Abstract groups and atlas

Abstract groups are defined using a presentation, without direct reference to a concrete realization. They are thus used when user-defined groups are recognized.

• AbstractGroup is a finite group whose elements are written as words composed of products of generators.

• Atlas is the infrastructure that recognizes groups. In particular, it is used to tip the user of the type of the group they just constructed. It is also used in exact decompositions, to retrieve the character table of a particular finite group.

• AtlasEntry contains a sketch of the structure of a finite group known by RepLAB. It is used to speed up the recognition process.

AbstractGroup

class replab.AbstractGroup(generatorNames, relators, varargin)

Bases: replab.gen.FiniteGroup

Finite group defined using a presentation: a set of generators and a set of relations among generators

The name of generator always start with a letter (a-z or A-Z), and then contain either letters (a-z or A-Z), digits (0-9) or underscores. Words in the generators are strings (charstring) containing generators separated by spaces, that sequence being understood as a product of generators.

In a word, generators can be taken to an integer power, using the syntax x^n where x is the name of a generator and n is an integer. In the word syntax, we optionally accept explicit multiplication operators (*), division operators (then x/y is understood as x y^-1), parenthesis, and commutators [x, y] = x^-1 y^-1 x y.

The abstract group is defined by a set of relations, that define equivalence relations between words. For example, a cyclic group is defined by the generator x and the relation x^n = 1 where n is the order of the group. Then we deduce the relations x^(n+m) = x^m and that x^-m = x^(n-m) for any integer m.

Informally, the abstract group is the “largest” group with the sets of generators subject to these relations. Note that RepLAB rewrites all relations as lhs = 1, and those left hand sides are named “relators”.

See https://en.wikipedia.org/wiki/Presentation_of_a_group for a detailed discussion.

Internally, RepLAB computes a realization of the abstract group as a permutation group, and delegates the computations to that permutation group.

Example

>>> [G, x] = replab.AbstractGroup.fromPresentation('< x | x^3 = 1 >');
>>> x
    'x'
>>> G.compose(x, x)
    'x^2'
>>> G.order
    3

Example

>>> G = replab.AbstractGroup({'x'}, {'x^3'});
>>> G.order
    3

Abstract groups can also be created by isomorphisms from an explicit group.

Example

>>> S3 = replab.S(3);
>>> G = S3.withGeneratorNames({'s', 't'});
>>> f = G.abstractIsomorphism;
>>> f.imageElement([2 3 1])
    's'

Class members list

Properties

• generators – Group generators

• identity – Monoid identity element

• inAtlas – Whether this group is already part of the atlas

• name – Group name

• nice – Nice object where computations are done

• niceIsomorphism – Order-preserving isomorphism from a group containing self to a group containing nice

• type – Type of the contained elements

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

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

Laws

• check – Checks the consistency of this object

• checkAndContinue – Checks the consistency of this object

• laws – Returns the laws that this object obeys

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

Unique ID

• eq – Equality test

• id – Returns the unique ID of this object (deprecated)

• ne – Non-equality test

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

• abstractIsomorphism – Returns an isomorphism to an abstract group

• automorphismGroup – Returns the automorphism group of this group

• commutingMorphismsMorphism – Constructs a morphism from morphisms with commuting images

• conjugatingAutomorphism – Returns the morphism that corresponds to left conjugation by an element

• findIsomorphism – Finds an isomorphism from this finite group to another finite group, if it exists

• findIsomorphisms – Finds all the isomorphisms from this finite group to another finite group

• findMorphisms – Finds all the morphisms from this finite group to another finite group

• innerAutomorphism – Returns the inner automorphism given using conjugation by the given element

• isIsomorphicTo – Returns whether this group is isomorphic to another group

• isMorphismByImages – Checks whether the given images describe a group morphism

• isomorphismByFunction – Constructs an isomorphism to a finite group using an image function

• isomorphismByFunctions – Constructs a group isomorphism using preimage/image functions

• isomorphismByImages – Constructs an isomorphism to a group using images of generators

• morphismByFunction – Constructs a group morphism using an image function

• morphismByImages – Constructs a morphism to a group using images of generators

• orderPreservingPermutationIsomorphism – Returns an isomorphism from this group to a permutation group, which preserves element order

