Group product constructions¶

Product constructions enable the user to construct new groups from the groups provided in RepLAB. The product constructions below also provide ways to construct standard representations that arise from representations of the factor groups.

DirectProductGroup represents the (outer) direct product of compact groups, the elements are represented by row cell arrays.
SemidirectProductGroup represents an outer semidirect product arising from the action of a group on another.
WreathProductGroup describes the wreath product of a compact group by a permutation group.

DirectProductGroup¶

class replab.DirectProductGroup¶

Bases: replab.CompactGroup

Describes an external direct product of compact groups

This is an abstract base class. Use DirectProductGroup.make or CompactGroup.directProduct to construct an instance.

Constructors are defined in subclasses, as for other group products.

Class members list

Properties

factors – Factor groups
identity – Monoid identity element

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Monoid operations

areCommuting – Returns whether the two given elements commute
compose – Composes two monoid/group elements
composeAll – Composes a sequence of monoid elements
composeN – Computes the power of a given element by repeated squaring
isIdentity – Tests if the given element is the identity

Morphisms

automorphismGroup – Returns the automorphism group of this group
commutingFactorMorphismsMorphism – Constructs a morphism from morphisms of the factors with commuting images
commutingFactorMorphismsMorphismFun – Constructs a morphism from morphisms of the factors with commuting images (function version)
commutingMorphismsMorphism – Constructs a morphism from morphisms with commuting images
injection – Returns the morphism embedding the i-th factor into the direct product
innerAutomorphism – Returns the inner automorphism given using conjugation by the given element
isomorphismByFunctions – Constructs a group isomorphism using preimage/image functions
morphismByFunction – Constructs a group morphism using an image function
projection – Returns the morphism projecting this group into its i-th factor

Group methods

commutator – Returns the commutator of two group elements
composeFlat – Returns the composition of elements from an array picked according to the given indices/letters
composeWithInverse – Returns the composition of an element with the inverse of another element
inverse – Computes the inverse of an element
leftConjugate – Returns the left conjugate of a group element

Representations

commutingFactorRepsRep – Constructs a representation from commuting representations of the factors
commutingFactorRepsRepFun – Constructs a representation from commuting representations of the factors (function version)
commutingRepsRep – Constructs a representation from commuting representations
directSumFactorRep – Constructs a direct sum representation
directSumFactorRepFun – Constructs a direct sum representation from a function that maps factors to representations
directSumRep – Computes the direct sum of representations on this group
tensorFactorRep – Constructs a tensor product representation
tensorFactorRepFun – Constructs a direct sum representation from a function that maps factors to representations
tensorRep – Computes the tensor product of representations
trivialRep – Returns the trivial representation of this group on a finite dimensional vector space

Group construction

directPower – Returns the direct product of this group with itself a number of times
directProduct – Returns the direct product of groups
semidirectProduct – Describes an external semidirect product of groups

Group properties

hasReconstruction – Returns whether the group has a “reconstruction”
maximalTorusDimension – Returns, if available, the dimension of the maximal torus contained in the connected component of this group
reconstruction – Returns a reconstruction of the group used to speed up group averaging

Factor manipulation

factor – Returns a factor in this direct product
nFactors – Returns the number of factors in this direct product

General

make –
torusBlocks –

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

areCommuting(self, x, y)¶: Documentation in replab.Monoid.areCommuting()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

automorphismGroup(self)¶: Documentation in replab.Group.automorphismGroup()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

commutator(self, x, y, startWithInverse)¶: Documentation in replab.Group.commutator()

commutingMorphismsMorphism(self, target, morphisms)¶: Documentation in replab.Group.commutingMorphismsMorphism()

commutingRepsRep(self, field, dimension, reps)¶: Documentation in replab.CompactGroup.commutingRepsRep()

compose(self, x, y)¶: Documentation in replab.Monoid.compose()

composeAll(self, elements)¶: Documentation in replab.Monoid.composeAll()

composeFlat(self, array, letters)¶: Documentation in replab.Group.composeFlat()

composeN(self, x, n)¶: Documentation in replab.Monoid.composeN()

composeWithInverse(self, x, y)¶: Documentation in replab.Group.composeWithInverse()

directPower(self, n)¶: Documentation in replab.CompactGroup.directPower()

directProduct(varargin)¶: Documentation in replab.CompactGroup.directProduct()

