Characters¶

Character table based methods are used in RepLAB to provide exact results when requested, at the price of a major slowdown.

Character represents a character of a finite group (not necessarily irreducible).
CharacterTable is the base class
ComplexCharacterTable is the usual textbook character table of a finite group.
RealCharacterTable is the character table of the real irreducible representations of a finite group.

See also ConjugacyClasses and ConjugacyClass.

Character¶

class replab.Character(conjugacyClasses, values)¶

Bases: replab.Obj

Describes a character

Example

>>> D = replab.PermutationGroup.dihedral(6); % doctest: +cyclotomic
>>> rep = D.repByImages('R', 1, 'preimages', {[6 5 4 3 2 1], [2 3 4 5 6 1]}, 'images', {1 -1});
>>> c = replab.Character.fromRep(rep);
>>> g = [2 3 4 5 6 1];
>>> c.value(g)
    -1
>>> c1 = c + c;
>>> c1.value(g)
    -2
>>> c2 = c * c;
>>> c2.value(g)
    1

Class members list

Properties

conjugacyClasses – List of conjugacy classes for which the order of values is defined
group – Group on which this class function is defined
values – Values of the character over the conjugacy classes

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

Character –
dot – Computes the inner product of this character with another character
dotRep – Computes the approximate inner product of this character with the character of an approximate representation
forClasses – Returns the character with conjugacy classes reordered
frobeniusSchurIndicator – Returns the Frobenius-Schur indicator corresponding to this character
fromApproximateRep – Computes an exact character from an approximate representation using Dixon’s trick
fromExactRep – Computes an exact character from an approximate representation using Dixon’s trick
fromRep – Computes an exact character from a representation (approximate or not)
kernel – Returns the kernel of this character
mtimes –
plus –
value – Returns the value of the class function over a group element or conjugacy class

Inherited elements

Character()¶: 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()

mtimes()¶: No documentation

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

plus()¶: No documentation

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

Own members

conjugacyClasses¶

List of conjugacy classes for which the order of values is defined

Type: ConjugacyClasses

group¶

Group on which this class function is defined

Type: FiniteGroup

values¶

Values of the character over the conjugacy classes

Type: cyclotomic (1,*)

dot(self, rhs)¶

Computes the inner product of this character with another character

Parameters: rhs (Character) – Character
Returns: Value of the dot product
Return type: cyclotomic

dotRep(self, rhs)¶

Computes the approximate inner product of this character with the character of an approximate representation

Parameters: rhs (Rep) – Approximate representation
Returns: Value of the dot product
Return type: double

forClasses(self, newConjugacyClasses)¶: Returns the character with conjugacy classes reordered

frobeniusSchurIndicator(self)¶: Returns the Frobenius-Schur indicator corresponding to this character

static fromApproximateRep(rep)¶

Computes an exact character from an approximate representation using Dixon’s trick

See J. D. Dixon, “Computing irreducible representations of groups”, Math. Comp., vol. 24, no. 111, pp. 707–712, 1970 https://www.ams.org/mcom/1970-24-111/S0025-5718-1970-0280611-6/

Note that the function has complexity in \(O(N e^2)\) where \(N\) is the number of conjugacy classes and \(e\) is the group exponent.

Parameters: rep (Rep) – Approximate representation
Returns: Exact character
Return type: Character

static fromExactRep(rep)¶

Computes an exact character from an approximate representation using Dixon’s trick

See J. D. Dixon, “Computing irreducible representations of groups”, Math. Comp., vol. 24, no. 111, pp. 707–712, 1970 https://www.ams.org/mcom/1970-24-111/S0025-5718-1970-0280611-6/

Note that the function has complexity in \(O(N e^2)\) where \(N\) is the number of conjugacy classes and \(e\) is the group exponent.