• permutationIsomorphism – Returns an isomorphism from this group to a permutation group

• regularIsomorphism – Returns an isomorphism from this group to a permutation group whose degree matches the order of this group

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

Elements

• elementOrder – Returns the order of a group element

• smallGeneratingSet – Returns a small set of elements generating the group

Relations to other sets

• hasSameTypeAs – Returns if this finite set has the same type as the given finite set

Image under isomorphism

• imap – Returns the image of this finite set under an isomorphism

General

• AbstractGroup – Creates an abstract group from generator names and relators

• compatibleWithNiceIsomorphism – Returns whether the given finite isomorphism is compatible with this object isomorphism

• fromPresentation – Creates an abstract group from a presentation string

• mtimes –

• simplify – Attempts to simplify the given word

Representations

• commutingRepsRep – Constructs a representation from commuting representations

• directSumRep – Computes the direct sum of representations on this group

• permutationRep – Constructs a permutation representation of this group

• regularRep – Returns the left regular representation of this group

• repByImages – Constructs a finite dimensional representation of this group from preimages/images pairs

• signedPermutationRep – Returns a real signed permutation representation of 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

• abelianInvariants – Returns the group abelian invariants

• exponent – Returns the group exponent

• fastRecognize – Attempts to recognize this group in the standard atlas

• hasReconstruction – Returns whether the group has a “reconstruction”

• isCommutative – Returns whether this group is commutative

• isCyclic – Returns whether this group is a cyclic group

• isSimple – Returns whether this group is simple

• isTrivial – Tests whether this group is trivial

• knownOrder – Returns whether the order of this group has already been computed

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

• order – Returns the group order

• presentation – Returns a presentation of this group as a string

• recognize – Attempts to recognize this group in the standard atlas

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

Isomorphic groups

• abstractGroup – Returns an abstract group isomorphic to this finite gorup

• permutationGroup – Returns the permutation group isomorphic to this finite group

Subgroups

• center – Returns the center of this group

• centralizer – Returns the centralizer of the given object in this group

• derivedSeries – Returns the derived series of this group

• derivedSubgroup – Computes the derived subgroup of this group

• intersection – Computes the intersection of two groups

• randomProperSubgroup – Constructs a random proper subgroup of this group

• randomSubgroup – Constructs a random subgroup of this group

• subgroup – Constructs a subgroup of the current group from elements

• subgroupWithGenerators – Constructs a subgroup of the current group from generators

• trivialSubgroup – Returns the trivial subgroup of this group

Conjugacy classes and character table

• characterTable – Alias for complexCharacterTable

• complexCharacterTable – Returns the complex character table of this group

• conjugacyClass – Returns the conjugacy class corresponding to the given element

• conjugacyClasses – Returns the conjugacy classes of this group

• findLeftConjugations – Returns the set of all elements that left conjugates an element to another element

• knownComplexCharacterTable – Returns whether the complex character table of this group is already known or computed

• knownRealCharacterTable – Returns whether the real character table of this group is already known or computed

• realCharacterTable – Returns the real character table of this group

• setComplexCharacterTable – Sets the complex character table of this group

• setConjugacyClasses – Sets the conjugacy classes of this group

• setRealCharacterTable – Sets the real character table of this group

Construction of groups

• closure – Computes the group generated by the elements of this group and other elements/groups

• leftConjugateGroup – Returns the left conjugate of the current group by the given element

• normalClosure – Computes the normal closure of an object in the closure of this group and this object

• withGeneratorNames – Returns a modified copy of this finite group with renamed generators

Cosets

• doubleCoset – Returns a double coset

• doubleCosets – Returns the set of double cosets in this group by the given groups

• isNormalizedBy – Returns whether a given element/group normalizes this group

• leftCoset – Returns a left coset

• leftCosets – Returns the set of left cosets of the given subgroup in this group

• mldivide – Shorthand for rightCosets

• mrdivide – Shorthand for leftCosets

• normalCoset – Returns a normal coset

• normalCosets – Returns the set of normal cosets of the given subgroup in this group

