Colored Permutations#

Todo

Much of the colored permutations (and element) class can be generalized to \(G \wr S_n\)

class sage.combinat.colored_permutations.ColoredPermutation(parent, colors, perm)[source]#

Bases: MultiplicativeGroupElement

A colored permutation.

colors()[source]#

Return the colors of self.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: s1,s2,t = C.gens()
sage: x = s1*s2*t
sage: x.colors()
[1, 0, 0]

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> s1,s2,t = C.gens()
>>> x = s1*s2*t
>>> x.colors()
[1, 0, 0]
has_left_descent(i)[source]#

Return True if i is a left descent of self.

Let \(p = ((s_1, \ldots s_n), \sigma)\) be a colored permutation. We say \(p\) has a left \(n\)-descent if \(s_n > 0\). If \(i < n\), then we say \(p\) has a left \(i\)-descent if either

  • \(s_i \neq 0, s_{i+1} = 0\) and \(\sigma_i < \sigma_{i+1}\) or

  • \(s_i = s_{i+1}\) and \(\sigma_i > \sigma_{i+1}\).

This notion of a left \(i\)-descent is done in order to recursively construct \(w(p) = \sigma_i w(\sigma_i^{-1} p)\), where \(w(p)\) denotes a reduced word of \(p\).

EXAMPLES:

sage: C = ColoredPermutations(2, 4)
sage: s1,s2,s3,s4 = C.gens()
sage: x = s4*s1*s2*s3*s4
sage: [x.has_left_descent(i) for i in C.index_set()]
[True, False, False, True]

sage: C = ColoredPermutations(1, 5)
sage: s1,s2,s3,s4 = C.gens()
sage: x = s4*s1*s2*s3*s4
sage: [x.has_left_descent(i) for i in C.index_set()]
[True, False, False, True]

sage: C = ColoredPermutations(3, 3)
sage: x = C([[2,1,0],[3,1,2]])
sage: [x.has_left_descent(i) for i in C.index_set()]
[False, True, False]

sage: C = ColoredPermutations(4, 4)
sage: x = C([[2,1,0,1],[3,2,4,1]])
sage: [x.has_left_descent(i) for i in C.index_set()]
[False, True, False, True]

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(2), Integer(4))
>>> s1,s2,s3,s4 = C.gens()
>>> x = s4*s1*s2*s3*s4
>>> [x.has_left_descent(i) for i in C.index_set()]
[True, False, False, True]

>>> C = ColoredPermutations(Integer(1), Integer(5))
>>> s1,s2,s3,s4 = C.gens()
>>> x = s4*s1*s2*s3*s4
>>> [x.has_left_descent(i) for i in C.index_set()]
[True, False, False, True]

>>> C = ColoredPermutations(Integer(3), Integer(3))
>>> x = C([[Integer(2),Integer(1),Integer(0)],[Integer(3),Integer(1),Integer(2)]])
>>> [x.has_left_descent(i) for i in C.index_set()]
[False, True, False]

>>> C = ColoredPermutations(Integer(4), Integer(4))
>>> x = C([[Integer(2),Integer(1),Integer(0),Integer(1)],[Integer(3),Integer(2),Integer(4),Integer(1)]])
>>> [x.has_left_descent(i) for i in C.index_set()]
[False, True, False, True]
length()[source]#

Return the length of self in generating reflections.

This is the minimal numbers of generating reflections needed to obtain self.

EXAMPLES:

sage: C = ColoredPermutations(3, 3)
sage: x = C([[2,1,0],[3,1,2]])
sage: x.length()
7

sage: C = ColoredPermutations(4, 4)
sage: x = C([[2,1,0,1],[3,2,4,1]])
sage: x.length()
12

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(3), Integer(3))
>>> x = C([[Integer(2),Integer(1),Integer(0)],[Integer(3),Integer(1),Integer(2)]])
>>> x.length()
7

>>> C = ColoredPermutations(Integer(4), Integer(4))
>>> x = C([[Integer(2),Integer(1),Integer(0),Integer(1)],[Integer(3),Integer(2),Integer(4),Integer(1)]])
>>> x.length()
12
one_line_form()[source]#

Return the one line form of self.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: s1,s2,t = C.gens()
sage: x = s1*s2*t
sage: x
[[1, 0, 0], [3, 1, 2]]
sage: x.one_line_form()
[(1, 3), (0, 1), (0, 2)]

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> s1,s2,t = C.gens()
>>> x = s1*s2*t
>>> x
[[1, 0, 0], [3, 1, 2]]
>>> x.one_line_form()
[(1, 3), (0, 1), (0, 2)]
permutation()[source]#

Return the permutation of self.

This is obtained by forgetting the colors.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: s1,s2,t = C.gens()
sage: x = s1*s2*t
sage: x.permutation()
[3, 1, 2]

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> s1,s2,t = C.gens()
>>> x = s1*s2*t
>>> x.permutation()
[3, 1, 2]
reduced_word()[source]#

Return a word in the simple reflections to obtain self.

