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Algebra::PermutationGroup
/
Algebra::Permutation
This is the class of permutations.
The elements are assumed to be the instances of Permutation .
::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 .
::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]]
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.