + 1 .. dimA * i], [dimA * (k - 1) + 1 .. dimA * k]) <> zeroMat then + return fail; + fi; + od; + fi; + od; + return list; +end); + +############################################################################# +## +#M WreathProductElementListNC(, ) +## +InstallMethod( WreathProductElementListNC, "matrix wreath product", true, + [ IsMatrixGroup and HasWreathProductInfo, IsList ], 0, +function(G, list) + local info; + + info := WreathProductInfo(G); + # TODO: Remove `MatrixByBlockMatrix` when `BlockMatrix` supports the MatObj interface. + return MatrixByBlockMatrix(BlockMatrix(List([1 .. info.degI], i -> [i, i ^ list[info.degI + 1], list[i]]), info.degI, info.degI)); +end); + # tensor wreath -- dimension d^e This is not a faithful representation of # the tensor product if the matrix group has a center. DeclareGlobalFunction("TensorWreathProduct"); diff --git a/lib/gprdperm.gi b/lib/gprdperm.gi index ceddd36134..d3eb190f47 100644 --- a/lib/gprdperm.gi +++ b/lib/gprdperm.gi @@ -856,6 +856,82 @@ local info, proj, H, degI, degK, constPoints, projFunc; return proj; end); +############################################################################# +## +#M ListWreathProductElementNC( , ) +## +InstallMethod( ListWreathProductElementNC, "perm wreath product", true, + [ IsPermGroup and HasWreathProductInfo, IsObject, IsBool ], 0, +function(G, x, testDecomposition) + local info, list, h, f, degK, i, j, constPoints, imageComponents, comp, restPerm; + + info := WreathProductInfo(G); + # The top group element + h := x ^ Projection(G); + if h = fail then + return fail; + fi; + # The product of the base group elements + f := x * Image(Embedding(G, info.degI + 1), h) ^ (-1); + list := EmptyPlist(info!.degI + 1); + list[info.degI + 1] := h; + # Imprimitive Action, tuple (i, j) corresponds + # to point i + degK * (j - 1) + if IsBound(info.permimpr) and info.permimpr then + for i in [1 .. info.degI] do + restPerm := RESTRICTED_PERM(f, info.components[i], testDecomposition); + if restPerm = fail then + return fail; + fi; + list[i] := restPerm ^ info.perms[i]; + od; + # Primitive Action, tuple (t_1, ..., t_degI) corresponds + # to point Sum_{i=1}^degI t_i * degK ^ (i - 1) + elif IsBound(info.productType) and info.productType then + degK := NrMovedPoints(info.groups[1]); + # constPoints correspond to [1, 1, 1, ...], [2, 2, 2, ...], ... + constPoints := List([0 .. degK - 1], i -> Sum([0 .. info.degI - 1], + j -> i * degK ^ j)) + 1; + # imageComponents = [ [1 ^ f_1, 1 ^ f_2, 1 ^ f_3, ...], + # [2 ^ f_1, 2 ^ f_2, 2 ^ f_3, ...], ... ] + imageComponents := List(OnTuples(constPoints, f) - 1, + p -> CoefficientsQadic(p, degK) + 1); + # The qadic expansion has no "trailing" zeros. Thus we need to append them. + # For example if (1, ..., 1) ^ (f_1, ..., f_m) = (1, ..., 1), + # we have imageComponents[1] = CoefficientsQadic(0, degK) + 1 = []. + # Note that we append 1's instead of 0's, + # since we already transformed the result of the qadic expansion + # from [{0, ..., degK - 1}, ...] to [{1, ..., degK}, ...]. + for i in [1 .. degK] do + comp := imageComponents[i]; + Append(comp, ListWithIdenticalEntries(info.degI - Length(comp), 1)); + od; + for j in [1 .. info.degI] do + list[j] := PermList(List([1 .. degK], i -> imageComponents[i,j])); + if list[j] = fail then + return fail; + fi; + od; + else + ErrorNoReturn("Error: cannot determine which action ", + "was used for wreath product"); + fi; + return list; +end); + +############################################################################# +## +#M WreathProductElementListNC(, ) +## +InstallMethod( WreathProductElementListNC, "perm wreath product", true, + [ IsPermGroup and HasWreathProductInfo, IsList ], 