Base group structures¶

The following classes provide basic structures for monoids/groups. All objects below inherit from Str / Obj / Domain, and thus provide pretty printing infrastructure, a Domain.eqv equality test method, and a Domain.sample method to generate a random element.

Monoid: Structure with a binary associative operation, an identity element (property), and test for identity.
Group: A monoid with inverse elements.
CompactGroup: A Group with an associated Haar measure, which enables the construction of representations that can be handled by RepLAB.
AutomorphismGroup: Describes the automorphism group of a group
Action: Group can act on objects, and this describes the action of elements of a group on elements of a domain.

Monoid¶

class replab.Monoid¶

Bases: replab.Domain

Describes a monoid

See Monoid on Wikipedia

Class members list

Properties

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

Domain construction

lambda – Constructs a monoid from function handles

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

Inherited elements

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

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

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

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()

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()

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

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

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

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

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

identity¶

Monoid identity element

Type: element

areCommuting(self, x, y)¶

Returns whether the two given elements commute

Parameters

x (element) – Monoid/Group element
y (element) – Monoid/Group element

Returns

True if x y = y x, false otherwise

Return type

logical

compose(self, x, y)¶

Composes two monoid/group elements

Parameters

x (element) – Left hand side of the binary operation
y (element) – Right hand side of the binary operation

Returns

Result of the binary operation x op y

Return type

element

composeAll(self, elements)¶

Composes a sequence of monoid elements

For self.composeAll({x1 x2 ... xn}), returns x1*x2*...*xn (shown here in multiplaticative notation)

Parameters: elements (cell(1,*) of element) – Elements to compose
Returns: Composition of the given elements
Return type: element

composeN(self, x, n)¶

Computes the power of a given element by repeated squaring

Parameters

x (element) – Monoid/group element
n (integer) – Power; if this is not a group, we need n >= 0

Returns

The element x^n

Return type

element

isIdentity(self, x)¶

Tests if the given element is the identity

Parameters: x (element) – Monoid/Group element
Returns: True if x is the identity, false otherwise
Return type: logical

static lambda(header, eqvFun, sampleFun, composeFun, identity)¶

Constructs a monoid from function handles

Parameters

header (char) – Header display string
eqvFun (function_handle) – Handle implementing the eqv method
sampleFun (function_handle) – Handle implementing the sample method
composeFun (function_handle) – Handle implementing the compose method
identity (element) – Identity element of this monoid

Returns

The constructed monoid

Return type

Monoid

Group¶

class replab.Group¶

Bases: replab.Monoid

Describes a group

A group is a Monoid where each element has an inverse.

Class members list

Properties

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

Domain construction

lambda – Constructs a group from function handles

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
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

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()

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()

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

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

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

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()

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

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

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

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

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

automorphismGroup(self)¶: Returns the automorphism group of this group

commutator(self, x, y, startWithInverse)¶

Returns the commutator of two group elements

Parameters

x (element) – First element
y (element) – Second element
startWithInverse (logical) – True if we compute x^-1 y^-1 x y, false if we compute x y x^-1 y^-1

Returns

Commutator

Return type

element

commutingMorphismsMorphism(self, target, morphisms)¶

Constructs a morphism from morphisms with commuting images

Parameters

target (Group) – Target group
morphisms (cell(1,*) of replab.Morphism) – Morphisms whose images commute in target

Returns

The computed morphism

Return type

Morphism

composeFlat(self, array, letters)¶

Returns the composition of elements from an array picked according to the given indices/letters

Computes composeAll({array(letters(1)) ... array(letters(n))}) with the convention that

Parameters

letters (integer(1,*)) – Letters composing word
array (cell(1,*) of elements) – Elements to compose

Returns

The result of the composition

Return type

element

composeWithInverse(self, x, y)¶

Returns the composition of an element with the inverse of another element

Convenience method that can be overriden for speed optimizations

Parameters

x (element) – First element
y (element) – Second element

Returns

the result of x * y^-1 in multiplicative notation

Return type

element

innerAutomorphism(self, by)¶

Returns the inner automorphism given using conjugation by the given element

Parameters: by (element) – Conjugating element
Returns: Group automorphism (returns a FiniteMorphism if this group is finite)
Return type: Morphism

inverse(self, x)¶

Computes the inverse of an element

Given x, returns xInv such that

x xInv = identity

Example

>>> G = replab.S(3);
>>> g = [2 3 1];
>>> ginv = G.inverse(g);
>>> G.isIdentity(G.compose(g, ginv))
    1

