Continuous groups

RepLAB provides constructions of standard continuous groups.

First, we consider torus-like groups.

• TorusGroup: Describes a compact, connected, abelian Lie group (i.e. a torus group).

• T: Describes the standard torus group of given rank.

Then, we have the classical compact groups.

• ClassicalCompactGroup: Class implementing all the groups below.

• O: Constructs the group of (real) orthogonal matrices of a given size.

• SO: Constructs the group of (real) special orthogonal matrices of a given size.

• U: Constructs the group of (complex) unitary matrices of a given size.

• SU: Constructs the group of (complex) special unitary matrices of a given size.

• Sp: Constructs the compact symplectic group of a given size, which encodes using complex numbers the generalization of the unitary group to quaternions.

TorusGroup

class replab.TorusGroup(E, names)

Bases: replab.CompactGroup

Describes a compact, connected, abelian Lie group

Its elements are column vectors of angles expressed in turns (see https://en.wikipedia.org/wiki/Turn_(angle)), i.e. real numbers between 0 and 1, 1 not included. The group binary operation is addition modulo 1, the identity is the turn 0, and the inverse operation maps any non-zero turn x to 1-x.

When there are no additional restrictions on the elements of a given torus group, it is the standard torus group, and it can be constructed with the function replab.T.

Torus morphisms from integer matrices: https://math.stackexchange.com/questions/2229088/automorphisms-of-a-torus. Let s = [s1;...;sn] be the element of a (standard) torus of dimension n and t = [t1;...;tm] the element of a (standard) torus of dimension m. A torus morphism f is given by a m x n integer matrix M, with the image of s being written t = f(s) = mod(M * s, 1).

Let the coefficients [t1; t2; ...; tn] be the coefficients of this column vector, where n is the torus group dimension. The coefficients may or may not obey equations of the form mod(a1*t1 + a2*t2 + ... + an*tn, 1) == 0, where the coefficients a1 ... an are integers. Those equations are collected as rows in the E integer matrix. Equivalently, the torus group obeying those relations is the kernel of the torus morphism encoded in E.

How do we deal with those equations? We use the Hermite normal form of the matrix E to extract a pair of morphisms from T (whose elements obey the equations) to a standard torus S (whose dimension is then the rank of T). The morphism projection from T to S is surjective, and the morphism injection from S to T is injective, and while projection(injection(s)) = s. Both morphisms are given by integer matrices.

The defining representation of this group (definingRep) is the n x n matrix with diagonal elements u1 … un.

Automorphisms of a torus group are given by invertible square integer matrices M such that E*M*injection = 0.

The explicit group elements (for example, the value returned by sample) are written using the additive convention; while the equations are written multiplicatively in symbolic form.

Class members list

Properties

• E – Equations obeyed by the elements of this torus

• identity – Monoid identity element

• injection – Injection map from maximal torus to this torus

• n – Torus dimension

• names – Names of coefficients in the multiplicative convention

• projection – Projection map from this torus to a maximal torus

• r – Torus rank

Prettyprinting

• additionalFields – Returns the name/value pairs corresponding to additional fields to be printed

• disp – Standard MATLAB/Octave display method

• headerStr – Tiny single line description of the current object type

• hiddenFields – Returns the names of the fields that are not printed as a row vector

• longStr – Multi-line description of the current object

• shortStr – Single line text description of the current object

Property cache

• cache – Sets the value of the designated property in the cache

• cached – Returns the cached property if it exists, computing it if necessary

• cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet

• cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet

• inCache – Returns whether the value of the given property has already been computed

Laws

• check – Checks the consistency of this object

• checkAndContinue – Checks the consistency of this object

• laws – Returns the laws that this object obeys

Unique ID

• eq – Equality test

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

• isequal – Equality test

• ne – Non-equality test

Test helpers

• assertEqv – Compares two elements for equality

• assertNotEqv – Compares two elements for inequality

Sampling and equality test

• eqv – Tests domain elements for equality/equivalence

• sample – Samples an element from this set

Monoid operations

• areCommuting – Returns whether the two given elements commute

• compose – Composes two monoid/group elements

• composeAll – Composes a sequence of monoid elements

• composeN – Computes the power of a given element by repeated squaring

• isIdentity – Tests if the given element is the identity

Morphisms

• automorphism – Constructs an automorphism of this torus from an integer matrix describing a torus map