Parameters: rep (Rep) – Approximate representation
Returns: Exact character
Return type: Character

static fromRep(rep)¶

Computes an exact character from a representation (approximate or not)

Parameters: rep (Rep) – Representation, may be approximate or exact
Returns: Character
Return type: Character

kernel(self)¶

Returns the kernel of this character

The kernel of this character \(\chi\) is defined as all \(g \in G\) such that \(\chi(g) = \chi(1)\) where \(1\) is the identity.

Returns: A subgroup of group
Return type: FiniteGroup

value(self, arg)¶

Returns the value of the class function over a group element or conjugacy class

Parameters: element (element of group or ConjugacyClass) – Element to compute the value of
Returns: Class function value
Return type: cyclotomic

CharacterTable¶

class replab.CharacterTable(group, field, classes, values, varargin)¶

Bases: replab.Obj

Describes the real or complex character table of a finite group

This contains several pieces of information. Some of these pieces, such as the character values, are essentially unique for a given group (up to permutation of characters and conjugacy classes). Other pieces of information, such as the names of conjugacy classes/irreducible representations may depend on the context (the mathematical or physical are, chemistry, etc).

The character table can optionally store the explicit images of the irreducible representations. Of course, these irreps are defined up to the choice of basis. For example, if the character table is of type ComplexCharacterTable, it can contain irreducible representations of real-type, those images are not necessarily written using real coefficients, even if they could.

The arguments passed to the constructor for irreducible representations (irreps) or generalized Kronecker coefficients (kronecker) can include function handles; it that case, the corresponding objects are only constructed when they are requested and the computed value is cached.

Class members list

Properties

classNames – Names of conjugacy classes
classes – Conjugacy classes of group
field – Whether the character table is real or complex
group – Group represented by character table
irrepNames – Names of the irreducible representations/characters
values – Character values

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

CharacterTable – Constructs a character table
defaultClassNames – Constructs default names for conjugacy classes
defaultIrrepNames – Constructs default names for irreducible representations
kronecker – Returns the Kronecker coefficients corresponding to this character table
overC – Returns whether the character table is defined over the complex field
overR – Returns whether the character table is defined over the real field

Characters

character – Returns an irreducible character in this character table
characters – Returns the characters in this table
linearCharacterIndices – Returns the indices of the linear characters
multiplicities – Calculate the multiplicities of the irreducible characters in this table in a given representation or character
nCharacters – Returns the number of characters in this character table
trivialCharacterIndex – Returns the row index of the trivial character

Representations

decomposeRep – Computes the decomposition of a representation into irreducibles using the explicit irreps contained in this table
hasIrrep – Returns whether the irreducible representation that corresponds to the character of given index is available
hasIrreps – Returns whether explicit constructions are available for all irreducible representations
irrep – Returns the irreducible representation that corresponds to the character of given index
irreps – Returns a cell vector of the irreducible representations corresponding to the characters in this table
nIrreps – Returns the number of

Conjugacy classes

identityConjugacyClassIndex – Returns the index of the conjugacy class containing the identity
nClasses – Returns the number of conjugacy classes in this character table

Transformations of the character table

imap – Maps the character table under an isomorphism

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

classNames¶

Names of conjugacy classes

Type: cell(1,nClasses) of charstring

classes¶

Conjugacy classes of group

Type: ConjugacyClasses

field¶

Whether the character table is real or complex

Type: {‘R’, ‘C’}

group¶

Group represented by character table

Type: replab.FiniteGroup

irrepNames¶

Names of the irreducible representations/characters

Type: cell(1,nIrreps) of charstring

irreps_¶

Explicit representations (can contain empty values)

Type: cell(1, nClasses) of [] or Rep or function_handle

kronecker_¶

Kronecker coefficients

Type: integer(*,*,*) or function_handle

values¶

Character values

Type: cyclotomic (nIrreps,nClasses)

CharacterTable(group, field, classes, values, varargin)¶

Constructs a character table

Parameters

group (FiniteGroup) – Group represented by character table
field ({'R', 'C'}) – Field over which the characters are defined
classes (ConjugacyClasses) – Conjugacy classes of group
values (cyclotomic (nClasses, nClasses)) – Character values
irreps – Explicit matrix representations (can contain empty values)
classNames – Names of conjugacy classes
irrepNames – Names of irreducible representations
kronecker – Kronecker coefficients

