Finite group representation#

Some utility function for representations of finite groups

Most of the functions are just GAP wrappers.

surface_dynamics.misc.group_representation.conjugacy_class_matrix(cl, d)[source]#

Return the matrix associated to a given conjugacy class of a permutation group.

The result is a d times d matrix that is invariant under conjugation by the group action on ZZ^d. It is used to produce the projection matrices on the isotypic subspaces.

EXAMPLES:

sage: from surface_dynamics.misc.group_representation import conjugacy_class_matrix

sage: G = QuaternionGroup()
sage: Ggap = libgap(G)
sage: cls = Ggap.ConjugacyClasses()
sage: m = conjugacy_class_matrix(cls[2], 8)
sage: m
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]

sage: for cl in cls:
....:      m = conjugacy_class_matrix(cl, 8)
....:      for g in G:
....:          g = g.matrix()
....:          assert g*m*~g == m

sage: A = AlternatingGroup(5)
sage: Agap = libgap(A)
sage: Ggap = Agap.Action(Agap)
sage: gens = libgap.GeneratorsOfGroup(Ggap).sage()
sage: G = PermutationGroup(gens)
sage: cls = Ggap.ConjugacyClasses()
sage: for cl in cls:
....:      m = conjugacy_class_matrix(cls[2], 60)
....:      for _ in range(20):
....:           g = G.random_element().matrix()
....:           assert g*m*~g == m

surface_dynamics.misc.group_representation.isotypic_projection_matrix(G, d, chi, deg, conj_mats=None, floating_point=False)[source]#

Return an isotypic projection matrix

INPUT:

G – a permutation group
d – (integer) the domain of the group is {1, 2, ldots, d}
chi – (tuple) real or complex character
deg – (integer) degree of the character
conj_mats – (optional list) matrices of the conjugacy classes
floating_point – whether to return matrices with floating point entries instead
of elements in the cyclotomic field

Recall the formula for the projection as given in Theorem 8 in [Ser67]. If G is a permutation group, then

\[\pi_\chi = \sum_{g \in G} \overline_{\chi(g)} g\]

EXAMPLES:

sage: from surface_dynamics.misc.group_representation import real_characters, isotypic_projection_matrix
sage: G = AlternatingGroup(5)
sage: T,deg = real_characters(G)
sage: isotypic_projection_matrix(G, 5, T[0], deg[0])
[1/5 1/5 1/5 1/5 1/5]
[1/5 1/5 1/5 1/5 1/5]
[1/5 1/5 1/5 1/5 1/5]
[1/5 1/5 1/5 1/5 1/5]
[1/5 1/5 1/5 1/5 1/5]
sage: isotypic_projection_matrix(G, 5, T[1], deg[1])
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
sage: isotypic_projection_matrix(G, 5, T[2], deg[2])
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
sage: isotypic_projection_matrix(G, 5, T[3], deg[3])
[ 4/5 -1/5 -1/5 -1/5 -1/5]
[-1/5  4/5 -1/5 -1/5 -1/5]
[-1/5 -1/5  4/5 -1/5 -1/5]
[-1/5 -1/5 -1/5  4/5 -1/5]
[-1/5 -1/5 -1/5 -1/5  4/5]
sage: isotypic_projection_matrix(G, 5, T[3], deg[3], floating_point=True)
array([[ 0.8, -0.2, -0.2, -0.2, -0.2],
       [-0.2,  0.8, -0.2, -0.2, -0.2],
       [-0.2, -0.2,  0.8, -0.2, -0.2],
       [-0.2, -0.2, -0.2,  0.8, -0.2],
       [-0.2, -0.2, -0.2, -0.2,  0.8]])
sage: isotypic_projection_matrix(G, 5, T[4], deg[4])
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]

sage: sum(isotypic_projection_matrix(G, 5, T[i], deg[i]) for i in range(5)).is_one()
True

surface_dynamics.misc.group_representation.real_characters(G)[source]#

Return a pair (table of characters, character degrees) for the group G.

OUTPUT:

table of characters - a list of characters. Each character is represented as the list of its values on conjugacy classes. The order of conjugacy classes is the same as the one returned by GAP.
degrees - the list of degrees of the characters

EXAMPLES:

sage: from surface_dynamics.misc.group_representation import real_characters
sage: T, deg = real_characters(AlternatingGroup(5))
sage: T
[(1, 1, 1, 1, 1),
 (3, -1, 0, -E(5) - E(5)^4, -E(5)^2 - E(5)^3),
 (3, -1, 0, -E(5)^2 - E(5)^3, -E(5) - E(5)^4),
 (4, 0, 1, -1, -1),
 (5, 1, -1, 0, 0)]
sage: set(parent(x) for chi in T for x in chi)
{Universal Cyclotomic Field}
sage: deg
[1, 3, 3, 4, 5]

sage: T, deg = real_characters(CyclicPermutationGroup(6))
sage: T
[(1, 1, 1, 1, 1, 1),
 (1, -1, 1, -1, 1, -1),
 (2, -1, -1, 2, -1, -1),
 (2, 1, -1, -2, -1, 1)]
sage: deg
[1, 1, 1, 1]

surface_dynamics.misc.group_representation.real_isotypic_projection_matrices(G, d)[source]#

Return the real projections

EXAMPLES:

sage: from surface_dynamics.misc.group_representation import real_isotypic_projection_matrices

sage: G = AlternatingGroup(5)
sage: mats = real_isotypic_projection_matrices(G, 6)
sage: sum(mats).is_one()
True

sage: G = CyclicPermutationGroup(6)
sage: mats = real_isotypic_projection_matrices(G, 6)
sage: sum(mats).is_one()
True

sage: G = QuaternionGroup()
sage: mats = real_isotypic_projection_matrices(G, 8)
sage: sum(mats).is_one()
True

surface_dynamics.misc.group_representation.regular_conjugacy_class_matrix(cl, G)[source]#: Return the matrix of the conjugacy class cl associated to the regular representation of G.