• automorphismGroup – Returns the automorphism group of this group

• commutingMorphismsMorphism – Constructs a morphism from morphisms with commuting images

• endomorphism – Constructs an endomorphism of this torus from an integer matrix describing a torus map

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

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

• morphismByFunction – Constructs a group morphism using an image function

• torusMorphism – Constructs a morphism from this torus group to another torus group from an integer matrix

Group methods

• commutator – Returns the commutator of two group elements

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

• composeWithInverse – Returns the composition of an element with the inverse of another element

• inverse – Computes the inverse of an element

• leftConjugate – Returns the left conjugate of a group element

Representations

• commutingRepsRep – Constructs a representation from commuting representations

• definingRep – Returns the defining representation of this torus group

• diagonalRep – Returns a diagonal representation of this torus group

• diagonalRepWith – Returns a diagonal rep of this torus group

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

• tensorRep – Computes the tensor product of representations

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

Group construction

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

• directProduct – Returns the direct product of groups

• semidirectProduct – Describes an external semidirect product of groups

Group properties

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

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

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

General

• TorusGroup – Constructs a torus group obeying the given equations, with optional element names

• equation – Returns the i-th equation defining this torus group in monomial form

• equations – Returns all equations that this torus group obeys in monomial form

• nEquations – Returns the number of equations defining this torus group

• subgroup – Returns a subgroup of this torus that obeys the equations given in integer matrix form

• subgroupWith – Returns a subgroup of this torus that obeys the equations given in monomial form

• torusRepImage – Returns the sparse diagonal matrix corresponding to the given torus element

• torusRepRealImage – Returns the sparse block-diagonal matrix corresponding to the given torus element

• withNames – Returns a copy of this torus group with the names of the unit complex coefficients replaced

Group constructions

• semidirectProductByFiniteGroup – Returns the semidirect product of a finite group and this torus, with the finite group action defined by images

• semidirectProductFromRep – Returns the semidirect product of a finite group on the torus elements using one of its integer representations

Inherited elements

additionalFields(self)

Documentation in replab.Str.additionalFields()

areCommuting(self, x, y)

Documentation in replab.Monoid.areCommuting()

assertEqv(self, x, y, context)

Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)

Documentation in replab.Domain.assertNotEqv()

automorphismGroup(self)

Documentation in replab.Group.automorphismGroup()

cache(self, name, value, handleExisting)

Documentation in replab.Obj.cache()

cached(self, name, fun)

Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)

Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)

Documentation in replab.Obj.cachedOrEmpty()

check(self)

Documentation in replab.Obj.check()

checkAndContinue(self)

Documentation in replab.Obj.checkAndContinue()

commutator(self, x, y, startWithInverse)

Documentation in replab.Group.commutator()

commutingMorphismsMorphism(self, target, morphisms)

Documentation in replab.Group.commutingMorphismsMorphism()

commutingRepsRep(self, field, dimension, reps)

Documentation in replab.CompactGroup.commutingRepsRep()

compose(self, x, y)

Documentation in replab.Monoid.compose()

composeAll(self, elements)

Documentation in replab.Monoid.composeAll()

composeFlat(self, array, letters)

Documentation in replab.Group.composeFlat()

composeN(self, x, n)

Documentation in replab.Monoid.composeN()

composeWithInverse(self, x, y)

Documentation in replab.Group.composeWithInverse()

directPower(self, n)

Documentation in replab.CompactGroup.directPower()

directProduct(varargin)

Documentation in replab.CompactGroup.directProduct()

directSumRep(self, field, reps)

Documentation in replab.CompactGroup.directSumRep()

disp(self)

Documentation in replab.Str.disp()

eq(self, rhs)

Documentation in replab.Obj.eq()

eqv(self, t, u)