• rightCoset – Returns a right coset

• rightCosets – Returns the set of right cosets of the given subgroup in this group

Generators-related methods

• factorizeFlat – Factorizes an element or a coset as a flat sequence in the generators

• factorizeWord – Factorizes an element or a coset as a word in the generators

• flatToWord – Prints a word described by letters into a string

• generator – Returns the i-th group generator

• generatorInverse – Returns the inverse of the i-th group generator

• generatorNames – Returns the names of the group generators

• imageFlat – Returns the image of a flat word in the group generators

• imageWord – Computes the image of a word in the group generators

• knownRelators – Returns whether the relators of this group are known

• nGenerators – Returns the number of group generators

• relatorsFlat – Returns the relators of a presentation of this finite group in the flat format

• relatorsWord – Returns the relators of a presentation of this finite group

• wordToFlat – Parses a word into generator letters

Relations to other groups

• isNormalSubgroupOf – Returns whether this group is a normal subgroup of another group

• isSubgroupOf – Returns whether this group is a subgroup of another group

Bugfix for Octave method selection

• contains – Tests whether this set contains the given element

• elements – Returns a cell array containing all the elements of this set

• elementsSequence – Returns a sequence corresponding to this set

• isequal – Tests finite sets for equality

• nElements – Returns the size of this set

• representative – Returns the minimal element of this set under the type ordering

• setProduct – Returns a description of this set as a product of sets

Inherited elements

abelianInvariants(self)

Documentation in replab.FiniteGroup.abelianInvariants()

abstractGroup(self)

Documentation in replab.FiniteGroup.abstractGroup()

abstractIsomorphism(self)

Documentation in replab.FiniteGroup.abstractIsomorphism()

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()

center(self)

Documentation in replab.FiniteGroup.center()

centralizer(self, arg)

Documentation in replab.FiniteGroup.centralizer()

characterTable(self)

Documentation in replab.FiniteGroup.characterTable()

check(self)

Documentation in replab.Obj.check()

checkAndContinue(self)

Documentation in replab.Obj.checkAndContinue()

closure(self, varargin)

Documentation in replab.FiniteGroup.closure()

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()

compatibleWithNiceIsomorphism(self, iso)

Documentation in replab.gen.FiniteSet.compatibleWithNiceIsomorphism()

complexCharacterTable(self)

Documentation in replab.FiniteGroup.complexCharacterTable()

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()

conjugacyClass(self, g, varargin)

Documentation in replab.FiniteGroup.conjugacyClass()

conjugacyClasses(self)

Documentation in replab.FiniteGroup.conjugacyClasses()

conjugatingAutomorphism(self, by)

Documentation in replab.FiniteGroup.conjugatingAutomorphism()

contains(self, el)

Documentation in replab.FiniteSet.contains()

derivedSeries(self)

Documentation in replab.FiniteGroup.derivedSeries()

derivedSubgroup(self)

Documentation in replab.FiniteGroup.derivedSubgroup()

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()

doubleCoset(self, element, rightSubgroup, varargin)

Documentation in replab.FiniteGroup.doubleCoset()

doubleCosets(self, leftSubgroup, rightSubgroup)

Documentation in replab.FiniteGroup.doubleCosets()

elementOrder(self, g)

Documentation in replab.FiniteGroup.elementOrder()

elements(self)

Documentation in replab.FiniteSet.elements()

elementsSequence(self)

Documentation in replab.FiniteSet.elementsSequence()

eq(self, rhs)

Documentation in replab.Obj.eq()

eqv(self, t, u)

Documentation in replab.Domain.eqv()

exponent(self)

Documentation in replab.FiniteGroup.exponent()

factorizeFlat(self, elementOrCoset)

Documentation in replab.FiniteGroup.factorizeFlat()

factorizeWord(self, elementOrCoset)

Documentation in replab.FiniteGroup.factorizeWord()

fastRecognize(self)

Documentation in replab.FiniteGroup.fastRecognize()

findIsomorphism(self, to)

Documentation in replab.FiniteGroup.findIsomorphism()

findIsomorphisms(self, to, varargin)

Documentation in replab.FiniteGroup.findIsomorphisms()

findLeftConjugations(self, s, t, varargin)