EXAMPLES:

sage: C = ColoredPermutations(3, 3)
sage: x = C([[2,1,0],[3,1,2]])
sage: x.reduced_word()
[2, 1, 3, 2, 1, 3, 3]

sage: C = ColoredPermutations(4, 4)
sage: x = C([[2,1,0,1],[3,2,4,1]])
sage: x.reduced_word()
[2, 1, 4, 3, 2, 1, 4, 3, 2, 4, 4, 3]

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(3), Integer(3))
>>> x = C([[Integer(2),Integer(1),Integer(0)],[Integer(3),Integer(1),Integer(2)]])
>>> x.reduced_word()
[2, 1, 3, 2, 1, 3, 3]

>>> C = ColoredPermutations(Integer(4), Integer(4))
>>> x = C([[Integer(2),Integer(1),Integer(0),Integer(1)],[Integer(3),Integer(2),Integer(4),Integer(1)]])
>>> x.reduced_word()
[2, 1, 4, 3, 2, 1, 4, 3, 2, 4, 4, 3]
to_matrix()[source]#

Return a matrix of self.

The colors are mapped to roots of unity.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: s1,s2,t = C.gens()
sage: x = s1*s2*t*s2; x.one_line_form()
[(1, 2), (0, 1), (0, 3)]
sage: M = x.to_matrix(); M                                                  # needs sage.rings.number_field
[    0     1     0]
[zeta4     0     0]
[    0     0     1]

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> s1,s2,t = C.gens()
>>> x = s1*s2*t*s2; x.one_line_form()
[(1, 2), (0, 1), (0, 3)]
>>> M = x.to_matrix(); M                                                  # needs sage.rings.number_field
[    0     1     0]
[zeta4     0     0]
[    0     0     1]

The matrix multiplication is in the opposite order:

sage: M == s2.to_matrix()*t.to_matrix()*s2.to_matrix()*s1.to_matrix()       # needs sage.rings.number_field
True

>>> from sage.all import *
>>> M == s2.to_matrix()*t.to_matrix()*s2.to_matrix()*s1.to_matrix()       # needs sage.rings.number_field
True
class sage.combinat.colored_permutations.ColoredPermutations(m, n)[source]#

Bases: ShephardToddFamilyGroup

The group of \(m\)-colored permutations on \(\{1, 2, \ldots, n\}\).

Let \(S_n\) be the symmetric group on \(n\) letters and \(C_m\) be the cyclic group of order \(m\). The \(m\)-colored permutation group on \(n\) letters is given by \(P_n^m = C_m \wr S_n\). This is also the complex reflection group \(G(m, 1, n)\).

We define our multiplication by

\[((s_1, \ldots s_n), \sigma) \cdot ((t_1, \ldots, t_n), \tau) = ((s_1 t_{\sigma(1)}, \ldots, s_n t_{\sigma(n)}), \tau \sigma).\]

EXAMPLES:

sage: C = ColoredPermutations(4, 3); C
4-colored permutations of size 3
sage: s1,s2,t = C.gens()
sage: (s1, s2, t)
([[0, 0, 0], [2, 1, 3]], [[0, 0, 0], [1, 3, 2]], [[0, 0, 1], [1, 2, 3]])
sage: s1*s2
[[0, 0, 0], [3, 1, 2]]
sage: s1*s2*s1 == s2*s1*s2
True
sage: t^4 == C.one()
True
sage: s2*t*s2
[[0, 1, 0], [1, 2, 3]]

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3)); C
4-colored permutations of size 3
>>> s1,s2,t = C.gens()
>>> (s1, s2, t)
([[0, 0, 0], [2, 1, 3]], [[0, 0, 0], [1, 3, 2]], [[0, 0, 1], [1, 2, 3]])
>>> s1*s2
[[0, 0, 0], [3, 1, 2]]
>>> s1*s2*s1 == s2*s1*s2
True
>>> t**Integer(4) == C.one()
True
>>> s2*t*s2
[[0, 1, 0], [1, 2, 3]]

We can also create a colored permutation by passing an iterable consisting of tuples consisting of (color, element):

sage: x = C([(2,1), (3,3), (3,2)]); x
[[2, 3, 3], [1, 3, 2]]

>>> from sage.all import *
>>> x = C([(Integer(2),Integer(1)), (Integer(3),Integer(3)), (Integer(3),Integer(2))]); x
[[2, 3, 3], [1, 3, 2]]

or a list of colors and a permutation:

sage: C([[3,3,1], [1,3,2]])
[[3, 3, 1], [1, 3, 2]]
sage: C(([3,3,1], [1,3,2]))
[[3, 3, 1], [1, 3, 2]]

>>> from sage.all import *
>>> C([[Integer(3),Integer(3),Integer(1)], [Integer(1),Integer(3),Integer(2)]])
[[3, 3, 1], [1, 3, 2]]
>>> C(([Integer(3),Integer(3),Integer(1)], [Integer(1),Integer(3),Integer(2)]))
[[3, 3, 1], [1, 3, 2]]

There is also the natural lift from permutations:

sage: P = Permutations(3)
sage: C(P.an_element())
[[0, 0, 0], [3, 1, 2]]

>>> from sage.all import *
>>> P = Permutations(Integer(3))
>>> C(P.an_element())
[[0, 0, 0], [3, 1, 2]]

A colored permutation:

sage: C(C.an_element()) == C.an_element()
True

>>> from sage.all import *
>>> C(C.an_element()) == C.an_element()
True

REFERENCES:

class sage.combinat.colored_permutations.ShephardToddFamilyGroup(m, p, n)[source]#

Bases: UniqueRepresentation, Parent

The Shephard-Todd family complex reflection group \(G(m, p, n)\) realized as a subgroup of colored permutations.

A general complex reflection group is a subgroup of \(GL(V)\), where \(V\) is a \(\CC\) vector space, that is generated by reflections, diagonalizable matrices with at most one eigenvalue not equal to \(1\). The group of colored permutations \(G(m, 1, n)\) are the generalized permutation matrices whose entries are \(m\)-th roots of unity. For \(p | m\), the group \(G(m, p, n)\) is the index \(p\) subgroup such that the product of the entries is a \(m/p\)-th root of unity.