directSumRep(self, field, reps)¶: Documentation in replab.CompactGroup.directSumRep()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

hasReconstruction(self)¶: Documentation in replab.CompactGroup.hasReconstruction()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

identity¶: Documentation in replab.Monoid.identity

inCache(self, name)¶: Documentation in replab.Obj.inCache()

innerAutomorphism(self, by)¶: Documentation in replab.Group.innerAutomorphism()

inverse(self, x)¶: Documentation in replab.Group.inverse()

isIdentity(self, x)¶: Documentation in replab.Monoid.isIdentity()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

isomorphismByFunctions(self, target, preimageElementFun, imageElementFun, torusMap)¶: Documentation in replab.Group.isomorphismByFunctions()

laws(self)¶: Documentation in replab.Obj.laws()

leftConjugate(self, by, on)¶: Documentation in replab.Group.leftConjugate()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

make()¶: No documentation

maximalTorusDimension(self)¶: Documentation in replab.CompactGroup.maximalTorusDimension()

morphismByFunction(self, target, imageElementFun, torusMap)¶: Documentation in replab.Group.morphismByFunction()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

reconstruction(self)¶: Documentation in replab.CompactGroup.reconstruction()

sample(self)¶: Documentation in replab.Domain.sample()

semidirectProduct(self, N, phi)¶: Documentation in replab.CompactGroup.semidirectProduct()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

tensorRep(self, field, reps)¶: Documentation in replab.CompactGroup.tensorRep()

torusBlocks()¶: No documentation

trivialRep(self, field, dimension)¶: Documentation in replab.CompactGroup.trivialRep()

Own members

factors¶

Factor groups

Type: cell(1,*) of subclass of CompactGroup

commutingFactorMorphismsMorphism(self, target, factorMorphisms)¶

Constructs a morphism from morphisms of the factors with commuting images

Parameters

target (CompactGroup) – Target group
factorMorphisms (cell(1,*) of Morphism) – Morphisms for each of the factor groups

Returns

The computed morphism

Return type

Morphism

commutingFactorMorphismsMorphismFun(self, target, fun)¶

Constructs a morphism from morphisms of the factors with commuting images (function version)

Parameters

target (CompactGroup) – Target group
fun (function_handle) – A function valid for each factor group that maps the group and its index to one of its morphisms

Returns

The computed morphism

Return type

Morphism

commutingFactorRepsRep(self, field, dimension, factorReps)¶

Constructs a representation from commuting representations of the factors

Parameters

field ({'R', 'C'}) – Field
dimension (integer) – Dimension of the representation
factorReps (cell(1,*) of Rep) – Representations for each of the factor groups (factorReps{i} is a representation of factor(i))

Returns

A representation computed from the product of representations

Return type

replab.Rep

commutingFactorRepsRepFun(self, field, dimension, fun)¶

Constructs a representation from commuting representations of the factors (function version)

Parameters

field ({'R', 'C'}) – Field
dimension (integer) – Dimension of the representation
fun (function_handle) – A function valid for each factor group that maps the group and its index to one of its representations

Returns

A representation computed from the product of representations

Return type

replab.Rep

directSumFactorRep(self, field, factorReps)¶

Constructs a direct sum representation

Parameters

field ({'R', 'C'}) – Field
factorReps (cell(1,*) of Rep) – Representations for each of the factor groups (factorReps{i} is a representation of factor(i))

Returns

A direct sum representation

Return type

replab.Rep

directSumFactorRepFun(self, field, fun)¶

Constructs a direct sum representation from a function that maps factors to representations

Parameters

field ({'R', 'C'}) – Field
fun (function_handle) – A function valid for each factor group that maps the group and its index to one of its representations

Returns

A direct sum representation

Return type

replab.Rep

factor(self, i)¶

Returns a factor in this direct product

Parameters: i (integer) – Factor idnex
Returns: The i-th factor
Return type: Group

injection(self, i)¶

Returns the morphism embedding the i-th factor into the direct product

Example

>>> S2 = replab.S(2);
>>> D = S2.directProduct(S2);
>>> m = D.injection(1);
>>> D.eqv({[2 1] [1 2]}, m.imageElement([2 1]))
    1

