RepLAB: A Computational/Numerical Approach to Representation Theory
647
>> dec = rep.decomposition
Complex similar representation (irreducible decomposition)
dimension: 4 // field: ’C’ // group: 2 x 2 unitary matrices
basis.(:,1): [-0.49064+0.50919i, 5.3749e-17+1.4048e-17i, ...
...
basis.(:,4): [-0.69962+0.10261i, 1.8576e-17-1.6639e-17i, ...
component(1): Isotypic component C(1) (trivial)
component(2): Isotypic component C(3)
>> dec.nice % we ask for rational basis recovery
Complex similar representation (irreducible decomposition)
...
basis.(:,1): [1, 0, 0, 1].’
basis.(:,2): [0, 0, 1, 0].’
basis.(:,3): [0, 1, 0, 0].’
basis.(:,4): [1, 0, 0, -1].’
Most SDP solvers encode complex-valued problems using real matrices with
overhead. It is thus more efficient to block diagonalize real SDPs over the real field.
RepLAB identifies the three types of division algebras that appear in irreducible real
representations (real-type ’R’, complex-type ’C’, quaternion-type ’H’). Below
we demonstrate a cyclic group and the quaternion group.
>> C3 = replab.S(3).cyclicSubgroup; decC3 = C3.naturalRep.decomposition
Orthogonal similar representation (irreducible decomposition)
dimension: 3 // field: ’R’ // ...
component(1): Isotypic component R(1) (trivial)
component(2): Isotypic component (harmonized) C(2)
>>> qminus = [-1 -2 -3 -4]; qi = [2 -1 4 -3]; qj = [3 -4 -1 2];
>>> Q = replab.signed.Permutations(4).subgroup({qminus qi qj});
>>> decQ = Q.naturalRep.decomposition
Orthogonal irreducible similar representation (irreducible decomposition)
dimension: 4 // field: ’R’ // ...
component(1): Isotypic component (harmonized) H(4)
RepLAB standardizes the matrix encodings of the complex and quaternion division
algebras: for example, decC3.component(2).sample returns the representation of a random group element, of the form [a -b; b a] for some a and b.
RepLAB contains out-of-the-box support for a variety of groups. Those groups
can be combined using standard constructions (direct, semidirect, and wreath
products, which are relevant in nonlocality [30]). RepLAB is compatible with the
free MATLAB clone Octave [31]. It extends the toolbox YALMIP [32] by providing
invariant SDP variables and constraints. It can also block diagonalize data in an
extension of the SeDuMi SDP solver format [33].
2 Algorithm
The decomposition of ρ implies four subtasks. First, we sample generic elements
from the commutant C ρ . Second, we compute the eigendecomposition of a generic
Hermitian matrix from C ρ to obtain invariant subspaces. Third, we find which
irreducible subrepresentations are equivalent and harmonize their bases. Fourth,
only for real subrepresentations, we identify the representation type and express
its division algebra in a standard encoding.
647
>> dec = rep.decomposition
Complex similar representation (irreducible decomposition)
dimension: 4 // field: ’C’ // group: 2 x 2 unitary matrices
basis.(:,1): [-0.49064+0.50919i, 5.3749e-17+1.4048e-17i, ...
...
basis.(:,4): [-0.69962+0.10261i, 1.8576e-17-1.6639e-17i, ...
component(1): Isotypic component C(1) (trivial)
component(2): Isotypic component C(3)
>> dec.nice % we ask for rational basis recovery
Complex similar representation (irreducible decomposition)
...
basis.(:,1): [1, 0, 0, 1].’
basis.(:,2): [0, 0, 1, 0].’
basis.(:,3): [0, 1, 0, 0].’
basis.(:,4): [1, 0, 0, -1].’
Most SDP solvers encode complex-valued problems using real matrices with
overhead. It is thus more efficient to block diagonalize real SDPs over the real field.
RepLAB identifies the three types of division algebras that appear in irreducible real
representations (real-type ’R’, complex-type ’C’, quaternion-type ’H’). Below
we demonstrate a cyclic group and the quaternion group.
>> C3 = replab.S(3).cyclicSubgroup; decC3 = C3.naturalRep.decomposition
Orthogonal similar representation (irreducible decomposition)
dimension: 3 // field: ’R’ // ...
component(1): Isotypic component R(1) (trivial)
component(2): Isotypic component (harmonized) C(2)
>>> qminus = [-1 -2 -3 -4]; qi = [2 -1 4 -3]; qj = [3 -4 -1 2];
>>> Q = replab.signed.Permutations(4).subgroup({qminus qi qj});
>>> decQ = Q.naturalRep.decomposition
Orthogonal irreducible similar representation (irreducible decomposition)
dimension: 4 // field: ’R’ // ...
component(1): Isotypic component (harmonized) H(4)
RepLAB standardizes the matrix encodings of the complex and quaternion division
algebras: for example, decC3.component(2).sample returns the representation of a random group element, of the form [a -b; b a] for some a and b.
RepLAB contains out-of-the-box support for a variety of groups. Those groups
can be combined using standard constructions (direct, semidirect, and wreath
products, which are relevant in nonlocality [30]). RepLAB is compatible with the
free MATLAB clone Octave [31]. It extends the toolbox YALMIP [32] by providing
invariant SDP variables and constraints. It can also block diagonalize data in an
extension of the SeDuMi SDP solver format [33].
2 Algorithm
The decomposition of ρ implies four subtasks. First, we sample generic elements
from the commutant C ρ . Second, we compute the eigendecomposition of a generic
Hermitian matrix from C ρ to obtain invariant subspaces. Third, we find which
irreducible subrepresentations are equivalent and harmonize their bases. Fourth,
only for real subrepresentations, we identify the representation type and express
its division algebra in a standard encoding.