By the (Chevalley-)Shephard-Todd classification of irreducible finite complex reflection groups, the groups \(G(m, p, n)\) (with \(G(2, 2, 2)\) being exceptionally reducible since it is the Klein four group) form the only infinite family with an additional 34 exceptional groups \(G_k\), where \(4 \leq k \leq 37\). To avoid ambiguities, we refer to \(G(m, p, n)\) as the Shephard-Todd family complex reflection group.

INPUT:

  • m – positive integer

  • p – positive integer dividing m

  • n – positive integer

REFERENCES:

EXAMPLES:

sage: groups.misc.ShephardToddFamily(6, 1, 4)
6-colored permutations of size 4
sage: groups.misc.ShephardToddFamily(6, 2, 4)
Complex reflection group G(6, 2, 4)
sage: groups.misc.ShephardToddFamily(6, 3, 4)
Complex reflection group G(6, 3, 4)
sage: groups.misc.ShephardToddFamily(6, 6, 4)
Complex reflection group G(6, 6, 4)

>>> from sage.all import *
>>> groups.misc.ShephardToddFamily(Integer(6), Integer(1), Integer(4))
6-colored permutations of size 4
>>> groups.misc.ShephardToddFamily(Integer(6), Integer(2), Integer(4))
Complex reflection group G(6, 2, 4)
>>> groups.misc.ShephardToddFamily(Integer(6), Integer(3), Integer(4))
Complex reflection group G(6, 3, 4)
>>> groups.misc.ShephardToddFamily(Integer(6), Integer(6), Integer(4))
Complex reflection group G(6, 6, 4)
Element[source]#

alias of ColoredPermutation

as_permutation_group()[source]#

Return the permutation group corresponding to self.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.as_permutation_group()                                              # needs sage.groups
Complex reflection group G(4, 1, 3) as a permutation group

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.as_permutation_group()                                              # needs sage.groups
Complex reflection group G(4, 1, 3) as a permutation group
cardinality()[source]#

Return the cardinality of self.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.cardinality()
384
sage: C.cardinality() == 4**3 * factorial(3)
True

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.cardinality()
384
>>> C.cardinality() == Integer(4)**Integer(3) * factorial(Integer(3))
True
codegrees()[source]#

Return the codegrees of self.

Let \(G\) be a complex reflection group. The codegrees \(d_1^* \leq d_2^* \leq \cdots \leq d_{\ell}^*\) of \(G\) can be defined by:

\[\prod_{i=1}^{\ell} (q - d_i^* - 1) = \sum_{g \in G} \det(g) q^{\dim(V^g)},\]

where \(V\) is the natural complex vector space that \(G\) acts on and \(\ell\) is the rank().

If \(m = 1\), then we are in the special case of the symmetric group and the codegrees are \((n-2, n-3, \ldots 1, 0)\). Otherwise the degrees are \(((n-1)m, (n-2)m, \ldots, m, 0)\).

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.codegrees()
(8, 4, 0)
sage: S = ColoredPermutations(1, 3)
sage: S.codegrees()
(1, 0)
sage: G = groups.misc.ShephardToddFamily(6, 2, 3)
sage: G.codegrees()
(12, 6, 0)

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.codegrees()
(8, 4, 0)
>>> S = ColoredPermutations(Integer(1), Integer(3))
>>> S.codegrees()
(1, 0)
>>> G = groups.misc.ShephardToddFamily(Integer(6), Integer(2), Integer(3))
>>> G.codegrees()
(12, 6, 0)
coxeter_matrix()[source]#

Return the Coxeter matrix of self.

When the group is imprimitive and not a Coxeter group, this returns None.

EXAMPLES:

sage: C = ColoredPermutations(3, 4)
sage: C.coxeter_matrix()                                                    # needs sage.modules
[1 3 2 2]
[3 1 3 2]
[2 3 1 4]
[2 2 4 1]

sage: C = ColoredPermutations(1, 4)
sage: C.coxeter_matrix()                                                    # needs sage.modules
[1 3 2]
[3 1 3]
[2 3 1]

sage: G = groups.misc.ShephardToddFamily(2, 2, 3)
sage: G.coxeter_matrix()
[1 3 3]
[3 1 2]
[3 2 1]

sage: G = groups.misc.ShephardToddFamily(2, 2, 2)
sage: G.coxeter_matrix()
[1 2]
[2 1]

sage: G = groups.misc.ShephardToddFamily(2, 2, 1)
sage: G.coxeter_matrix()
[1]

sage: G = groups.misc.ShephardToddFamily(5, 5, 1)
sage: G.coxeter_matrix()
[]

sage: G = groups.misc.ShephardToddFamily(4, 4, 2)
sage: G.coxeter_matrix()
[1 4]
[4 1]

sage: G = groups.misc.ShephardToddFamily(7, 7, 2)
sage: G.coxeter_matrix()
[1 7]
[7 1]

sage: G = groups.misc.ShephardToddFamily(6, 3, 1)
sage: G.coxeter_matrix() is None
True
sage: G = groups.misc.ShephardToddFamily(6, 3, 4)
sage: G.coxeter_matrix() is None
True

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(3), Integer(4))
>>> C.coxeter_matrix()                                                    # needs sage.modules
[1 3 2 2]
[3 1 3 2]
[2 3 1 4]
[2 2 4 1]

