Differences

This shows you the differences between two versions of the page.

@@ Line 1: / Line 1: @@
+====== Permutations and permutation groups ======
+<html>
+<div class="text-right" style="font-size: 15px; ">
+</html>
+Next topic: [[documentation:guide:programming_with_redberry]]
+<html>
+<span class="glyphicon glyphicon-arrow-right"></span>
+</div>
+</html>
+----
+====Permutations====
+Permutations play a very important role in tensor algebra. For example, [[Symmetries of tensors|symmetries of tensors]] are specified in terms of permutations of indices. Besides, permutations and permutation groups are used in many routines and algorithms with tensors (e.g. [[documentation:ref:Symmetrize]] transformation or [[mappings of indices]] are uses algorithms with  permutation groups).
+A single [[documentation:ref:Permutation]] in Redberry can be inputted using ''.p'' property both in cycle and one-line notation:
+<sxh groovy; gutter: false>
+//permutation in one-line notation
+def p1 = [0, 2, 5, 6, 7, 1, 3, 4].p
+//same permutation in cycle notation
+def p2 = [[1, 2, 5], [4, 7], [3, 6]].p
+assert p1 == p2
+</sxh>
+[[documentation:ref:Permutation]] can represent both permutational symmetry and antisymmetry. In order to convert symmetry to antisymmetry and vice versa, one can use minus:
+<sxh groovy; gutter: false>
+//symmetry
+def sym = [[0, 2, 5], [6, 7]].p
+//antisymmetry
+def asym = -sym
+</sxh>
+One can use [[documentation:ref:Permutation]] to permute a list of objects via ''%%>>%%'' operator:
+<sxh groovy; gutter: true>
+def p = [[0, 1], [2, 3]].p
+println p >> [10, 9, 8, 7]
+</sxh>
+<sxh plain; gutter: false>
+   > [9, 10, 7, 8]
+</sxh>
+<sxh groovy; gutter: true; first-line: 3>
+println p >> ['a', 'b', 'c', 'd', 'e']
+</sxh>
+<sxh plain; gutter: false>
+   > [b, a, d, c, e]
+</sxh>
+In order to make a composition of permutations one can simply use multiplication:
+<sxh groovy; gutter: true>
+def perm1 = [[0, 5, 4], [1, 3]].p
+def perm2 = [[0, 1], [2, 3]].p
+println perm1*perm2
+</sxh>
+<sxh plain; gutter: false>
+   > +[[0, 5, 4, 1, 2, 3]]
+</sxh>
+<sxh groovy; gutter: true; first-line: 4>
+println perm2*perm1
+</sxh>
+<sxh plain; gutter: false>
+   > +[[0, 3, 2, 1, 5, 4]]
+</sxh>
+The result of applying composition ''perm1*perm2'' is equivalent to applying ''perm2'' then ''perm1'':
+<sxh groovy; gutter: true; first-line: 5>
+def list = ['a', 'b', 'c', 'd', 'e', 'f']
+assert (perm1 * perm2) >> list == perm1 >> (perm2 >> list)
+assert (perm2 * perm1) >> list == perm2 >> (perm1 >> list)
+</sxh>
+See [[documentation:ref:Permutation]] for further details.
+====Permutation groups====
+In order to [[Symmetries of tensors|specify symmetries of tensors]] one should specify a generating set of permutations. Redberry uses generating set to create a [[documentation:ref:PermutationGroup]] which is used in all further routines. Redberry can handle permutation groups  with degree in the range of a few thousand, hence working with groups with more than $10^{1000}$  elements.
+ In order to create [[documentation:ref:PermutationGroup]] directly, one can do:
+<sxh groovy; gutter: true>
+def gen1 = [[1, 4, 5], [3, 6]].p
+def gen2 = -[[2, 3], [1, 5]].p
+def group = Group(gen1, gen2)
+</sxh>
+Giving a [[documentation:ref:PermutationGroup]], one can e.g. check its order (i.e. number of all elements in group) or enumerate all its element:
+<sxh groovy; gutter: true; first-line: 4>
+//print number of all group elements
+println group.order()
+</sxh>
+<sxh plain; gutter: false>
+   > 36
+</sxh>
+<sxh groovy; gutter: true; first-line: 6>
+//enumerate all elements
+def all = []
+for (def perm in group)
+    all << perm
+//find all antisymmetries
+def assyms = group.findAll { perm -> perm.antisymmetry() }
+println assyms
+</sxh>
+<sxh plain; gutter: false>
+   > [-[[4,5]], -[[3,6],[4,5]], -[[2,3],[4,5]], -[[2,6,3],[4,5]], -[[2,6],[4,5]],
+          -[[2,3,6],[4,5]], -[[1,4],[3,6]], -[[1,4]], -[[1,4],[2,3,6]], -[[1,4],[2,6]],
+          -[[1,4],[2,6,3]], -[[1,4],[2,3]], -[[1,5],[2,3]], -[[1,5],[2,6,3]], -[[1,5]],
+          -[[1,5],[3,6]], -[[1,5],[2,3,6]], -[[1,5],[2,6]]]
+</sxh>
+[[documentation:ref:PermutationGroup]] provides a wide range of specialized methods including membership testing, coset enumeration, searching for centralizers, stabilizers, etc. For example, to check whether some permutation belongs to a group one can do
+<sxh groovy; gutter: true; first-line: 13>
+def perm1 = [[1, 2, 5]].p
+println group.membershipTest(perm1)
+</sxh>
+<sxh plain; gutter: false>
+   > false
+</sxh>
+<sxh groovy; gutter: true; first-line: 15>
+def perm2 = [[1, 4, 5]].p
+println group.membershipTest(perm2)
+</sxh>
+<sxh plain; gutter: false>
+   > true
+</sxh>
+To find a subgroup that stabilizes a set of points (i.e. map set on itself "as a whole") one can do:
+<sxh groovy; gutter: true; first-line: 17>
+println group.setwiseStabilizer(1, 3, 5)
+</sxh>
+<sxh plain; gutter: false>
+   > Group( +[[2, 6]], -[[1, 5]] )
+</sxh>
+See [[documentation:ref:PermutationGroup]]  for further details.
+====See also====
+  * Related guides: [[documentation:guide:symmetries_of_tensors]]
+  * Related reference material: [[documentation:ref:permutation]], [[documentation:ref:permutationgroup]]