Parameters: i (integer) – Factor index
Returns: The injection morphism
Return type: Morphism

nFactors(self)¶

Returns the number of factors in this direct product

Returns: Number of factors
Return type: integer

projection(self, i)¶

Returns the morphism projecting this group into its i-th factor

Example

>>> S2 = replab.S(2);
>>> D = S2.directProduct(S2);
>>> m = D.projection(1);
>>> S2.eqv([2 1], m.imageElement({[2 1] [1 2]}))
    1

Parameters: i (integer) – Factor index
Returns: The projection
Return type: Morphism

tensorFactorRep(self, field, factorReps)¶

Constructs a tensor product representation

Parameters

field ({'R', 'C'}) – Field
factorReps (cell(1,*) of Rep)) – Representations for each of the factor groups (factorReps{i} is a representation of factor(i))

Returns

A tensor representation

Return type

replab.Rep

tensorFactorRepFun(self, field, fun)¶

Constructs a direct sum representation from a function that maps factors to representations

Parameters

field ({'R', 'C'}) – Field
fun (function_handle) – A function valid for each factor group that maps the group and its index to one of its representations

Returns

A tensor representation

Return type

replab.Rep

SemidirectProductGroup¶

class replab.SemidirectProductGroup¶

Bases: replab.Group

Semidirect product of compact groups through the external/outer construction

This is an abstract base class. Call CompactGroup.semidirectProduct or make to construct an instance.

Construction and multiplication rule

An outer semidirect product constructs a group from two given groups, with its set of elements being the Cartesian product of those two groups, and the multiplication operation given by the action of one group on the other.

Formally, we have two groups, $H$ and $N$, with $H$ acting on $N$ in the following way. We have a group action $\phi: H \times N \rightarrow N$, obeying the following:

$\phi(h_1, \phi(h_2, n)) = \phi(h_1 h_2, n)$
$\phi(h, n_1) \phi(h, n_2) = \phi(h, n_1 n_2)$

for $h,h_1,h_2 \in H$ and $n,n_1,n_2 \in N$.

Equivalently, this action corresponds to a group morphism $\phi: H \rightarrow (N \rightarrow N)$ or $\phi: H \rightarrow \operatorname{Aut}(N)$ from $H$ into the automorphism group of $N$.

We now construct the semidirect product $G = H \ltimes N$. An element of the product is written $G = (h, n)$, where $h \in H$ and $n \in N$.

Note that compared to other references such as Wikipedia ( https://en.wikipedia.org/wiki/Semidirect_product ), we write the element of the acting group $H$ first, and this changes the rule for multiplying group elements.

The multiplication rule of the group is $(h_1, n_1) \cdot (h_2, n_2) = (h_1 h_2, \phi(h_2^{-1}, n_1) n_2)$. The inverse of $(h, n)$ is given by $(h^{-1}, \phi(h, n))$.

How do we interpret this multiplication rule? First, we want to interpret the elements of $H$ and $N$ as “being part” of $G$.

We can embed elements of $H$ into $G$ by writing the injection $f(h) = (h, 1_N)$ where $1_N$ is the identity of $N$. We can embed elements of $N$ into $G$ by writing the injection $g(n) = (1_H, n)$ where $1_H$ is the identity of $H$.

Both $f$ and $g$ are morphisms: $f(h_1) f(h_2) = f(h_1 h_2)$ and $g(n_1) g(n_2) = g(n_1 n_2)$.

Those those injections are available using the methods Hinjection and Ninjection.

Noting that $(h,n) = f(h) g(n)$, we write:

$(h_1,n_1) (h_2,n_2) = f(h_1) g(n_1) f(h_2) g(n_2) = h_1 n_1 h_2 n_2$, with the $f()$ and $g()$ injections implicit.

If we could swap $n_1$ and $h_2$, as is the case with a direct product, we could rewrite the product as a pair of elements of $H$ and $N$ respectively. Now, we write $h_1 n_1 h_2 n_2 = h_1 h_2 h_2^{-1} n_1 h_2 n_2$, and we define $h_2^{-1} n_1 h_2 = \phi(h_2^{-1}, n_1)$.

Thus, $h_1 n_1 h_2 n_2 = (h_1 h_2) (\phi(h_2^{-1}, n_1) n_2) = (h_1 h_2, \phi(h_2^{-1}, n_1) n_2)$.

Thus, we interpret that group action $\phi$ as defining the conjugation of elements of $N$ by elements of $H$.

Representations

Representations of semidirect products can be constructed using the semidirectProductRep method, given compatible representations of the groups $H$ and $N$.