>>> C = ColoredPermutations(Integer(1), Integer(4))
>>> C.coxeter_matrix()                                                    # needs sage.modules
[1 3 2]
[3 1 3]
[2 3 1]

>>> G = groups.misc.ShephardToddFamily(Integer(2), Integer(2), Integer(3))
>>> G.coxeter_matrix()
[1 3 3]
[3 1 2]
[3 2 1]

>>> G = groups.misc.ShephardToddFamily(Integer(2), Integer(2), Integer(2))
>>> G.coxeter_matrix()
[1 2]
[2 1]

>>> G = groups.misc.ShephardToddFamily(Integer(2), Integer(2), Integer(1))
>>> G.coxeter_matrix()
[1]

>>> G = groups.misc.ShephardToddFamily(Integer(5), Integer(5), Integer(1))
>>> G.coxeter_matrix()
[]

>>> G = groups.misc.ShephardToddFamily(Integer(4), Integer(4), Integer(2))
>>> G.coxeter_matrix()
[1 4]
[4 1]

>>> G = groups.misc.ShephardToddFamily(Integer(7), Integer(7), Integer(2))
>>> G.coxeter_matrix()
[1 7]
[7 1]

>>> G = groups.misc.ShephardToddFamily(Integer(6), Integer(3), Integer(1))
>>> G.coxeter_matrix() is None
True
>>> G = groups.misc.ShephardToddFamily(Integer(6), Integer(3), Integer(4))
>>> G.coxeter_matrix() is None
True
degrees()[source]#

Return the degrees of self.

The degrees of a complex reflection group are the degrees of the fundamental invariants of the ring of polynomial invariants.

If \(m = 1\), then we are in the special case of the symmetric group and the degrees are \((2, 3, \ldots, n, n+1)\). Otherwise the degrees are \((m, 2m, \ldots, nm)\).

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.degrees()
(4, 8, 12)
sage: S = ColoredPermutations(1, 3)
sage: S.degrees()
(2, 3)
sage: G = groups.misc.ShephardToddFamily(6, 2, 3)
sage: G.degrees()
(6, 9, 12)

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.degrees()
(4, 8, 12)
>>> S = ColoredPermutations(Integer(1), Integer(3))
>>> S.degrees()
(2, 3)
>>> G = groups.misc.ShephardToddFamily(Integer(6), Integer(2), Integer(3))
>>> G.degrees()
(6, 9, 12)

We now check that the product of the degrees is equal to the cardinality of self:

sage: prod(C.degrees()) == C.cardinality()
True
sage: prod(S.degrees()) == S.cardinality()
True
sage: prod(G.degrees()) == G.cardinality()
True

>>> from sage.all import *
>>> prod(C.degrees()) == C.cardinality()
True
>>> prod(S.degrees()) == S.cardinality()
True
>>> prod(G.degrees()) == G.cardinality()
True
fixed_point_polynomial(q=None)[source]#

The fixed point polynomial of self.

The fixed point polynomial \(f_G\) of a complex reflection group \(G\) is counting the dimensions of fixed points subspaces:

\[f_G(q) = \sum_{w \in W} q^{\dim V^w}.\]

Furthermore, let \(d_1, d_2, \ldots, d_{\ell}\) be the degrees of \(G\), where \(\ell\) is the rank(). Then the fixed point polynomial is given by

\[f_G(q) = \prod_{i=1}^{\ell} (q + d_i - 1).\]

INPUT:

  • q – (default: the generator of ZZ['q']) the parameter \(q\)

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.fixed_point_polynomial()
q^3 + 21*q^2 + 131*q + 231

sage: S = ColoredPermutations(1, 3)
sage: S.fixed_point_polynomial()
q^2 + 3*q + 2

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.fixed_point_polynomial()
q^3 + 21*q^2 + 131*q + 231

>>> S = ColoredPermutations(Integer(1), Integer(3))
>>> S.fixed_point_polynomial()
q^2 + 3*q + 2
gens()[source]#

Return the generators of self.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.gens()
([[0, 0, 0], [2, 1, 3]],
 [[0, 0, 0], [1, 3, 2]],
 [[0, 0, 1], [1, 2, 3]])

sage: S = SignedPermutations(4)
sage: S.gens()
([2, 1, 3, 4], [1, 3, 2, 4], [1, 2, 4, 3], [1, 2, 3, -4])

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.gens()
([[0, 0, 0], [2, 1, 3]],
 [[0, 0, 0], [1, 3, 2]],
 [[0, 0, 1], [1, 2, 3]])

>>> S = SignedPermutations(Integer(4))
>>> S.gens()
([2, 1, 3, 4], [1, 3, 2, 4], [1, 2, 4, 3], [1, 2, 3, -4])
index_set()[source]#

Return the index set of self.

EXAMPLES:

sage: C = ColoredPermutations(3, 4)
sage: C.index_set()
(1, 2, 3, 4)

sage: C = ColoredPermutations(1, 4)
sage: C.index_set()
(1, 2, 3)

sage: G = groups.misc.ShephardToddFamily(6, 6, 4)
sage: G.index_set()
(1, 2, 3, 4)

sage: G = groups.misc.ShephardToddFamily(6, 2, 4)
sage: G.index_set()
(1, 2, 3, 4, 5)

sage: G = groups.misc.ShephardToddFamily(6, 6, 1)
sage: G.index_set()
()

sage: G = groups.misc.ShephardToddFamily(6, 2, 1)
sage: G.index_set()
(1,)

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(3), Integer(4))
>>> C.index_set()
(1, 2, 3, 4)

>>> C = ColoredPermutations(Integer(1), Integer(4))
>>> C.index_set()
(1, 2, 3)

>>> G = groups.misc.ShephardToddFamily(Integer(6), Integer(6), Integer(4))
>>> G.index_set()
(1, 2, 3, 4)

>>> G = groups.misc.ShephardToddFamily(Integer(6), Integer(2), Integer(4))
>>> G.index_set()
(1, 2, 3, 4, 5)

>>> G = groups.misc.ShephardToddFamily(Integer(6), Integer(6), Integer(1))
>>> G.index_set()
()

>>> G = groups.misc.ShephardToddFamily(Integer(6), Integer(2), Integer(1))
>>> G.index_set()
(1,)
is_well_generated()[source]#

Return if self is a well-generated complex reflection group.