Documentation in replab.Domain.eqv()

hasReconstruction(self)

Documentation in replab.CompactGroup.hasReconstruction()

headerStr(self)

Documentation in replab.Str.headerStr()

hiddenFields(self)

Documentation in replab.Str.hiddenFields()

id(self)

Documentation in replab.Obj.id()

identity

Documentation in replab.Monoid.identity

inCache(self, name)

Documentation in replab.Obj.inCache()

innerAutomorphism(self, by)

Documentation in replab.Group.innerAutomorphism()

inverse(self, x)

Documentation in replab.Group.inverse()

isIdentity(self, x)

Documentation in replab.Monoid.isIdentity()

isequal(lhs, rhs)

Documentation in replab.Obj.isequal()

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

Documentation in replab.Group.isomorphismByFunctions()

laws(self)

Documentation in replab.Obj.laws()

leftConjugate(self, by, on)

Documentation in replab.Group.leftConjugate()

longStr(self, maxRows, maxColumns)

Documentation in replab.Str.longStr()

maximalTorusDimension(self)

Documentation in replab.CompactGroup.maximalTorusDimension()

morphismByFunction(self, target, imageElementFun, torusMap)

Documentation in replab.Group.morphismByFunction()

ne(self, rhs)

Documentation in replab.Obj.ne()

reconstruction(self)

Documentation in replab.CompactGroup.reconstruction()

sample(self)

Documentation in replab.Domain.sample()

semidirectProduct(self, N, phi)

Documentation in replab.CompactGroup.semidirectProduct()

shortStr(self, maxColumns)

Documentation in replab.Str.shortStr()

tensorRep(self, field, reps)

Documentation in replab.CompactGroup.tensorRep()

trivialRep(self, field, dimension)

Documentation in replab.CompactGroup.trivialRep()

Own members

E

Equations obeyed by the elements of this torus

Type

integer(*,n)

injection

Injection map from maximal torus to this torus

Type

integer(n,r)

n

Torus dimension

Type

integer

names

Names of coefficients in the multiplicative convention

Type

cell(1,*) of charstring

projection

Projection map from this torus to a maximal torus

Type

integer(r,n)

r

Torus rank

Type

integer

TorusGroup(E, names)

Constructs a torus group obeying the given equations, with optional element names

Parameters

• E (integer(*,n)) – Integer matrix describing the equations; its number of columns provide the torus dimension

• names (cell(1,n) of charstring, optional) – Names of the group element coefficients in their unit complex form

automorphism(self, varargin)

Constructs an automorphism of this torus from an integer matrix describing a torus map

This function accepts a variable number of arguments. Each argument must be of the form:

• An integer matrix with n columns, defining equations as in the call to subgroup,

• A charstring containing an equation in monomial form.

Those arguments will be parsed and concatenated vertically to form an integer matrix describing the torus map. That matrix must be invertible.

Returns

The constructed automorphism

Return type

Isomorphism

definingRep(self)

Returns the defining representation of this torus group

The defining representation of a torus group of dimension n is a n-dimensional representation, whose images are diagonal in the standard Euclidean basis.

Returns

Representation

Return type

Rep

diagonalRep(self, torusMap)

Returns a diagonal representation of this torus group

Constructs a representation using a torusMap, whose images are diagonal matrices. The diagonal elements are monomials of the phases of the torus group. Each row of torusMap corresponds to a diagonal element, and the columns give the integer exponents corresponding to every phase.

Parameters

torusMap (integer(d, n)) – Phase exponents

Returns

Representation

Return type

Rep

diagonalRepWith(self, varargin)

Returns a diagonal rep of this torus group

endomorphism(self, varargin)

Constructs an endomorphism of this torus from an integer matrix describing a torus map

This function accepts a variable number of arguments. Each argument must be of the form:

• An integer matrix with n columns, defining equations as in the call to subgroup,

• A charstring containing an equation in monomial form.

Those arguments will be parsed and concatenated vertically to form an integer matrix describing the torus map.