Implementation note

As semidirect product groups are used as a base for wreath product groups, the constructors are duplicated in subclasses as to keep a simple hierarchy of constructor calls.

Example

>>> N = replab.S(3);
>>> H = replab.PermutationGroup.cyclic(3);
>>> A = N.innerAutomorphism([2 3 1]);
>>> phi = H.morphismByImages(replab.AutomorphismGroup(N), 'images', {A});
>>> sd = H.semidirectProduct(N, @(h, n) phi.imageElement(h).imageElement(n));
>>> sd.laws.checkSilent
    1

Class members list

Properties

H – Group acting
N – Group acted upon
identity – Monoid identity element
phi – Action of H on N

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Monoid operations

areCommuting – Returns whether the two given elements commute
compose – Composes two monoid/group elements
composeAll – Composes a sequence of monoid elements
composeN – Computes the power of a given element by repeated squaring
isIdentity – Tests if the given element is the identity

Morphisms

Hinjection – Returns the morphism that expresses elements of H in the semidirect product
Hprojection – Returns the morphism that projects elements of the semidirect product into H
Ninjection – Returns the morphism that expresses elements of N in the semidirect product
automorphismGroup – Returns the automorphism group of this group
commutingMorphismsMorphism – Constructs a morphism from morphisms with commuting images
innerAutomorphism – Returns the inner automorphism given using conjugation by the given element
isomorphismByFunctions – Constructs a group isomorphism using preimage/image functions
morphismByFunction – Constructs a group morphism using an image function

Group methods

commutator – Returns the commutator of two group elements
composeFlat – Returns the composition of elements from an array picked according to the given indices/letters
composeWithInverse – Returns the composition of an element with the inverse of another element
inverse – Computes the inverse of an element
leftConjugate – Returns the left conjugate of a group element

SemidirectProductGroup creation

make – Constructs a semidirect product group from an action

Representations

semidirectProductRep – Constructs a representation by a product of representations of the two groups in the semidirect product construction

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

areCommuting(self, x, y)¶: Documentation in replab.Monoid.areCommuting()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

automorphismGroup(self)¶: Documentation in replab.Group.automorphismGroup()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

commutator(self, x, y, startWithInverse)¶: Documentation in replab.Group.commutator()

commutingMorphismsMorphism(self, target, morphisms)¶: Documentation in replab.Group.commutingMorphismsMorphism()

compose(self, x, y)¶: Documentation in replab.Monoid.compose()

composeAll(self, elements)¶: Documentation in replab.Monoid.composeAll()

composeFlat(self, array, letters)¶: Documentation in replab.Group.composeFlat()

composeN(self, x, n)¶: Documentation in replab.Monoid.composeN()

composeWithInverse(self, x, y)¶: Documentation in replab.Group.composeWithInverse()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

identity¶: Documentation in replab.Monoid.identity

inCache(self, name)¶: Documentation in replab.Obj.inCache()

innerAutomorphism(self, by)¶: Documentation in replab.Group.innerAutomorphism()

inverse(self, x)¶: Documentation in replab.Group.inverse()

isIdentity(self, x)¶: Documentation in replab.Monoid.isIdentity()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

isomorphismByFunctions(self, target, preimageElementFun, imageElementFun, torusMap)¶: Documentation in replab.Group.isomorphismByFunctions()

laws(self)¶: Documentation in replab.Obj.laws()

leftConjugate(self, by, on)¶: Documentation in replab.Group.leftConjugate()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

morphismByFunction(self, target, imageElementFun, torusMap)¶: Documentation in replab.Group.morphismByFunction()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

sample(self)¶: Documentation in replab.Domain.sample()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

Own members

H¶

Group acting

Type: replab.CompactGroup

N¶

Group acted upon

Type: replab.CompactGroup

phi¶

Action of H on N

Type: replab.Action

Hinjection(self)¶

Returns the morphism that expresses elements of H in the semidirect product

Returns: The morphism from H to this semidirect product group
Return type: Morphism

Hprojection(self)¶

Returns the morphism that projects elements of the semidirect product into H

Note that for a semidirect product G, the morphism G.Hinjection.andThen(G.Hprojection is the identity.