A complex reflection group \(G\) is well-generated if it is generated by \(\ell\) reflections. Equivalently, \(G\) is well-generated if \(d_i + d_i^* = d_{\ell}\) for all \(1 \leq i \leq \ell\).

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.is_well_generated()
True
sage: C = ColoredPermutations(2, 8)
sage: C.is_well_generated()
True
sage: C = ColoredPermutations(1, 4)
sage: C.is_well_generated()
True

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.is_well_generated()
True
>>> C = ColoredPermutations(Integer(2), Integer(8))
>>> C.is_well_generated()
True
>>> C = ColoredPermutations(Integer(1), Integer(4))
>>> C.is_well_generated()
True
matrix_group()[source]#

Return the matrix group corresponding to self.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.matrix_group()                                                      # needs sage.modules
Matrix group over Cyclotomic Field of order 4 and degree 2 with 3 generators (
[0 1 0]  [1 0 0]  [    1     0     0]
[1 0 0]  [0 0 1]  [    0     1     0]
[0 0 1], [0 1 0], [    0     0 zeta4]
)

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.matrix_group()                                                      # needs sage.modules
Matrix group over Cyclotomic Field of order 4 and degree 2 with 3 generators (
[0 1 0]  [1 0 0]  [    1     0     0]
[1 0 0]  [0 0 1]  [    0     1     0]
[0 0 1], [0 1 0], [    0     0 zeta4]
)
number_of_reflection_hyperplanes()[source]#

Return the number of reflection hyperplanes of self.

The number of reflection hyperplanes of a complex reflection group is equal to the sum of the codegrees plus the rank.

EXAMPLES:

sage: C = ColoredPermutations(1, 2)
sage: C.number_of_reflection_hyperplanes()
1
sage: C = ColoredPermutations(1, 3)
sage: C.number_of_reflection_hyperplanes()
3
sage: C = ColoredPermutations(4, 12)
sage: C.number_of_reflection_hyperplanes()
276

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(1), Integer(2))
>>> C.number_of_reflection_hyperplanes()
1
>>> C = ColoredPermutations(Integer(1), Integer(3))
>>> C.number_of_reflection_hyperplanes()
3
>>> C = ColoredPermutations(Integer(4), Integer(12))
>>> C.number_of_reflection_hyperplanes()
276
one()[source]#

Return the identity element of self.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.one()
[[0, 0, 0], [1, 2, 3]]

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.one()
[[0, 0, 0], [1, 2, 3]]
order()[source]#

Return the cardinality of self.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.cardinality()
384
sage: C.cardinality() == 4**3 * factorial(3)
True

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.cardinality()
384
>>> C.cardinality() == Integer(4)**Integer(3) * factorial(Integer(3))
True
random_element()[source]#

Return an element of self at random.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: s = C.random_element(); s  # random
[[0, 2, 1], [2, 1, 3]]
sage: s in C
True

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> s = C.random_element(); s  # random
[[0, 2, 1], [2, 1, 3]]
>>> s in C
True
rank()[source]#

Return the rank of self.

The rank of a complex reflection group is equal to the dimension of the complex vector space the group acts on.

EXAMPLES:

sage: C = ColoredPermutations(4, 12)
sage: C.rank()
12
sage: C = ColoredPermutations(7, 4)
sage: C.rank()
4
sage: C = ColoredPermutations(1, 4)
sage: C.rank()
3

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(12))
>>> C.rank()
12
>>> C = ColoredPermutations(Integer(7), Integer(4))
>>> C.rank()
4
>>> C = ColoredPermutations(Integer(1), Integer(4))
>>> C.rank()
3
simple_reflection(i)[source]#

Return the i-th simple reflection of self.

EXAMPLES:

sage: C = ColoredPermutations(4, 3)
sage: C.gens()
([[0, 0, 0], [2, 1, 3]], [[0, 0, 0], [1, 3, 2]], [[0, 0, 1], [1, 2, 3]])
sage: C.simple_reflection(2)
[[0, 0, 0], [1, 3, 2]]
sage: C.simple_reflection(3)
[[0, 0, 1], [1, 2, 3]]

sage: S = SignedPermutations(4)
sage: S.simple_reflection(1)
[2, 1, 3, 4]
sage: S.simple_reflection(4)
[1, 2, 3, -4]

sage: G = groups.misc.ShephardToddFamily(4, 2, 3)
sage: list(G.simple_reflections())
[[[0, 0, 0], [2, 1, 3]],
 [[0, 0, 0], [1, 3, 2]],
 [[0, 1, 3], [1, 3, 2]],
 [[0, 0, 2], [1, 2, 3]]]

sage: G = groups.misc.ShephardToddFamily(8, 4, 1)
sage: G.simple_reflections()
Finite family {1: [[4], [1]]}

sage: G = groups.misc.ShephardToddFamily(8, 8, 1)
sage: G.simple_reflections()
Finite family {}