Documentation in replab.FiniteGroup.findLeftConjugations()

findMorphisms(self, to, varargin)

Documentation in replab.FiniteGroup.findMorphisms()

flatToWord(self, letters)

Documentation in replab.FiniteGroup.flatToWord()

generator(self, i)

Documentation in replab.FiniteGroup.generator()

generatorInverse(self, i)

Documentation in replab.FiniteGroup.generatorInverse()

generatorNames(self)

Documentation in replab.FiniteGroup.generatorNames()

generators

Documentation in replab.FiniteGroup.generators

hasReconstruction(self)

Documentation in replab.CompactGroup.hasReconstruction()

hasSameTypeAs(self, rhs)

Documentation in replab.FiniteSet.hasSameTypeAs()

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

imageFlat(self, letters)

Documentation in replab.FiniteGroup.imageFlat()

imageWord(self, word)

Documentation in replab.FiniteGroup.imageWord()

imap(self, f)

Documentation in replab.FiniteSet.imap()

inCache(self, name)

Documentation in replab.Obj.inCache()

innerAutomorphism(self, by)

Documentation in replab.Group.innerAutomorphism()

intersection(self, G)

Documentation in replab.FiniteGroup.intersection()

inverse(self, x)

Documentation in replab.Group.inverse()

isCommutative(self)

Documentation in replab.FiniteGroup.isCommutative()

isCyclic(self)

Documentation in replab.FiniteGroup.isCyclic()

isIdentity(self, x)

Documentation in replab.Monoid.isIdentity()

isIsomorphicTo(self, target)

Documentation in replab.FiniteGroup.isIsomorphicTo()

isMorphismByImages(self, target, varargin)

Documentation in replab.FiniteGroup.isMorphismByImages()

isNormalSubgroupOf(self, rhs)

Documentation in replab.FiniteGroup.isNormalSubgroupOf()

isNormalizedBy(self, element)

Documentation in replab.FiniteGroup.isNormalizedBy()

isSimple(self)

Documentation in replab.FiniteGroup.isSimple()

isSubgroupOf(self, rhs)

Documentation in replab.FiniteGroup.isSubgroupOf()

isTrivial(self)

Documentation in replab.FiniteGroup.isTrivial()

isequal(self, rhs)

Documentation in replab.FiniteSet.isequal()

isomorphismByFunction(self, target, imageElementFun, varargin)

Documentation in replab.FiniteGroup.isomorphismByFunction()

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

Documentation in replab.Group.isomorphismByFunctions()

isomorphismByImages(self, target, varargin)

Documentation in replab.FiniteGroup.isomorphismByImages()

knownComplexCharacterTable(self)

Documentation in replab.FiniteGroup.knownComplexCharacterTable()

knownOrder(self)

Documentation in replab.FiniteGroup.knownOrder()

knownRealCharacterTable(self)

Documentation in replab.FiniteGroup.knownRealCharacterTable()

knownRelators(self)

Documentation in replab.FiniteGroup.knownRelators()

laws(self)

Documentation in replab.Obj.laws()

leftConjugate(self, by, on)

Documentation in replab.Group.leftConjugate()

leftConjugateGroup(self, by)

Documentation in replab.FiniteGroup.leftConjugateGroup()

leftCoset(self, element, varargin)

Documentation in replab.FiniteGroup.leftCoset()

leftCosets(self, subgroup)

Documentation in replab.FiniteGroup.leftCosets()

longStr(self, maxRows, maxColumns)

Documentation in replab.Str.longStr()

maximalTorusDimension(self)

Documentation in replab.CompactGroup.maximalTorusDimension()

mldivide(self, supergroup)

Documentation in replab.FiniteGroup.mldivide()

morphismByFunction(self, target, imageElementFun, torusMap)

Documentation in replab.Group.morphismByFunction()

morphismByImages(self, target, varargin)

Documentation in replab.FiniteGroup.morphismByImages()

mrdivide(self, subgroup)

Documentation in replab.FiniteGroup.mrdivide()

mtimes()

No documentation

nElements(self)

Documentation in replab.FiniteSet.nElements()

nGenerators(self)