Returns: The morphism from the semidirect product to H
Return type: Morphism

Ninjection(self)¶

Returns the morphism that expresses elements of N in the semidirect product

Returns: The morphism from N to this semidirect product group
Return type: Morphism

static make(phi)¶

Constructs a semidirect product group from an action

Parameters: phi (replab.Action) – Action of a compact group on another compact group
Returns: A specialized instance of SemidirectProductGroup
Return type: SemidirectProductGroup

semidirectProductRep(self, Hrep, Nrep)¶

Constructs a representation by a product of representations of the two groups in the semidirect product construction

Those representations Hrep and Nrep must obey the following law.

Let phi be the semidirect product group uncurried homomorphism with h \in H and n \in N:

Nrep.image(phi(h, n)) == Hrep.image(h) * Nrep.image(n) * Hrep.inverseImage(h).

Parameters

Hrep (Rep) – Representation of H
Nrep (Rep) – Representation of N

Returns

Group representation

Return type

Rep

WreathProductGroup¶

class replab.WreathProductGroup¶

Bases: replab.SemidirectProductGroup

Wreath product group

This group contains:

a permutation group H acting on $n$ elements,
$n$ copies of a compact group A (the base group), collected in the direct product group H,

and those two groups are related through a semidirect product construction (see SemidirectProductGroup), where the permutation group acts on H by permuting the copies of A.

Note that compared to Wikipedia https://en.wikipedia.org/wiki/Wreath_product we write the elements of this wreath product as a tuple $(h, n)$ where the element $h \in H$ is written first, followed by $n = (a_1, ..., a_n)$ with $n \in N$ and $a_i \in A$.

The permutation group $H$ acts on $N$ as follows. Let $h \in H$ be a permutation and $n = (a_1, ..., a_n)$, and $h(n)$ the result of the action of $h$ on $n$. Then $h(n)_{h(i)} = a_i$, so that $h(n)_i = a_{h^{-1}(i)}$.

According to the literature (see also Wikipedia above), a wreath product defines two canonical actions, the imprimitive wreath product action and the primitive wreath product action.

Correspondingly, given a representation of the base group A, we provide methods to construct “imprimitive” and “primitive$ representations of this wreath product group using imprimitiveRep and primitiveRep.

Class members list

Properties

A – Base factor group
H – Group acting
N – Group acted upon
identity – Monoid identity element
n – Number of copies of the base group
phi – Action of H on N

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Monoid operations

areCommuting – Returns whether the two given elements commute
compose – Composes two monoid/group elements
composeAll – Composes a sequence of monoid elements
composeN – Computes the power of a given element by repeated squaring
isIdentity – Tests if the given element is the identity

Morphisms

Hinjection – Returns the morphism that expresses elements of H in the semidirect product
Hprojection – Returns the morphism that projects elements of the semidirect product into H
Ninjection – Returns the morphism that expresses elements of N in the semidirect product
automorphismGroup – Returns the automorphism group of this group
commutingMorphismsMorphism – Constructs a morphism from morphisms with commuting images
innerAutomorphism – Returns the inner automorphism given using conjugation by the given element
isomorphismByFunctions – Constructs a group isomorphism using preimage/image functions
morphismByFunction – Constructs a group morphism using an image function

Group methods

commutator – Returns the commutator of two group elements
composeFlat – Returns the composition of elements from an array picked according to the given indices/letters
composeWithInverse – Returns the composition of an element with the inverse of another element
inverse – Computes the inverse of an element
leftConjugate – Returns the left conjugate of a group element

SemidirectProductGroup creation

make – Constructs a wreath product group

Representations

imprimitiveRep – Returns an imprimitive representation of this wreath product
imprimitiveRepFun – Returns an imprimitive representation of this wreath product
primitiveRep – Returns a primitive representation of this wreath product
primitiveRepFun – Returns an primitive representation of this wreath product
semidirectProductRep – Constructs a representation by a product of representations of the two groups in the semidirect product construction

Permutation actions

imprimitivePermutation – Returns the permutation corresponding to the canonical imprimitive action
primitivePermutation – Returns the permutation corresponding to the canonical primitive action

Inherited elements

H¶: Documentation in replab.SemidirectProductGroup.H

Hinjection(self)¶: Documentation in replab.SemidirectProductGroup.Hinjection()

Hprojection(self)¶: Documentation in replab.SemidirectProductGroup.Hprojection()

N¶: Documentation in replab.SemidirectProductGroup.N

Ninjection(self)¶: Documentation in replab.SemidirectProductGroup.Ninjection()

