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[index] Algebra::PermutationGroup / Algebra::Permutation

Algebra::PermutationGroup

This is the class of permutations. The elements are assumed to be the instances of Permutation.

File Name:

permutation-group.rb

SuperClass:

Group

Class Methods:

::new(u, [g0, [g1, ...]]): Returns the group with unit u, whcih consists of g0, g1, ....
::unit_group(d): Return the unit group of degree d.
::unity(n): Retunrs the unity of degree n.
::perm(a): Returns the permuation represented by the array a.
::symmetric(n): Returns the simmetric group of degree n
::alternate(n): Returns the alternative group of dgree n.

Algebra::Permutation

File Name:

permutation-group.rb

SuperClass:

Object

Included Module:

Enumerable
Powers

Class Methods:

::new(x)

Returns the permutaiont represented by the array x.

::[[n0, [n1, [n2, ..., ]]]]

Returns the permutation [n0, n1, n2, ..., ].

Example:

a = Permutation[1, 2, 0]
p a**2 #=> [2, 0, 1]
p a**3 #=> [0, 1, 2]

::unity(d)

Returns the unity of degree d.

::cyclic2perm(c, n)

Returns the Permutation represented by c : the array of arrays of cyclic permutations, where n is the degree. This method is the inverse of decompose_cyclic.

Example:

Permutation.cyclic2perm([[1,6,5,4], [2,3]], 7) #=> [0, 6, 3, 2, 1, 4, 5]
Permutation[0, 6, 3, 2, 1, 4, 5].decompose_cyclic #=> [[1,6,5,4], [2,3]]

Methods:

unity: Returns the unity.
perm: Returns the array which represents self
degree: Returns the degree
size: Alias of degree.
each: Iterates for each entry.
eql?(other): Returns true if self is equal to other.
==: Alias of eql?.
hash: Returns the hash number.
[i]: Returns the number to which i is transferrd.
call: Alias of [].
index(i): Returns the number from which i is transferred.
right_act(other): Returns the value multiplied by other from right. It follows (g.right_act(h))[x] == h[g[x]].
*: Alias of right_act
left_act(other): Returns the value multiplied by other from left. It follows (g.left_act(h))[x] == g[h[x]].
inverse: Returns the inverse element.
inv: Alias of inverse.
sign: Returns the sign of self.
conjugate(g): Returns the conjugate by g: g * self * g.inv.
decompose_cyclic: Returns the array of arrays of cyclic permutations. This is the inverse of ::cyclic2perm(c, n).
to_map: Returns the Map object of self.
decompose_transposition: Decompose into the array of the transpositions.