Parameters: x (element) – Group element to compute the inverse of
Returns: Inverse of x
Return type: element

isomorphismByFunctions(self, target, preimageElementFun, imageElementFun, torusMap)¶

Constructs a group isomorphism using preimage/image functions

Parameters

target (replab.Group) – Target group
preimageElementFun (function_handle) – Returns the source element for a target element
imageElementFun (function_handle) – Returns the target element for a source element
torusMap (integer(*,*) or [], optional) – Torus map used in the Morphism construction, default: []

Returns

Constructed isomorphism (returns a FiniteIsomorphism if this group is finite)

Return type

Isomorphism

static lambda(header, eqvFun, sampleFun, composeFun, identity, inverseFun)¶

Constructs a group from function handles

Parameters

header (char) – Header display string
eqvFun (function_handle) – Handle implementing the parent Domain.eqv method
sampleFun (function_handle) – Handle implementing the parent Domain.sample method
composeFun (function_handle) – Handle implementing the parent Monoid.compose method
identity (element) – Identity element of this monoid
inverseFun (function_handle) – Handle implementing the inverse method

Returns

The constructed group

Return type

replab.Group

leftConjugate(self, by, on)¶

Returns the left conjugate of a group element

Convenience method that can be overriden for speed optimizations

Parameters

by (element) – Element conjugating
on (element) – Element conjugated

Returns

left conjugate, i.e. by * on * by^-1 in multiplicative notation

Return type

element

morphismByFunction(self, target, imageElementFun, torusMap)¶

Constructs a group morphism using an image function

Parameters

target (replab.Group) – Target group
imageElementFun (function_handle) – Returns the target element for a source element
torusMap (integer(*,*) or [], optional) – Torus map used in the Morphism construction, default: []

CompactGroup¶

class replab.CompactGroup¶

Bases: replab.Group

A group equipped with a Haar measure

Class members list

Properties

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

Domain construction

lambda – Constructs a compact group from function handles

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
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

Representations

commutingRepsRep – Constructs a representation from commuting representations
directSumRep – Computes the direct sum of representations on this group
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

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

commutingRepsRep(self, field, dimension, reps)¶

Constructs a representation from commuting representations

Parameters

field ({'R', 'C'}) – Field
dimension (integer) – Dimension of the representation
reps (cell(1,*) of Rep) – Commuting representations with the given field and dimension

Returns

A representation computed from the product of representations

Return type

replab.Rep

directPower(self, n)¶

Returns the direct product of this group with itself a number of times

Parameters

n (integer) – Number of copies

Returns

The direct product self x ...(n times)... x self: The return type is specialized as in directProduct.

Return type

DirectProductGroup

directProduct(varargin)¶

Returns the direct product of groups

Example

>>> S3 = replab.S(3);
>>> G = S3.directProduct(S3, S3);
>>> G.order
    216

Parameters

varargin – Variable number of arguments of type CompactGroup

Returns

The direct product group: If all groups are of type FiniteGroup, the return type FiniteGroup as well.

Return type

DirectProductGroup

directSumRep(self, field, reps)¶

Computes the direct sum of representations on this group

Parameters

field ({'R', 'C'}) – Field
reps (cell(1,*) of Rep) – Representation of this group over the given field

Returns

Direct sum of the representations

Return type

Rep

hasReconstruction(self)¶

Returns whether the group has a “reconstruction”

Returns: True if the call to reconstruction would succeed
Return type: logical

static lambda(header, eqvFun, sampleFun, composeFun, identity, inverseFun)¶

Constructs a compact group from function handles

Parameters

header (char) – Header display string
eqvFun (function_handle) – Handle implementing the eqv method
sampleFun (function_handle) – Handle implementing the sample method
composeFun (function_handle) – Handle implementing the compose method
identity (element) – Identity element of this monoid
inverseFun (function_handle) – Handle implementing the inverse method

Returns

The constructed compact group

Return type

replab.CompactGroup

maximalTorusDimension(self)¶

Returns, if available, the dimension of the maximal torus contained in the connected component of this group

Returns: Maximal torus dimension if available, or [] if no reconstruction is available
Return type: integer or []

reconstruction(self)¶

Returns a reconstruction of the group used to speed up group averaging

When G is a compact Lie group, we consider the chain of subgroups G >= G0 >= T, where:

G0 is the connected component of the group around the identity; we also know that the number of left cosets of G0 in G is finite; see the group of components, or, for example: https://mathoverflow.net/questions/378160/improved-classification-of-compact-lie-groups .