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(4), Integer(3))
>>> C.gens()
([[0, 0, 0], [2, 1, 3]], [[0, 0, 0], [1, 3, 2]], [[0, 0, 1], [1, 2, 3]])
>>> C.simple_reflection(Integer(2))
[[0, 0, 0], [1, 3, 2]]
>>> C.simple_reflection(Integer(3))
[[0, 0, 1], [1, 2, 3]]

>>> S = SignedPermutations(Integer(4))
>>> S.simple_reflection(Integer(1))
[2, 1, 3, 4]
>>> S.simple_reflection(Integer(4))
[1, 2, 3, -4]

>>> G = groups.misc.ShephardToddFamily(Integer(4), Integer(2), Integer(3))
>>> list(G.simple_reflections())
[[[0, 0, 0], [2, 1, 3]],
 [[0, 0, 0], [1, 3, 2]],
 [[0, 1, 3], [1, 3, 2]],
 [[0, 0, 2], [1, 2, 3]]]

>>> G = groups.misc.ShephardToddFamily(Integer(8), Integer(4), Integer(1))
>>> G.simple_reflections()
Finite family {1: [[4], [1]]}

>>> G = groups.misc.ShephardToddFamily(Integer(8), Integer(8), Integer(1))
>>> G.simple_reflections()
Finite family {}
class sage.combinat.colored_permutations.SignedPermutation(parent, colors, perm)[source]#

Bases: ColoredPermutation

A signed permutation.

cycle_type()[source]#

Return a pair of partitions of len(self) corresponding to the signed cycle type of self.

A cycle is a tuple \(C = (c_0, \ldots, c_{k-1})\) with \(\pi(c_i) = c_{i+1}\) for \(0 \leq i < k\) and \(\pi(c_{k-1}) = c_0\). If \(C\) is a cycle, \(\overline{C} = (-c_0, \ldots, -c_{k-1})\) is also a cycle. A cycle is negative, if \(C = \overline{C}\) up to cyclic reordering. In this case, \(k\) is necessarily even and the length of \(C\) is \(k/2\). A positive cycle is a pair \(C \overline{C}\), its length is \(k\).

Let \(\alpha\) be the partition whose parts are the lengths of the positive cycles and let \(\beta\) be the partition whose parts are the lengths of the negative cycles. Then \((\alpha, \beta)\) is the cycle type of \(\pi\).

EXAMPLES:

sage: G = SignedPermutations(7)
sage: pi = G([2, -1, 4, -6, -5, -3, 7])
sage: pi.cycle_type()
([3, 1], [2, 1])

sage: G = SignedPermutations(5)
sage: all(pi.cycle_type().size() == 5 for pi in G)
True
sage: set(pi.cycle_type() for pi in G) == set(PartitionTuples(2, 5))
True

>>> from sage.all import *
>>> G = SignedPermutations(Integer(7))
>>> pi = G([Integer(2), -Integer(1), Integer(4), -Integer(6), -Integer(5), -Integer(3), Integer(7)])
>>> pi.cycle_type()
([3, 1], [2, 1])

>>> G = SignedPermutations(Integer(5))
>>> all(pi.cycle_type().size() == Integer(5) for pi in G)
True
>>> set(pi.cycle_type() for pi in G) == set(PartitionTuples(Integer(2), Integer(5)))
True
has_left_descent(i)[source]#

Return True if i is a left descent of self.

EXAMPLES:

sage: S = SignedPermutations(4)
sage: s1,s2,s3,s4 = S.gens()
sage: x = s4*s1*s2*s3*s4
sage: [x.has_left_descent(i) for i in S.index_set()]
[True, False, False, True]

>>> from sage.all import *
>>> S = SignedPermutations(Integer(4))
>>> s1,s2,s3,s4 = S.gens()
>>> x = s4*s1*s2*s3*s4
>>> [x.has_left_descent(i) for i in S.index_set()]
[True, False, False, True]
order()[source]#

Return the multiplicative order of the signed permutation.

EXAMPLES:

sage: pi = SignedPermutations(7)([2,-1,4,-6,-5,-3,7])
sage: pi.to_cycles(singletons=False)
[(1, 2, -1, -2), (3, 4, -6), (-3, -4, 6), (5, -5)]
sage: pi.order()
12

>>> from sage.all import *
>>> pi = SignedPermutations(Integer(7))([Integer(2),-Integer(1),Integer(4),-Integer(6),-Integer(5),-Integer(3),Integer(7)])
>>> pi.to_cycles(singletons=False)
[(1, 2, -1, -2), (3, 4, -6), (-3, -4, 6), (5, -5)]
>>> pi.order()
12
to_cycles(singletons=True, use_min=True, negative_cycles=True)[source]#

Return the signed permutation self as a list of disjoint cycles.

The cycles are returned in the order of increasing smallest elements, and each cycle is returned as a tuple which starts with its smallest positive element.

INPUT:

  • singletons – (default: True) whether to include singleton cycles or not

  • use_min – (default: True) if False, the cycles are returned in the order of increasing largest (not smallest) elements, and each cycle starts with its largest element

  • negative_cycles – (default: True) if False, for any two cycles \(C^{\pm} = \{\pm c_1, \ldots, \pm c_k\}\) such that \(C^+ \neq C^-\), this does not include the cycle \(C^-\)

Warning

The arugment negative_cycles does not refer to the usual definition of a negative cycle; see cycle_type().