Returns

The constructed endomorphism

Return type

Morphism

equation(self, i)

Returns the i-th equation defining this torus group in monomial form

Parameters

i (integer) – Equation index

Returns

Equation in monomial form (whose value must be equal to one)

Return type

charstring

equations(self)

Returns all equations that this torus group obeys in monomial form

Returns

Equations defining this torus group in monomial form

Return type

cell(1,*) of charstring

nEquations(self)

Returns the number of equations defining this torus group

Returns

Number of equations

Return type

integer

semidirectProductByFiniteGroup(self, group, varargin)

Returns the semidirect product of a finite group and this torus, with the finite group action defined by images

Parameters

• group (FiniteGroup) – Finite group acting on this torus group

• preimages – Preimages of the morphism which generate group, defaults to group.generators

• images – Action on the torus for the given preimages, defaults to a permutation action if group is a permutation group

Kwtype preimages

cell(1, *) of group elements

Kwtype images

cell(1, *) of target elements

Returns

Semidirect product of group and this torus group

Return type

SemidirectProductGroup

semidirectProductFromRep(self, rep)

Returns the semidirect product of a finite group on the torus elements using one of its integer representations

Invertible integer matrices represent automorphisms of a torus group of the same dimension; thus, given a group G, a representation of G with integer images defines a morphism from G to the automorphism group of a torus T.

One can then use this morphism to construct an outer semidirect product of G acting on T.

Parameters

rep (Rep) – Group representation with integer coefficients

Returns

Semidirect product of rep.group and this torus group

Return type

SemidirectProductGroup

subgroup(self, newE)

Returns a subgroup of this torus that obeys the equations given in integer matrix form

Parameters

E (integer(*, n)) – New equations

Returns

Torus group satisfying the equations of this group and the additional ones given

Return type

TorusGroup

subgroupWith(self, varargin)

Returns a subgroup of this torus that obeys the equations given in monomial form

This function accepts a variable number of arguments. Each argument must be of the form:

• An integer matrix with n columns, defining equations as in the call to subgroup,

• A charstring containing an equation in monomial form. If the equation has no right-hand side, the right-hand side is assumed to be the identity.

Returns

Torus group satisfying the equations of this group and the additional ones given

Return type

TorusGroup

torusMorphism(self, target, varargin)

Constructs a morphism from this torus group to another torus group from an integer matrix

After the target argument, this function accepts a variable number of arguments. Each of these arguments must be of the form:

• An integer matrix with n columns, defining equations as in the call to subgroup,

• A charstring containing an equation in monomial form.

Parameters

target (TorusGroup) – Torus group target of the morphism

Returns

A morphism from this torus group to a torus group of dimension n1

Return type

Morphism

static torusRepImage(t)

Returns the sparse diagonal matrix corresponding to the given torus element

Equivalent to torusGroup.definingRep.image(t), without constructing additional objects.

Parameters

t (double(n,1)) – Torus group element

Returns

Diagonal matrix corresponding to the torus element

Return type

double(n,n), sparse

static torusRepRealImage(t)

Returns the sparse block-diagonal matrix corresponding to the given torus element

The element t(i) will correspond to a 2x2 block: [c s; -s c] where c = cos(2*pi*t(i)) and s = sin(2*pi*t(i)).

Parameters

t (double(n,1)) – Torus group element

Returns

Block diagonal matrix of 2x2 real matrices

Return type

double(2*n,2*n)

withNames(self, newNames)

Returns a copy of this torus group with the names of the unit complex coefficients replaced

Parameters

newNames (cell(1,n) of charstring) – New names of the complex coefficients

Returns

Updated group

Return type

TorusGroup

T

replab.T(n)

Returns the standard torus group of given rank

Parameters

n (integer) – Torus group rank

Returns

Standard torus group

Return type

replab.TorusGroup

ClassicalCompactGroup

class replab.ClassicalCompactGroup(n, algebra, isSpecial)

Bases: replab.CompactGroup

Describes a classical compact group of square real/complex/quaternionic matrices

Group elements are matrices whose coefficients are encoded using double floating-point arithmetic. Of course, they will only be approximately orthogonal and thus do not represent directly a group element.