Documentation in replab.FiniteGroup.nGenerators()

ne(self, rhs)

Documentation in replab.Obj.ne()

nice

Documentation in replab.gen.FiniteSet.nice

niceIsomorphism

Documentation in replab.gen.FiniteSet.niceIsomorphism

normalClosure(self, obj)

Documentation in replab.FiniteGroup.normalClosure()

normalCoset(self, element, varargin)

Documentation in replab.FiniteGroup.normalCoset()

normalCosets(self, subgroup)

Documentation in replab.FiniteGroup.normalCosets()

order(self)

Documentation in replab.FiniteGroup.order()

orderPreservingPermutationIsomorphism(self)

Documentation in replab.FiniteGroup.orderPreservingPermutationIsomorphism()

permutationGroup(self)

Documentation in replab.FiniteGroup.permutationGroup()

permutationIsomorphism(self)

Documentation in replab.FiniteGroup.permutationIsomorphism()

permutationRep(self, dimension, varargin)

Documentation in replab.FiniteGroup.permutationRep()

presentation(self)

Documentation in replab.FiniteGroup.presentation()

randomProperSubgroup(self, nSteps)

Documentation in replab.FiniteGroup.randomProperSubgroup()

randomSubgroup(self)

Documentation in replab.FiniteGroup.randomSubgroup()

realCharacterTable(self)

Documentation in replab.FiniteGroup.realCharacterTable()

recognize(self)

Documentation in replab.FiniteGroup.recognize()

reconstruction(self)

Documentation in replab.CompactGroup.reconstruction()

regularIsomorphism(self)

Documentation in replab.FiniteGroup.regularIsomorphism()

regularRep(self)

Documentation in replab.FiniteGroup.regularRep()

relatorsFlat(self)

Documentation in replab.FiniteGroup.relatorsFlat()

relatorsWord(self)

Documentation in replab.FiniteGroup.relatorsWord()

repByImages(self, field, dimension, varargin)

Documentation in replab.FiniteGroup.repByImages()

representative(self)

Documentation in replab.FiniteSet.representative()

rightCoset(self, element, varargin)

Documentation in replab.FiniteGroup.rightCoset()

rightCosets(self, subgroup)

Documentation in replab.FiniteGroup.rightCosets()

sample(self)

Documentation in replab.Domain.sample()

semidirectProduct(self, N, phi)

Documentation in replab.CompactGroup.semidirectProduct()

setComplexCharacterTable(self, table)

Documentation in replab.FiniteGroup.setComplexCharacterTable()

setConjugacyClasses(self, classes)

Documentation in replab.FiniteGroup.setConjugacyClasses()

setProduct(self)

Documentation in replab.FiniteSet.setProduct()

setRealCharacterTable(self, table)

Documentation in replab.FiniteGroup.setRealCharacterTable()

shortStr(self, maxColumns)

Documentation in replab.Str.shortStr()

signedPermutationRep(self, dimension, varargin)

Documentation in replab.FiniteGroup.signedPermutationRep()

smallGeneratingSet(self)

Documentation in replab.FiniteGroup.smallGeneratingSet()

subgroup(self, elements, varargin)

Documentation in replab.FiniteGroup.subgroup()

subgroupWithGenerators(self, generators, varargin)

Documentation in replab.FiniteGroup.subgroupWithGenerators()

tensorRep(self, field, reps)

Documentation in replab.CompactGroup.tensorRep()

trivialRep(self, field, dimension)

Documentation in replab.CompactGroup.trivialRep()

trivialSubgroup(self)

Documentation in replab.FiniteGroup.trivialSubgroup()

type

Documentation in replab.FiniteSet.type

withGeneratorNames(self, newNames)

Documentation in replab.FiniteGroup.withGeneratorNames()

wordToFlat(self, word)

Documentation in replab.FiniteGroup.wordToFlat()

Own members

inAtlas

Whether this group is already part of the atlas

Type

logical

name

Group name

Type

charstring

AbstractGroup(generatorNames, relators, varargin)

Creates an abstract group from generator names and relators

Parameters

• generatorNames (cell(1,*) of charstring) – Generator names

• relators (cell(1,*) of charstring, optional) – Relators

• name – Group name, optional

• order – Group order

• permutationGenerators – Realization of the generators using permutations

• inAtlas – Whether this group is part of the atlas

Kwtype name

charstring, optional

Kwtype order

integer, optional

Kwtype permutationGenerators

cell(1,*) of permutation

Kwtype inAtlas

logical, optional

static fromPresentation(str, varargin)

Creates an abstract group from a presentation string

Returns the finite group generators as additional output arguments.