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

areCommuting(self, x, y)¶: Documentation in replab.Monoid.areCommuting()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

automorphismGroup(self)¶: Documentation in replab.Group.automorphismGroup()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

commutator(self, x, y, startWithInverse)¶: Documentation in replab.Group.commutator()

commutingMorphismsMorphism(self, target, morphisms)¶: Documentation in replab.Group.commutingMorphismsMorphism()

compose(self, x, y)¶: Documentation in replab.Monoid.compose()

composeAll(self, elements)¶: Documentation in replab.Monoid.composeAll()

composeFlat(self, array, letters)¶: Documentation in replab.Group.composeFlat()

composeN(self, x, n)¶: Documentation in replab.Monoid.composeN()

composeWithInverse(self, x, y)¶: Documentation in replab.Group.composeWithInverse()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

identity¶: Documentation in replab.Monoid.identity

inCache(self, name)¶: Documentation in replab.Obj.inCache()

innerAutomorphism(self, by)¶: Documentation in replab.Group.innerAutomorphism()

inverse(self, x)¶: Documentation in replab.Group.inverse()

isIdentity(self, x)¶: Documentation in replab.Monoid.isIdentity()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

isomorphismByFunctions(self, target, preimageElementFun, imageElementFun, torusMap)¶: Documentation in replab.Group.isomorphismByFunctions()

laws(self)¶: Documentation in replab.Obj.laws()

leftConjugate(self, by, on)¶: Documentation in replab.Group.leftConjugate()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

morphismByFunction(self, target, imageElementFun, torusMap)¶: Documentation in replab.Group.morphismByFunction()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

phi¶: Documentation in replab.SemidirectProductGroup.phi

sample(self)¶: Documentation in replab.Domain.sample()

semidirectProductRep(self, Hrep, Nrep)¶: Documentation in replab.SemidirectProductGroup.semidirectProductRep()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

Own members

A¶

Base factor group

Type: CompactGroup

n¶

Number of copies of the base group

Type: integer

imprimitivePermutation(self, w, phiA)¶

Returns the permutation corresponding to the canonical imprimitive action

See https://en.wikipedia.org/wiki/Wreath_product

Parameters

w (element) – Wreath product group element to compute the image of
phi (function_handle, optional) – Morphism from elements of A to permutations If omitted default to identity, valid only when A is a permutation group

Returns

The permutation corresponding to the imprimitive action of w

Return type

permutation

imprimitiveRep(self, factorRep)¶

Returns an imprimitive representation of this wreath product

It acts on a space of dimension self.n * factorRep.dimension, which is a direct sum of copies of factorRep. The permutation group acts by permuting the blocks.

Parameters: factorRep (Rep) – A representation of the base factor group A
Returns: The corresponding imprimitive representation
Return type: Rep

imprimitiveRepFun(self, fun)¶

Returns an imprimitive representation of this wreath product

See imprimitiveRep.

Parameters: fun (function_handle) – A function that returns a representation of A when called on A as in Arep = fun(self.A)
Returns: The corresponding imprimitive representation
Return type: Rep

static make(H, A)¶

Constructs a wreath product group

Parameters

H (PermutationGroup) – Complement group that permutes the factors
A (CompactGroup) – Base factor group of which the direct product composes the base

Returns

A specialized instance of WreathProductGroup

Return type

WreathProductGroup

primitivePermutation(self, w, phiA)¶

Returns the permutation corresponding to the canonical primitive action

See https://en.wikipedia.org/wiki/Wreath_product

Parameters

w (element) – Wreath product group element to compute the image of
phi (function_handle, optional) – Morphism from elements of A to permutations If omitted default to identity, valid only when A is a permutation group

Returns

The permutation corresponding to the primitive action of w

Return type

permutation

primitiveRep(self, factorRep)¶

Returns a primitive representation of this wreath product

It acts on a space of dimension factorRep.dimension^self.n, which is a tensor product of copies of factorRep. The permutation group acts by permuting tensor indices.

Parameters: factorRep (Rep) – A representation of the base factor group A
Returns: The corresponding primitive representation
Return type: Rep

primitiveRepFun(self, fun)¶

Returns an primitive representation of this wreath product

See primitiveRep

Parameters: fun (function_handle) – A function that returns a representation of A when called on A as in Arep = fun(self.A)
Returns: The corresponding primitive representation
Return type: Rep