Kwtype irreps

cell(1,*) of [] or replab.Rep or function_handle, optional

Kwtype classNames

cell(1,*) of charstring, optional

Kwtype irrepNames

cell(1,*) of charstring, optional

Kwtype kronecker

integer(*,*,*) or function_handle, optional

character(self, ind)¶

Returns an irreducible character in this character table

Parameters: ind (integer) – Index of the character
Returns: Indexed character
Return type: Character

characters(self)¶

Returns the characters in this table

Returns: List of characters contained in this character table
Return type: cell(1,*) of Character

decomposeRep(self, rep)¶

Computes the decomposition of a representation into irreducibles using the explicit irreps contained in this table

Parameters: rep (Rep) – Representation to decompose, with rep.field == self.field
Returns: Irreducible decomposition
Return type: Irreducible

static defaultClassNames(elementOrders)¶

Constructs default names for conjugacy classes

Parameters: elementOrders (integer(1,*)) – Orders of the conjugacy classes elements
Returns: Conjugacy class names
Return type: cell(1,*) of charstring

static defaultIrrepNames(n)¶

Constructs default names for irreducible representations

Parameters: n (integer) – Number of irreducible representations
Returns: Irreducible representation names
Return type: cell(1,*) of charstring

hasIrrep(self, ind)¶

Returns whether the irreducible representation that corresponds to the character of given index is available

Parameters: ind (integer) – Index of the character
Returns: Whether the irreducible representation is available
Return type: logical

hasIrreps(self)¶: Returns whether explicit constructions are available for all irreducible representations

identityConjugacyClassIndex(self)¶

Returns the index of the conjugacy class containing the identity

Returns: Index of (the column containing) the identity conjugacy class
Return type: integer

imap(self, f)¶

Maps the character table under an isomorphism

Example

>>> D6a = replab.PermutationGroup.of([3 2 1], [2 3 1]); % doctest: +cyclotomic
>>> D6b = replab.PermutationGroup.of([1 4 3 2], [1 3 4 2]);
>>> f = D6a.isomorphismByImages(D6b, 'preimages', D6a.generators, 'images', D6b.generators);
>>> Ca = D6a.characterTable;
>>> Cb = Ca.imap(f);
>>> Cb.laws.checkSilent;

Parameters: f (FiniteIsomorphism) – Isomorphism with self.group.isSubgroupOf(f.source)
Returns: The character table of the subgroup in the image of the isomorphism
Return type: CharacterTable

irrep(self, ind)¶

Returns the irreducible representation that corresponds to the character of given index

Parameters: ind (integer) – Index of the character
Returns: An irreducible representation with coefficients in the cyclotomic field
Return type: RepByImages

irreps(self)¶

Returns a cell vector of the irreducible representations corresponding to the characters in this table

Returns: Irreducible representations
Return type: cell(1,nIrreps) of RepByImages

kronecker(self)¶

Returns the Kronecker coefficients corresponding to this character table

This returns an integer matrix \(K\) such that \(K(i,j,k)\) is the multiplicity of the \(i\)-th irrep in the product of the \(j\)-th and \(k\)-th irrep.

Returns: Kronecker coefficients
Return type: integer(*,*,*)

linearCharacterIndices(self)¶

Returns the indices of the linear characters

The linear characters correspond to representation of dimension 1.

Returns: Indices of the linear characters
Return type: integer(1,*)

multiplicities(self, arg)¶

Calculate the multiplicities of the irreducible characters in this table in a given representation or character

The ordering of coefficients corresponds to the order of irreducible representations in this character table.

Note that this method is optimized when the representation is a tensor product.