Additional arguments are passed to the AbstractGroup constructor.

Example

>>> [G, x] = replab.AbstractGroup.fromPresentation('< x | x^3 = 1 >');
>>> [G, x, y] = replab.AbstractGroup.fromPresentation('< x, y | x^3 = y^2 = x y x y^-1 = 1 >');

Parameters

str (charstring) – Single-line description string

Returns

The parsed abstract group

Return type

AbstractGroup

simplify(self, word)

Attempts to simplify the given word

Parameters

word (charstring) – Word to simplify

Returns

Simplified word

Return type

charstring

Atlas

class replab.Atlas

An atlas of finite groups

Class members list

General

• computeAndAdd – Computes the irreps of the given group and write the corresponding JSON file to the atlas

• read –

• recognize – Attempts to identify the given group

Inherited elements

read()

No documentation

Own members

static computeAndAdd(G)

Computes the irreps of the given group and write the corresponding JSON file to the atlas

If the group is already recognized by RepLAB, an error is thrown.

Note that the FiniteGroup.recognize result is cached, so to be recognized, the given group G must be constructed again in user code for the new atlas entry to be found.

Parameters

G (FiniteGroup) – Finite group not present in the atlas

static recognize(group)

Attempts to identify the given group

Returns

A result in case of positive identification; or [] if unrecognized.

Return type

replab.FiniteIsomorphism or []

AtlasEntry

class replab.AtlasEntry(filename, md5, order, derivedSeriesAbelianInvariants, varargin)

Bases: replab.Obj

Describes an Atlas entry available in the replab/atlas directory

Note: the JSON file must contain only ASCII characters; check if our parser replab.util.parseJSON supports escape sequences.

Class members list

Properties

• derivedSeriesAbelianInvariants – Abelian invariants of the group derived series

• filename – Filename, excluding path, including .json extension

• md5 – MD5 hash in hexadecimal notation, without spaces and letters uppercase

• order – Group order

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

• AtlasEntry –

• canMatch – Returns whether this atlas entry can match the given group

• data – Returns the parsed data

• fromFile – Constructs an AtlasEntry from a JSON file

• group – Returns the group corresponding to this atlas entry

• groupFromJSONData – Computes the group stored in this entry from the known JSON data

• json – Returns the JSON data as text

• match – Attemps to match the given group to this group

Inherited elements

AtlasEntry()

No documentation

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

derivedSeriesAbelianInvariants

Abelian invariants of the group derived series

Type

cell(1,*) of integer(1,*)

filename

Filename, excluding path, including .json extension

Type

charstring

md5

MD5 hash in hexadecimal notation, without spaces and letters uppercase

Type

charstring

order

Group order

Type

vpi

canMatch(self, group)

Returns whether this atlas entry can match the given group

data(self)

Returns the parsed data

Returns

Parsed JSON data

Return type

struct or cell array

static fromFile(filename)

Constructs an AtlasEntry from a JSON file

This method parses the JSON enough to read the group order and the abelian invariants. The group itself is constructed lazily when requested.

Parameters

filename (charstring) – Filename, excluding path but including the .json extension

Returns

The parsed atlas entry

Return type

AtlasEntry

group(self)

Returns the group corresponding to this atlas entry

Returns

Group

Return type

AbstractGroup

static groupFromJSONData(data)

Computes the group stored in this entry from the known JSON data

json(self)

Returns the JSON data as text

Returns

Data

Return type

charstring

match(self, group)

Attemps to match the given group to this group

Returns, if it exists, the isomorphism from the stored atlas group to the given group

Returns

The isomorphism if it exists or []

Return type

FiniteIsomorphism or []