Named discrete groups

RepLAB provides constructions of standard discrete groups.

General (signed) permutation groups are the two groups below.

• SymmetricGroup: Describes the symmetric group on a given domain.

• S: Shorthand for the construction of a symmetric group.

• SignedSymmetricGroup: Describes the signed symmetric group.

Then we distinguish particular groups.

• A: Describes the alternating group on a given domain.

• C: Describes the cyclic group on a given domain.

• D: Describes the dihedral group on a given domain.

• KleinFourGroup: Describes the Klein four-group

• QuaternionGroup: Describes the quaternion group of order 8.

SymmetricGroup

class replab.SymmetricGroup(domainSize)

Bases: replab.PermutationGroup

The symmetric group S(n), i.e. all permutations over n = “domainSize” elements

Example

>>> S5 = replab.S(5);
>>> S5.order
   ans =
   120

Class members list

Properties

• generators – Group generators

• identity – Monoid identity element

• 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)

• isequal – Tests finite sets for equality

• 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

• contains – Tests whether this set contains the given element

• elementOrder – Returns the order of a group element

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

• elementsSequence – Returns a sequence corresponding to this set

• 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

• 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

• make – Constructs the symmetric over a given domain size

• mtimes –

Representations

• commutingRepsRep – Constructs a representation from commuting representations

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

• indexRelabelingRep – Representation that permutes the indices of a tensor

• irrep – Returns the irreducible representation of this symmetric group corresponding the given Young Diagram

• irrepDimension – Returns the dimension of the irreducible representation of a symmetric group corresponding a Young Diagram

• naturalRep – Returns the natural permutation representation of this permutation 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

• signRep – Returns the sign representation of this permutation group

• signedPermutationRep – Returns a real signed permutation representation of this group

• standardRep – Returns the standard representation of this permutation 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

Group internal description

• chain – Returns the stabilizer chain corresponding to this permutation group.

• factorization – Returns an object able to compute factorizations in the group generators

• lexChain – Returns the reduced stabilizer chain corresponding to this permutation group in lexicographic order

• partialChain – Returns the stabilizer chain corresponding to this permutation group if it can be computed quickly

Methods specific to permutation groups

• domainSize –

• generatorsAsMatrix – Returns the generators of this group concatenated in a matrix

• matrixFindPermutationsTo – Finds the permutations that send a matrix to another matrix

• matrixStabilizer – Returns the permutation subgroup that leaves a given matrix invariant by joint permutation of its rows and columns

• orbits – Returns the partition of the domain into orbits under this group

• orderedPartitionStabilizer – Computes the subgroup that leaves the given ordered partition invariant

• pointwiseStabilizer – Returns the subgroup that stabilizes the given set pointwise

• setwiseStabilizer – Returns the subgroup that stabilizes the given set as a set

• stabilizer – Returns the subgroup that stabilizes a given point

• unorderedPartitionStabilizer – Computes the subgroup that leaves the given unordered partition invariant

• vectorFindLexMinimal – Finds the lexicographic minimal vector under permutation of its coefficients by this group

• vectorFindPermutationsTo – Finds the permutations that send a vector to another vector

• vectorStabilizer – Returns the permutation subgroup that leaves a given vector invariant

• wreathProduct – Returns the wreath product of a compact group by this permutation group

Actions

• indexRelabelingMorphism – Returns the morphism the permutation action of this group on tensor coefficients

• indexRelabelingPermutation – Returns the permutation that acts by permuting tensor coefficients

• matrixAction – Returns the simultaneous action of permutations on both rows and columns of square matrices

• naturalAction – Returns the natural action of elements of this group on its domain

• vectorAction – Returns the action of permutations on column vectors

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()

chain(self)

Documentation in replab.PermutationGroup.chain()

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()

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()

domainSize()

No documentation

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()

factorization(self)

Documentation in replab.PermutationGroup.factorization()

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

generatorsAsMatrix(self)

Documentation in replab.PermutationGroup.generatorsAsMatrix()

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()

indexRelabelingMorphism(self, indexRange)

Documentation in replab.PermutationGroup.indexRelabelingMorphism()

indexRelabelingPermutation(self, g, indexRange)

Documentation in replab.PermutationGroup.indexRelabelingPermutation()

indexRelabelingRep(self, indexRange)

Documentation in replab.PermutationGroup.indexRelabelingRep()

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()

lexChain(self)

Documentation in replab.PermutationGroup.lexChain()

longStr(self, maxRows, maxColumns)

Documentation in replab.Str.longStr()

matrixAction(self)

Documentation in replab.PermutationGroup.matrixAction()

matrixFindPermutationsTo(self, S, T, SStabilizer, TStabilizer)

Documentation in replab.PermutationGroup.matrixFindPermutationsTo()

matrixStabilizer(self, matrix)

Documentation in replab.PermutationGroup.matrixStabilizer()

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()

naturalAction(self)

Documentation in replab.PermutationGroup.naturalAction()

naturalRep(self)

Documentation in replab.PermutationGroup.naturalRep()

ne(self, rhs)

Documentation in replab.Obj.ne()

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()

orbits(self)

Documentation in replab.PermutationGroup.orbits()

order(self)

Documentation in replab.FiniteGroup.order()

orderPreservingPermutationIsomorphism(self)

Documentation in replab.FiniteGroup.orderPreservingPermutationIsomorphism()

orderedPartitionStabilizer(self, partition)