Example

>>> G = replab.PermutationGroup.dihedral(3); % doctest: +cyclotomic
>>> ct = G.characterTable;
>>> rep2 = ct.irreps{2};
>>> rep3 = ct.irreps{3};
>>> rep = kron(rep2, rep3);
>>> isequal(ct.multiplicities(rep), [0 0 1])
    1

Example

>>> G = replab.S(5); % doctest: +cyclotomic
>>> ct = G.characterTable;
>>> S5 = ct.group;
>>> isequal(ct.multiplicities(S5.naturalRep), [1 0 1 0 0 0 0])
    1

Parameters: arg (Character or Rep) – Character or representation of group
Returns: Multiplicities of irreducible representations
Return type: (integer(1,*))

nCharacters(self)¶

Returns the number of characters in this character table

Returns: Number of characters
Return type: integer

nClasses(self)¶

Returns the number of conjugacy classes in this character table

Returns: Number of conjugacy classes
Return type: integer

nIrreps(self)¶: Returns the number of

overC(self)¶

Returns whether the character table is defined over the complex field

Returns: True if the character table is defined over the complex field
Return type: logical

overR(self)¶

Returns whether the character table is defined over the real field

Returns: True if the character table is defined over the real field
Return type: logical

trivialCharacterIndex(self)¶: Returns the row index of the trivial character

ComplexCharacterTable¶

class replab.ComplexCharacterTable(group, classes, values, varargin)¶

Bases: replab.CharacterTable

Describes the standard character table of a group

Example

>>> G = replab.PermutationGroup.dihedral(3); % doctest: +cyclotomic
>>> G.characterTable
    Class  1a   3a   2a
     Size   1    2    3

X.1 1 1 1 X.2 1 1 -1 X.3 2 -1 0

The character values are stored as elements of the cyclotomic field, using the cyclotomic class which requires external libraries and a Java Virtual Machine available.

Instances of ComplexCharacterTable are immutable.

See CharacterTable for additional details.

Class members list

Properties

classNames – Names of conjugacy classes
classes – Conjugacy classes of group
field – Whether the character table is real or complex
group – Group represented by character table
irrepNames – Names of the irreducible representations/characters
values – Character values

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

ComplexCharacterTable – Constructs a character table
conjugateCharacterIndices – For each character, returns the index of the character conjugate to it
fromIrreps – Creates a complex character table from a complete list of irreducible exact representations
fromRealCharacterTable – Creates a complex character table from a real character table
kronecker – Returns the Kronecker coefficients corresponding to this character table
overC – Returns whether the character table is defined over the complex field
overR – Returns whether the character table is defined over the real field

Characters

character – Returns an irreducible character in this character table
characters – Returns the characters in this table
linearCharacterIndices – Returns the indices of the linear characters
multiplicities – Calculate the multiplicities of the irreducible characters in this table in a given representation or character
nCharacters – Returns the number of characters in this character table
trivialCharacterIndex – Returns the row index of the trivial character

Representations

decomposeRep – Computes the decomposition of a representation into irreducibles using the explicit irreps contained in this table
hasIrrep – Returns whether the irreducible representation that corresponds to the character of given index is available
hasIrreps – Returns whether explicit constructions are available for all irreducible representations
irrep – Returns the irreducible representation that corresponds to the character of given index
irreps – Returns a cell vector of the irreducible representations corresponding to the characters in this table
nIrreps – Returns the number of

Conjugacy classes

identityConjugacyClassIndex – Returns the index of the conjugacy class containing the identity
nClasses – Returns the number of conjugacy classes in this character table

Transformations of the character table

imap – Maps the character table under an isomorphism

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()

character(self, ind)¶: Documentation in replab.CharacterTable.character()

characters(self)¶: Documentation in replab.CharacterTable.characters()

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

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

classNames¶: Documentation in replab.CharacterTable.classNames

classes¶: Documentation in replab.CharacterTable.classes

decomposeRep(self, rep)¶: Documentation in replab.CharacterTable.decomposeRep()

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

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

field¶: Documentation in replab.CharacterTable.field

group¶: Documentation in replab.CharacterTable.group

hasIrrep(self, ind)¶: Documentation in replab.CharacterTable.hasIrrep()

hasIrreps(self)¶: Documentation in replab.CharacterTable.hasIrreps()