EXAMPLES:

sage: pi = SignedPermutations(7)([2,-1,4,-6,-5,-3,7])
sage: pi.to_cycles()
[(1, 2, -1, -2), (3, 4, -6), (-3, -4, 6), (5, -5), (7,), (-7,)]
sage: pi.to_cycles(singletons=False)
[(1, 2, -1, -2), (3, 4, -6), (-3, -4, 6), (5, -5)]
sage: pi.to_cycles(negative_cycles=False)
[(1, 2, -1, -2), (3, 4, -6), (5, -5), (7,)]
sage: pi.to_cycles(singletons=False, negative_cycles=False)
[(1, 2, -1, -2), (3, 4, -6), (5, -5)]
sage: pi.to_cycles(use_min=False)
[(7,), (-7,), (6, -3, -4), (-6, 3, 4), (5, -5), (2, -1, -2, 1)]
sage: pi.to_cycles(singletons=False, use_min=False)
[(6, -3, -4), (-6, 3, 4), (5, -5), (2, -1, -2, 1)]

>>> from sage.all import *
>>> pi = SignedPermutations(Integer(7))([Integer(2),-Integer(1),Integer(4),-Integer(6),-Integer(5),-Integer(3),Integer(7)])
>>> pi.to_cycles()
[(1, 2, -1, -2), (3, 4, -6), (-3, -4, 6), (5, -5), (7,), (-7,)]
>>> pi.to_cycles(singletons=False)
[(1, 2, -1, -2), (3, 4, -6), (-3, -4, 6), (5, -5)]
>>> pi.to_cycles(negative_cycles=False)
[(1, 2, -1, -2), (3, 4, -6), (5, -5), (7,)]
>>> pi.to_cycles(singletons=False, negative_cycles=False)
[(1, 2, -1, -2), (3, 4, -6), (5, -5)]
>>> pi.to_cycles(use_min=False)
[(7,), (-7,), (6, -3, -4), (-6, 3, 4), (5, -5), (2, -1, -2, 1)]
>>> pi.to_cycles(singletons=False, use_min=False)
[(6, -3, -4), (-6, 3, 4), (5, -5), (2, -1, -2, 1)]
to_matrix()[source]#

Return a matrix of self.

EXAMPLES:

sage: S = SignedPermutations(4)
sage: s1,s2,s3,s4 = S.gens()
sage: x = s4*s1*s2*s3*s4
sage: M = x.to_matrix(); M                                                  # needs sage.modules
[ 0  1  0  0]
[ 0  0  1  0]
[ 0  0  0 -1]
[-1  0  0  0]

>>> from sage.all import *
>>> S = SignedPermutations(Integer(4))
>>> s1,s2,s3,s4 = S.gens()
>>> x = s4*s1*s2*s3*s4
>>> M = x.to_matrix(); M                                                  # needs sage.modules
[ 0  1  0  0]
[ 0  0  1  0]
[ 0  0  0 -1]
[-1  0  0  0]

The matrix multiplication is in the opposite order:

sage: m1,m2,m3,m4 = [g.to_matrix() for g in S.gens()]                       # needs sage.modules
sage: M == m4 * m3 * m2 * m1 * m4                                           # needs sage.modules
True

>>> from sage.all import *
>>> m1,m2,m3,m4 = [g.to_matrix() for g in S.gens()]                       # needs sage.modules
>>> M == m4 * m3 * m2 * m1 * m4                                           # needs sage.modules
True
class sage.combinat.colored_permutations.SignedPermutations(n)[source]#

Bases: ColoredPermutations

Group of signed permutations.

The group of signed permutations is also known as the hyperoctahedral group, the Coxeter group of type \(B_n\), and the 2-colored permutation group. Thus it can be constructed as the wreath product \(S_2 \wr S_n\).

EXAMPLES:

sage: S = SignedPermutations(4)
sage: s1,s2,s3,s4 = S.group_generators()
sage: x = s4*s1*s2*s3*s4; x
[-4, 1, 2, -3]
sage: x^4 == S.one()
True

>>> from sage.all import *
>>> S = SignedPermutations(Integer(4))
>>> s1,s2,s3,s4 = S.group_generators()
>>> x = s4*s1*s2*s3*s4; x
[-4, 1, 2, -3]
>>> x**Integer(4) == S.one()
True

This is a finite Coxeter group of type \(B_n\):

sage: S.canonical_representation()                                              # needs sage.modules
Finite Coxeter group over Number Field in a with defining polynomial x^2 - 2
 with a = 1.414213562373095? with Coxeter matrix:
[1 3 2 2]
[3 1 3 2]
[2 3 1 4]
[2 2 4 1]
sage: S.long_element()
[-1, -2, -3, -4]
sage: S.long_element().reduced_word()
[1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 4, 3, 4]

>>> from sage.all import *
>>> S.canonical_representation()                                              # needs sage.modules
Finite Coxeter group over Number Field in a with defining polynomial x^2 - 2
 with a = 1.414213562373095? with Coxeter matrix:
[1 3 2 2]
[3 1 3 2]
[2 3 1 4]
[2 2 4 1]
>>> S.long_element()
[-1, -2, -3, -4]
>>> S.long_element().reduced_word()
[1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 4, 3, 4]

We can also go between the 2-colored permutation group:

sage: C = ColoredPermutations(2, 3)
sage: S = SignedPermutations(3)
sage: S.an_element()
[-3, 1, 2]
sage: C(S.an_element())
[[1, 0, 0], [3, 1, 2]]
sage: S(C(S.an_element())) == S.an_element()
True
sage: S(C.an_element())
[-3, 1, 2]