Documentation in replab.PermutationGroup.orderedPartitionStabilizer()

partialChain(self)

Documentation in replab.PermutationGroup.partialChain()

permutationGroup(self)

Documentation in replab.FiniteGroup.permutationGroup()

permutationIsomorphism(self)

Documentation in replab.FiniteGroup.permutationIsomorphism()

permutationRep(self, dimension, varargin)

Documentation in replab.FiniteGroup.permutationRep()

pointwiseStabilizer(self, set)

Documentation in replab.PermutationGroup.pointwiseStabilizer()

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()

setwiseStabilizer(self, set)

Documentation in replab.PermutationGroup.setwiseStabilizer()

shortStr(self, maxColumns)

Documentation in replab.Str.shortStr()

signRep(self)

Documentation in replab.PermutationGroup.signRep()

signedPermutationRep(self, dimension, varargin)

Documentation in replab.FiniteGroup.signedPermutationRep()

smallGeneratingSet(self)

Documentation in replab.FiniteGroup.smallGeneratingSet()

stabilizer(self, p)

Documentation in replab.PermutationGroup.stabilizer()

standardRep(self)

Documentation in replab.PermutationGroup.standardRep()

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

unorderedPartitionStabilizer(self, partition)

Documentation in replab.PermutationGroup.unorderedPartitionStabilizer()

vectorAction(self)

Documentation in replab.PermutationGroup.vectorAction()

vectorFindLexMinimal(self, s, sStabilizer)

Documentation in replab.PermutationGroup.vectorFindLexMinimal()

vectorFindPermutationsTo(self, s, t, sStabilizer, tStabilizer)

Documentation in replab.PermutationGroup.vectorFindPermutationsTo()

vectorStabilizer(self, vector)

Documentation in replab.PermutationGroup.vectorStabilizer()

withGeneratorNames(self, newNames)

Documentation in replab.FiniteGroup.withGeneratorNames()

wordToFlat(self, word)

Documentation in replab.FiniteGroup.wordToFlat()

wreathProduct(self, A)

Documentation in replab.PermutationGroup.wreathProduct()

Own members

irrep(self, partition, form)

Returns the irreducible representation of this symmetric group corresponding the given Young Diagram

Example

>>> S5 = replab.S(5); % doctest: +cyclotomic
>>> rep = S5.irrep([3 2], 'specht');
>>> rep.dimension
      5
>>> rep = S5.irrep([3 2], 'seminormal');
>>> rep.dimension
      5
>>> rep = S5.irrep([3 2], 'orthogonal');
>>> rep.dimension
      5

Parameters

• partition (integer(1,*)) – The partition corresponding the Young Diagram, with elements listed in decreasing order (e.g [3 3 1] represents the partition of 7 elements: 7 = 3+3+1

• form ('specht', 'seminormal', or 'orthogonal') – The form the irrep takes. Default is ‘specht’.

Note: ‘specht’ is slower to construct than ‘seminormal’ or ‘orthogonal’ but, unlike them, has integer entries.

Returns

The corresponding irreducible representation

Return type

replab.Rep

static irrepDimension(partition)

Returns the dimension of the irreducible representation of a symmetric group corresponding a Young Diagram

Example

>>> S5 = replab.S(5);
>>> rep = S5.irrep([3 2]);
>>> rep.dimension
      5
>>> S5.irrepDimension([3 2])
      5

Parameters

partition (integer(1,*)) – The partition corresponding the Young Diagram, with elements listed in decreasing order (e.g [5 3 1] represents the partition of 9 elements 9 = 5+3+1

Returns

The dimension of the corresponding irreducible representation.

Return type

integer

static make(n)

Constructs the symmetric over a given domain size

This static method keeps the constructed copies of S(n) in cache.

Parameters

domainSize (integer) – Domain size, must be > 0

Returns

The constructed or cached symmetric group

Return type

SymmetricGroup

S

replab.S(domainSize)

Returns the symmetric group acting on a certain domain size

Alias for replab.PermutationGroup.symmetric

Parameters

domainSize (integer) – Domain size, must be >= 0

Returns

Symmetric group

Return type

replab.PermutationGroup

SignedSymmetricGroup

replab.SignedSymmetricGroup

A

replab.A(n)

Constructs the alternating group

Parameters

n (integer) – Group degree

Returns

The alternating group of degree n

Return type

replab.PermutationGroup

C

replab.C(n)

Describes cyclic permutations over n = “domainSize” elements

Parameters

n (integer) – Cyclic group order and domain size

Returns

The cyclic group of given order/domain size

Return type

replab.PermutationGroup

D

replab.D(n)

Constructs the dihedral group of order 2*n

This corresponds to the group of symmetries of the polygon with n vertices

Parameters

n (integer) – Half the dihedral group order

Returns

The dihedral group permuting the vertices of the n-gon

Return type

replab.PermutationGroup

KleinFourGroup

replab.KleinFourGroup()

Constructs the Klein Four-Group

This corresponds to the symmetry group of a non-square rectangle, and corresponds to the direct product S2 x S2.

Returns

The Klein four-group as a permutation gorup

Return type

replab.PermutationGroup

QuaternionGroup

replab.QuaternionGroup()

Returns the quaternion group as a signed permutation group

The quaternion group acts on the {1, j, k, l} basis elements.

Returns

The quaternion group

Return type

SignedPermutationGroup