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

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

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

identityConjugacyClassIndex(self)¶: Documentation in replab.CharacterTable.identityConjugacyClassIndex()

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

irrep(self, ind)¶: Documentation in replab.CharacterTable.irrep()

irrepNames¶: Documentation in replab.CharacterTable.irrepNames

irreps(self)¶: Documentation in replab.CharacterTable.irreps()

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

kronecker(self)¶: Documentation in replab.CharacterTable.kronecker()

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

linearCharacterIndices(self)¶: Documentation in replab.CharacterTable.linearCharacterIndices()

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

multiplicities(self, arg)¶: Documentation in replab.CharacterTable.multiplicities()

nCharacters(self)¶: Documentation in replab.CharacterTable.nCharacters()

nClasses(self)¶: Documentation in replab.CharacterTable.nClasses()

nIrreps(self)¶: Documentation in replab.CharacterTable.nIrreps()

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

overC(self)¶: Documentation in replab.CharacterTable.overC()

overR(self)¶: Documentation in replab.CharacterTable.overR()

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

trivialCharacterIndex(self)¶: Documentation in replab.CharacterTable.trivialCharacterIndex()

values¶: Documentation in replab.CharacterTable.values

Own members

ComplexCharacterTable(group, classes, values, varargin)¶

Constructs a character table

Parameters

group (FiniteGroup) – Group represented by character table
classes (ConjugacyClasses) – Conjugacy classes of group
values (cyclotomic (nClasses, nClasses)) – Character values
irreps – Explicit matrix representations (can contain empty values)
classNames – Names of conjugacy classes
irrepNames – Names of irreducible representations
kronecker – Kronecker coefficients

Kwtype irreps

cell(1,*) of [] or replab.RepByImages, optional

Kwtype classNames

cell(1,*) of charstring, optional

Kwtype irrepNames

cell(1,*) of charstring, optional

Kwtype kronecker

integer(*,*,*), optional

conjugateCharacterIndices(self)¶

For each character, returns the index of the character conjugate to it

Returns: Index of the conjugate character
Return type: integer(1,*)

static fromIrreps(group, irreps)¶

Creates a complex character table from a complete list of irreducible exact representations

Parameters

group (FiniteGroup) – Finite group
irreps (cell(1,*) of Rep) – Irreducible representations

Returns

The corresponding complex character table

Return type

ComplexCharacterTable

static fromRealCharacterTable(R)¶

Creates a complex character table from a real character table

This works only if its real irreducible representations are absolutely irreducible; though that fact is not checked by this method.

Parameters: R (RealCharacterTable) – Real character table
Returns: The corresponding complex character table
Return type: ComplexCharacterTable

imap(self, f)¶

Maps the character table under an isomorphism

Example

>>> D6a = replab.PermutationGroup.of([3 2 1], [2 3 1]); % doctest: +cyclotomic
>>> D6b = replab.PermutationGroup.of([1 4 3 2], [1 3 4 2]);
>>> f = D6a.isomorphismByImages(D6b, 'preimages', D6a.generators, 'images', D6b.generators);
>>> Ca = D6a.characterTable;
>>> Cb = Ca.imap(f);
>>> Cb.laws.checkSilent;

Parameters: f (FiniteIsomorphism) – Isomorphism with self.group.isSubgroupOf(f.source)
Returns: The character table of the subgroup in the image of the isomorphism
Return type: ComplexCharacterTable

RealCharacterTable¶

class replab.RealCharacterTable(group, classes, values, varargin)¶

Bases: replab.CharacterTable

Describes the real character table of a group

The character values are stored as elements of the cyclotomic field, using the cyclotomic class which requires external libraries and a Java Virtual Machine available.

Instances of RealCharacterTable are immutable.

For additional help, see CharacterTable.