>>> from sage.all import *
>>> C = ColoredPermutations(Integer(2), Integer(3))
>>> S = SignedPermutations(Integer(3))
>>> S.an_element()
[-3, 1, 2]
>>> C(S.an_element())
[[1, 0, 0], [3, 1, 2]]
>>> S(C(S.an_element())) == S.an_element()
True
>>> S(C.an_element())
[-3, 1, 2]

There is also the natural lift from permutations:

sage: P = Permutations(3)
sage: x = S(P.an_element()); x
[3, 1, 2]
sage: x.parent()
Signed permutations of 3

>>> from sage.all import *
>>> P = Permutations(Integer(3))
>>> x = S(P.an_element()); x
[3, 1, 2]
>>> x.parent()
Signed permutations of 3

REFERENCES:

Element[source]#

alias of SignedPermutation

conjugacy_class_representative(nu)[source]#

Return a permutation with (signed) cycle type nu.

EXAMPLES:

sage: G = SignedPermutations(4)
sage: for nu in PartitionTuples(2, 4):
....:     print(nu, G.conjugacy_class_representative(nu))
....:     assert nu == G.conjugacy_class_representative(nu).cycle_type(), nu
([4], []) [2, 3, 4, 1]
([3, 1], []) [2, 3, 1, 4]
([2, 2], []) [2, 1, 4, 3]
([2, 1, 1], []) [2, 1, 3, 4]
([1, 1, 1, 1], []) [1, 2, 3, 4]
([3], [1]) [2, 3, 1, -4]
([2, 1], [1]) [2, 1, 3, -4]
([1, 1, 1], [1]) [1, 2, 3, -4]
([2], [2]) [2, 1, 4, -3]
([2], [1, 1]) [2, 1, -3, -4]
([1, 1], [2]) [1, 2, 4, -3]
([1, 1], [1, 1]) [1, 2, -3, -4]
([1], [3]) [1, 3, 4, -2]
([1], [2, 1]) [1, 3, -2, -4]
([1], [1, 1, 1]) [1, -2, -3, -4]
([], [4]) [2, 3, 4, -1]
([], [3, 1]) [2, 3, -1, -4]
([], [2, 2]) [2, -1, 4, -3]
([], [2, 1, 1]) [2, -1, -3, -4]
([], [1, 1, 1, 1]) [-1, -2, -3, -4]

>>> from sage.all import *
>>> G = SignedPermutations(Integer(4))
>>> for nu in PartitionTuples(Integer(2), Integer(4)):
...     print(nu, G.conjugacy_class_representative(nu))
...     assert nu == G.conjugacy_class_representative(nu).cycle_type(), nu
([4], []) [2, 3, 4, 1]
([3, 1], []) [2, 3, 1, 4]
([2, 2], []) [2, 1, 4, 3]
([2, 1, 1], []) [2, 1, 3, 4]
([1, 1, 1, 1], []) [1, 2, 3, 4]
([3], [1]) [2, 3, 1, -4]
([2, 1], [1]) [2, 1, 3, -4]
([1, 1, 1], [1]) [1, 2, 3, -4]
([2], [2]) [2, 1, 4, -3]
([2], [1, 1]) [2, 1, -3, -4]
([1, 1], [2]) [1, 2, 4, -3]
([1, 1], [1, 1]) [1, 2, -3, -4]
([1], [3]) [1, 3, 4, -2]
([1], [2, 1]) [1, 3, -2, -4]
([1], [1, 1, 1]) [1, -2, -3, -4]
([], [4]) [2, 3, 4, -1]
([], [3, 1]) [2, 3, -1, -4]
([], [2, 2]) [2, -1, 4, -3]
([], [2, 1, 1]) [2, -1, -3, -4]
([], [1, 1, 1, 1]) [-1, -2, -3, -4]
long_element(index_set=None)[source]#

Return the longest element of self, or of the parabolic subgroup corresponding to the given index_set.

INPUT:

  • index_set – (optional) a subset (as a list or iterable) of the nodes of the indexing set

EXAMPLES:

sage: S = SignedPermutations(4)
sage: S.long_element()
[-1, -2, -3, -4]

>>> from sage.all import *
>>> S = SignedPermutations(Integer(4))
>>> S.long_element()
[-1, -2, -3, -4]
one()[source]#

Return the identity element of self.

EXAMPLES:

sage: S = SignedPermutations(4)
sage: S.one()
[1, 2, 3, 4]

>>> from sage.all import *
>>> S = SignedPermutations(Integer(4))
>>> S.one()
[1, 2, 3, 4]
random_element()[source]#

Return an element drawn uniformly at random.

EXAMPLES:

sage: C = SignedPermutations(7)
sage: s = C.random_element(); s # random
[7, 6, -4, -5, 2, 3, -1]
sage: s in C
True

>>> from sage.all import *
>>> C = SignedPermutations(Integer(7))
>>> s = C.random_element(); s # random
[7, 6, -4, -5, 2, 3, -1]
>>> s in C
True
simple_reflection(i)[source]#

Return the i-th simple reflection of self.

EXAMPLES:

sage: S = SignedPermutations(4)
sage: S.simple_reflection(1)
[2, 1, 3, 4]
sage: S.simple_reflection(4)
[1, 2, 3, -4]

>>> from sage.all import *
>>> S = SignedPermutations(Integer(4))
>>> S.simple_reflection(Integer(1))
[2, 1, 3, 4]
>>> S.simple_reflection(Integer(4))
[1, 2, 3, -4]