Instead, we postulate that the matrix encodes the nearest group element in Frobenius norm. For non-special groups, this element can be found as U = X*sqrtm(X'*X) where X is the approximate element.

While inverses should be exact (we take the conjugate transpose), the composition is only approximate. Thus, to be able to check the usual group laws, we compare group elements up to a certain tolerance, defined in replab.globals.matrixGroupTol.

Class members list

Properties

• algebra – Unital associative algebra over which the group is defined

• identity – Monoid identity element

• isSpecial – Whether the matrices should have determinant one; must be false when algebra = ‘H’

• n – Dimension of the matrices

Prettyprinting

• additionalFields – Returns the name/value pairs corresponding to additional fields to be printed

• disp – Standard MATLAB/Octave display method

• headerStr – Tiny single line description of the current object type

• hiddenFields – Returns the names of the fields that are not printed as a row vector

• longStr – Multi-line description of the current object

• shortStr – Single line text description of the current object

Property cache

• cache – Sets the value of the designated property in the cache

• cached – Returns the cached property if it exists, computing it if necessary

• cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet

• cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet

• inCache – Returns whether the value of the given property has already been computed

Laws

• check – Checks the consistency of this object

• checkAndContinue – Checks the consistency of this object

• laws – Returns the laws that this object obeys

Unique ID

• eq – Equality test

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

• isequal – Equality test

• ne – Non-equality test

Test helpers

• assertEqv – Compares two elements for equality

• assertNotEqv – Compares two elements for inequality

Sampling and equality test

• eqv – Tests domain elements for equality/equivalence

• sample – Samples an element from this set

Monoid operations

• areCommuting – Returns whether the two given elements commute

• compose – Composes two monoid/group elements

• composeAll – Composes a sequence of monoid elements

• composeN – Computes the power of a given element by repeated squaring

• isIdentity – Tests if the given element is the identity

Morphisms

• automorphismGroup – Returns the automorphism group of this group

• commutingMorphismsMorphism – Constructs a morphism from morphisms with commuting images

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

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

• morphismByFunction – Constructs a group morphism using an image function

Group methods

• commutator – Returns the commutator of two group elements

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

• composeWithInverse – Returns the composition of an element with the inverse of another element

• inverse – Computes the inverse of an element

• leftConjugate – Returns the left conjugate of a group element

Representations

• commutingRepsRep – Constructs a representation from commuting representations

• definingRep – Returns the defining representation of this unitary group

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

• tensorRep – Computes the tensor product of representations

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

Group construction

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

• directProduct – Returns the direct product of groups

• semidirectProduct – Describes an external semidirect product of groups

Group properties

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

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

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

General

• ClassicalCompactGroup – Constructs a classical compact group

Inherited elements

additionalFields(self)

Documentation in replab.Str.additionalFields()

areCommuting(self, x, y)

Documentation in replab.Monoid.areCommuting()

assertEqv(self, x, y, context)

Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)

Documentation in replab.Domain.assertNotEqv()

automorphismGroup(self)

Documentation in replab.Group.automorphismGroup()

cache(self, name, value, handleExisting)

Documentation in replab.Obj.cache()

cached(self, name, fun)

Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)

Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)

Documentation in replab.Obj.cachedOrEmpty()

check(self)

Documentation in replab.Obj.check()

checkAndContinue(self)

Documentation in replab.Obj.checkAndContinue()

commutator(self, x, y, startWithInverse)

Documentation in replab.Group.commutator()

commutingMorphismsMorphism(self, target, morphisms)

Documentation in replab.Group.commutingMorphismsMorphism()

commutingRepsRep(self, field, dimension, reps)

Documentation in replab.CompactGroup.commutingRepsRep()

compose(self, x, y)

Documentation in replab.Monoid.compose()

composeAll(self, elements)

Documentation in replab.Monoid.composeAll()

composeFlat(self, array, letters)

Documentation in replab.Group.composeFlat()

composeN(self, x, n)

Documentation in replab.Monoid.composeN()

composeWithInverse(self, x, y)