0, +function(G, list) + local info; + + info := WreathProductInfo(G); + return Product(List([1 .. info.degI + 1], i -> list[i] ^ Embedding(G, i))); +end); + ############################################################################# ## #F WreathProductProductAction( , ) wreath product in product action diff --git a/tst/testinstall/opers/ListWreathProductElement.tst b/tst/testinstall/opers/ListWreathProductElement.tst new file mode 100644 index 0000000000..3f990630b9 --- /dev/null +++ b/tst/testinstall/opers/ListWreathProductElement.tst @@ -0,0 +1,215 @@ +# +gap> START_TEST("ListWreathProductElement.tst"); + +# +# Perm Wreath Product In Imprimitive Action +# + +# +gap> K := SymmetricGroup(3);; +gap> H := SymmetricGroup(5);; +gap> G := WreathProduct(K, H);; + +# +gap> list := [(1,2), (1,2,3), (), (1,2,3), (), (1,2)(3,4)];; +gap> x := WreathProductElementList(G, list);; +gap> list = ListWreathProductElement(G, x); +true + +# +gap> list := [(1,2), (1,2,3), (), (1,2,3), (), (1,2,3)];; +gap> x := WreathProductElementList(G, list);; +gap> list = ListWreathProductElement(G, x); +true + +# +gap> x := (1,5,9,12,13)(2,4,8,11,15,3,6,7,10,14);; +gap> list := ListWreathProductElement(G, x);; +gap> x = WreathProductElementList(G, list); +true + +# Top Component fails +gap> x := (1,5);; +gap> x ^ Projection(G); +fail +gap> ListWreathProductElement(G, x); +fail + +# Base Component fails +gap> x := (1,2,3,4,5);; +gap> x ^ Projection(G); +() +gap> ListWreathProductElement(G, x); +fail + +# +# Perm Wreath Product In Product Action +# + +# +gap> K := SymmetricGroup(3);; +gap> H := SymmetricGroup(5);; +gap> G := WreathProductProductAction(K, H);; + +# +gap> list := [(1,2), (1,2,3), (), (1,2,3), (), (1,2)(3,4)];; +gap> x := WreathProductElementList(G, list);; +gap> list = ListWreathProductElement(G, x); +true + +# +gap> list := [(1,2), (1,2,3), (), (1,2,3), (), (1,2,3)];; +gap> x := WreathProductElementList(G, list);; +gap> list = ListWreathProductElement(G, x); +true + +# +gap> x := +> ( 1, 25, 24, 12, 17, 5)( 2, 7, 22, 21, 18, 14)( 3, 16, 23)( 4, 19, 27, 15, 11, 8)( 6, 10, 26)( 9, 13, 20) +> ( 28,106, 78,174, 44, 86, 55,187, 51, 93, 71,167)( 29, 88, 76,183, 45, 95, 56,169, 49,102, 72,176) +> ( 30, 97, 77,165, 43,104, 57,178, 50, 84, 70,185)( 31,100, 81,177, 38, 89, 58,181, 54, 96, 65,170) +> ( 32, 82, 79,186, 39, 98, 59,163, 52,105, 66,179)( 33, 91, 80,168, 37,107, 60,172, 53, 87, 64,188) +> ( 34,103, 75,180, 41, 83, 61,184, 48, 99, 68,164)( 35, 85, 73,189, 42, 92, 62,166, 46,108, 69,173) +> ( 36, 94, 74,171, 40,101, 63,175, 47, 90, 67,182)(109,160,240,201,125,140,217,214,132,147,233,194) +> (110,142,238,210,126,149,218,196,130,156,234,203)(111,151,239,192,124,158,219,205,131,138,232,212) +> (112,154,243,204,119,143,220,208,135,150,227,197)(113,136,241,213,120,152,221,190,133,159,228,206) +> (114,145,242,195,118,161,222,199,134,141,226,215)(115,157,237,207,122,137,223,211,129,153,230,191) +> (116,139,235,216,123,146,224,193,127,162,231,200)(117,148,236,198,121,155,225,202,128,144,229,209);; +gap> list := ListWreathProductElement(G, x);; +gap> x = WreathProductElementList(G, list); +true + +# Top Projection fails +gap> x := (4,16);; +gap> x ^ Projection(G); +fail +gap> ListWreathProductElement(G, x); +fail + +# Base Decomposition fails +gap> x := (1,2,3,4,5);; +gap> x ^ Projection(G); +() +gap> ListWreathProductElement(G, x); +fail + +# +# Matrix Wreath Product +# + +# +gap> K := GL(3,3);; +gap> H := SymmetricGroup(5);; +gap> G := WreathProduct(K, H);; + +# +gap> list := [K.1, K.2, One(K), K.2, One(K), (1,2)(3,4)];; +gap> x := WreathProductElementList(G, list);; +gap> list = ListWreathProductElement(G, x); +true + +# +gap> list := [K.1, K.2, One(K), K.2, One(K), (1,2,3)];; +gap> x := WreathProductElementList(G, list);; +gap> list = ListWreathProductElement(G, x); +true + +# +gap> x := [ +> [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), Z(3)^0, Z(3)^0 ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ Z(3)^0, Z(3)^0, Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ] ];; +gap> list := ListWreathProductElement(G, x);; +gap> x = WreathProductElementList(G, list); +true +gap> ConvertToMatrixRep(x);; +gap> list := ListWreathProductElement(G, x);; +gap> x = WreathProductElementList(G, list); +true + +# Top Projection fails +gap> x := [ +> [ Z(3), 0*Z(3), Z(3)^0, Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3), Z(3), Z(3)^0, 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0 ], +> [ Z(3), Z(3), Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0, Z(3), Z(3)^0, 0*Z(3), Z(3)^0, Z(3), Z(3) ], +> [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3), 0*Z(3), Z(3)^0, Z(3), Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3), Z(3)^0 ], +> [ Z(3), 0*Z(3), Z(3), 0*Z(3), Z(3), Z(3), Z(3)^0, Z(3), Z(3), Z(3), Z(3), Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], +> [ Z(3)^0, Z(3), Z(3)^0, 0*Z(3), 0*Z(3), Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, Z(3), 0*Z(3), Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), Z(3)^0, 0*Z(3), Z(3)^0, Z(3), 0*Z(3), Z(3), Z(3)^0, 0*Z(3) ], +> [ Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0, 0*Z(3), Z(3), Z(3)^0, Z(3)^0, 0*Z(3), Z(3), 0*Z(3), 0*Z(3), Z(3), Z(3), Z(3)^0 ], +> [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3), Z(3)^0, Z(3), Z(3)^0, 0*Z(3), 0*Z(3), Z(3), Z(3)^0, Z(3)^0, Z(3), Z(3), 0*Z(3) ], +> [ Z(3), Z(3)^0, Z(3)^0, Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], +> [ Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3), Z(3), Z(3), Z(3), Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ], +> [ 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3), Z(3)^0, Z(3), Z(3), 0*Z(3), 0*Z(3), Z(3), Z(3), Z(3)^0, Z(3) ], +> [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3), Z(3), Z(3)^0, Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0, Z(3) ], +> [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3), 0*Z(3), Z(3), 0*Z(3), Z(3), Z(3), 0*Z(3), Z(3)^0, Z(3)^0 ], +> [ 0*Z(3), 0*Z(3), Z(3), Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3), Z(3)^0, Z(3), Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, Z(3) ], +> [ 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3), 0*Z(3), Z(3)^0, Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3)^0 ] ];; +gap> x ^ Projection(G); +fail +gap> ListWreathProductElement(G, x); +fail +gap> ConvertToMatrixRep(x);; +gap> x ^ Projection(G); +fail +gap> ListWreathProductElement(G, x); +fail + +# Base Decomposition fails +gap> x := [ +> [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3), Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), Z(3), 0*Z(3), Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3) ], +> [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0 ] ];; +gap> x ^ Projection(G); +() +gap> ListWreathProductElement(G, x); +fail +gap> ConvertToMatrixRep(x);; +gap> x ^ Projection(G); +() +gap> ListWreathProductElement(G, x); +fail + +# +# Generic Wreath Product +# + +# +gap> K := DihedralGroup(12);; +gap> H := SymmetricGroup(5);; +gap> G := WreathProduct(K, H);; + +# +gap> list := [K.1, K.2, One(K), K.2, One(K), (1,2)(3,4)];; +gap> x := WreathProductElementList(G, list);; +gap> list = ListWreathProductElement(G, x); +true + +# +gap> x := G.1 * G.5 * G.2 * G.5 ^ ((G.4 * G.5) ^ 2) * G.2;; +gap> list := ListWreathProductElement(G, x);; +gap> x = WreathProductElementList(G, list); +true