We write R a (finite) set of left coset representatives. Thus any element g of G can be written g = r g0 where r \in R and g0 \in G0.
G0 contains a maximal torus T.

To average over the action of a representation rho of G on x, we write x1 = \int_G d\mu(g) \rho(g) x.

However, this is equivalent to x1 = 1/|R| \sum_{r \in R} \int_G0 d\mu(g) \rho(g) x.

We finally write x1 = 1/{R} \sum_{r \in R} \int_G0 d\mu(g) \rho(g) \int_T d\mu(t) \rho(t) x; again, the fact that we integrate first over the maximal torus subgroup does not affect the result. However, integration over the maximal torus can often be done very quickly, and the reconstruction speeds up the computation of various equivariant subspaces dramatically.

This method returns R as a SetProduct, and T as a Morphism from T to G.

semidirectProduct(self, N, phi)¶

Describes an external semidirect product of groups

See the construction in https://en.wikipedia.org/wiki/Semidirect_product

Let H = self be a group, N a group.

The semidirect product is defined using a homomorphism

`` phi: H -> Aut(N) ``

which we write here in uncurried form:

phi: H x N -> N ``, ``n1 = phi(h, n).

We describe this homomorphism by a function handle with parameters of type H x N, which returns an element of type N.

We write each semidirect group element as a cell array of two elements {h, n}.

The basic law of a semidirect product construction is the following. Let h \in H, n \in N, h1 = {h, N.identity}, n1 = {H.identity, n}. Then phi(h, n) == h1 n1 h1^-1.

The type of the return value depends on the most refined type at the intersection of the type of self and N, with possible types CompactGroup/FiniteGroup.

Parameters

N (CompactGroup) – Group acted upon
phi (function_handle) – Function describing a homomorphism as described above

Returns

Semidirect product group

Return type

SemidirectProductGroup

tensorRep(self, field, reps)¶

Computes the tensor product of representations

Parameters: reps (cell(1,*) of Rep) – Representation of the same group over the same field
Returns: Tensor product of the representations
Return type: Rep

trivialRep(self, field, dimension)¶

Returns the trivial representation of this group on a finite dimensional vector space

For convenience, either the representation can act on a real or complex vector space, and multiple copies of the 1-dimensional trivial representation can be included, when dimension > 1.

Parameters

field ({'R', 'C'}) – Whether the representation is real (R) or complex (C)
dimension (integer, optional) – Representation dimension Default value 1

Returns

An instance of the trivial representation

Return type

Rep

AutomorphismGroup¶

class replab.AutomorphismGroup(object)¶

Bases: replab.Group

Describes the automorphism group of a group

Class members list

Properties

identity – Monoid identity element
object – Group

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
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

General

AutomorphismGroup –

Inherited elements

AutomorphismGroup()¶: No documentation

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

object¶

Group

Type: Group

Action¶

class replab.Action¶

Bases: replab.Obj

A group action describing the action of a group on a set

Elements of the group G act upon elements of type P.

Class members list

Properties

G – Group acting
P – Set acted upon

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

General

lambda – Constructs an action from a function handle

Action methods

leftAction – Computes the left action of a group element on a set element
rightAction – Computes the right action of a group element on a set element

Inherited elements

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

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()

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

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

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

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

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

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

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

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

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

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

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

Own members

G¶

Group acting

Type: Group

P¶

Set acted upon

Type: Domain

static lambda(header, G, P, leftActionFun)¶

Constructs an action from a function handle

Parameters

G (Group) – Group acting
(Domain (P) – Set acted upon
leftActionFun (function_handle) – Handle implementing leftAction

Returns

The action

Return type

Action

leftAction(self, g, p)¶

Computes the left action of a group element on a set element

Returns the left action, often denoted by p1 = g(p) of G over P, which is compatible with group composition as in:

`` p2 = g(h(p)) = (g compose h)(p) ``

Parameters

g (element of G) – Group element acting
p (element of P) – Element acted upon

Returns

Result of the left action

Return type

element of P

rightAction(self, p, g)¶

Computes the right action of a group element on a set element

Returns the right action, often denoted p1 = p^g, compatible with the group composition as in

p2 = (p^g)^h = p^(g compose h)

Parameters

p (element of P) – Element acted upon
g (element of G) – Group element acting

Returns

Result of the right action

Return type

element of P