Documentation in replab.Group.composeWithInverse()

directPower(self, n)

Documentation in replab.CompactGroup.directPower()

directProduct(varargin)

Documentation in replab.CompactGroup.directProduct()

directSumRep(self, field, reps)

Documentation in replab.CompactGroup.directSumRep()

disp(self)

Documentation in replab.Str.disp()

eq(self, rhs)

Documentation in replab.Obj.eq()

eqv(self, t, u)

Documentation in replab.Domain.eqv()

hasReconstruction(self)

Documentation in replab.CompactGroup.hasReconstruction()

headerStr(self)

Documentation in replab.Str.headerStr()

hiddenFields(self)

Documentation in replab.Str.hiddenFields()

id(self)

Documentation in replab.Obj.id()

identity

Documentation in replab.Monoid.identity

inCache(self, name)

Documentation in replab.Obj.inCache()

innerAutomorphism(self, by)

Documentation in replab.Group.innerAutomorphism()

inverse(self, x)

Documentation in replab.Group.inverse()

isIdentity(self, x)

Documentation in replab.Monoid.isIdentity()

isequal(lhs, rhs)

Documentation in replab.Obj.isequal()

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

Documentation in replab.Group.isomorphismByFunctions()

laws(self)

Documentation in replab.Obj.laws()

leftConjugate(self, by, on)

Documentation in replab.Group.leftConjugate()

longStr(self, maxRows, maxColumns)

Documentation in replab.Str.longStr()

maximalTorusDimension(self)

Documentation in replab.CompactGroup.maximalTorusDimension()

morphismByFunction(self, target, imageElementFun, torusMap)

Documentation in replab.Group.morphismByFunction()

ne(self, rhs)

Documentation in replab.Obj.ne()

reconstruction(self)

Documentation in replab.CompactGroup.reconstruction()

sample(self)

Documentation in replab.Domain.sample()

semidirectProduct(self, N, phi)

Documentation in replab.CompactGroup.semidirectProduct()

shortStr(self, maxColumns)

Documentation in replab.Str.shortStr()

tensorRep(self, field, reps)

Documentation in replab.CompactGroup.tensorRep()

trivialRep(self, field, dimension)

Documentation in replab.CompactGroup.trivialRep()

Own members

algebra

Unital associative algebra over which the group is defined

Type

‘R’, ‘C’ or ‘H’

isSpecial

Whether the matrices should have determinant one; must be false when algebra = ‘H’

Type

logical

n

Dimension of the matrices

Type

integer

ClassicalCompactGroup(n, algebra, isSpecial)

Constructs a classical compact group

Parameters

• n (integer) – Size of the n x n matrices

• algebra ('R', 'C' or 'H') – Unital associatvie algebra over which the group is defined

• isSpecial (logical) – Whether the matrices should have det(X) == 1

definingRep(self, field)

Returns the defining representation of this unitary group

“Defining representation” is a loosely defined term designing the smallest faithful representation of a group. In our case, the defining representation of O(n) or SO(n) is a n-dimensional real representation, the defining representation of U(n) or SU(n) is a n-dimension complex representation, while the defining representation of Sp(n) is a 2n-dimension complex representation.

All these representations are unitary. By specifying the field, one can encode the representation in a different field: if the natural field is real (O(n), SO(n)), the representation can be complexified into a representation of the same dimension. If the natural field is complex (U(n), SU(n), Sp(n)), the complex representation can be encoded into a real representation, doubling its dimension.

Parameters

field ('R', 'C', optional) – Field over which to define the matrices, default: group dependent, see above

Returns

Defining representation

Return type

Rep

O

replab.O(d)

Returns the group of d x d orthonormal matrices

SO

replab.SO(d)

Returns the group of d x d orthonormal matrices with determinant one

U

replab.U(d)

Returns the group of d x d unitary complex matrices

SU

replab.SU(d)

Returns the group of d x d unitary complex matrices with determinant one

Sp

replab.Sp(d)

Returns the compact symplectic group whose elements are unitary quaternion matrices