Class members list

Properties

classNames – Names of conjugacy classes
classes – Conjugacy classes of group
field – Whether the character table is real or complex
group – Group represented by character table
irrepNames – Names of the irreducible representations/characters
values – Character values

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

RealCharacterTable – Constructs a character table
kronecker – Returns the Kronecker coefficients corresponding to this character table
overC – Returns whether the character table is defined over the complex field
overR – Returns whether the character table is defined over the real field

Characters

character – Returns an irreducible character in this character table
characters – Returns the characters in this table
linearCharacterIndices – Returns the indices of the linear characters
multiplicities – Calculate the multiplicities of the irreducible characters in this table in a given representation or character
nCharacters – Returns the number of characters in this character table
trivialCharacterIndex – Returns the row index of the trivial character

Representations

decomposeRep – Computes the decomposition of a representation into irreducibles using the explicit irreps contained in this table
hasIrrep – Returns whether the irreducible representation that corresponds to the character of given index is available
hasIrreps – Returns whether explicit constructions are available for all irreducible representations
irrep – Returns the irreducible representation that corresponds to the character of given index
irreps – Returns a cell vector of the irreducible representations corresponding to the characters in this table
nIrreps – Returns the number of

Conjugacy classes

identityConjugacyClassIndex – Returns the index of the conjugacy class containing the identity
nClasses – Returns the number of conjugacy classes in this character table

Transformations of the character table

imap – Maps the character table under an isomorphism

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()

character(self, ind)¶: Documentation in replab.CharacterTable.character()

characters(self)¶: Documentation in replab.CharacterTable.characters()

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

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

classNames¶: Documentation in replab.CharacterTable.classNames

classes¶: Documentation in replab.CharacterTable.classes

decomposeRep(self, rep)¶: Documentation in replab.CharacterTable.decomposeRep()

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

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

field¶: Documentation in replab.CharacterTable.field

group¶: Documentation in replab.CharacterTable.group

hasIrrep(self, ind)¶: Documentation in replab.CharacterTable.hasIrrep()

hasIrreps(self)¶: Documentation in replab.CharacterTable.hasIrreps()

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

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

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

identityConjugacyClassIndex(self)¶: Documentation in replab.CharacterTable.identityConjugacyClassIndex()

imap(self, f)¶: Documentation in replab.CharacterTable.imap()

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

irrep(self, ind)¶: Documentation in replab.CharacterTable.irrep()

irrepNames¶: Documentation in replab.CharacterTable.irrepNames

irreps(self)¶: Documentation in replab.CharacterTable.irreps()

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

kronecker(self)¶: Documentation in replab.CharacterTable.kronecker()

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

linearCharacterIndices(self)¶: Documentation in replab.CharacterTable.linearCharacterIndices()

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

multiplicities(self, arg)¶: Documentation in replab.CharacterTable.multiplicities()

nCharacters(self)¶: Documentation in replab.CharacterTable.nCharacters()

nClasses(self)¶: Documentation in replab.CharacterTable.nClasses()

nIrreps(self)¶: Documentation in replab.CharacterTable.nIrreps()

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

overC(self)¶: Documentation in replab.CharacterTable.overC()

overR(self)¶: Documentation in replab.CharacterTable.overR()

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

trivialCharacterIndex(self)¶: Documentation in replab.CharacterTable.trivialCharacterIndex()

values¶: Documentation in replab.CharacterTable.values

Own members

RealCharacterTable(group, classes, values, varargin)¶

Constructs a character table

Parameters

group (FiniteGroup) – Group represented by character table
classes (ConjugacyClasses) – Conjugacy classes of group
values (cyclotomic (nIrreps, nClasses)) – Character values
irreps – Explicit matrix representations (can contain empty values)
classNames – Names of conjugacy classes
irrepNames – Names of irreducible representations
kronecker – Kronecker coefficients

Kwtype irreps

cell(1,*) of [] or replab.RepByImages, optional

Kwtype classNames

cell(1,*) of charstring, optional

Kwtype irrepNames

cell(1,*) of charstring, optional

Kwtype kronecker

integer(*,*,*), optional