====== SimpleIndices ======
----

====Basics====
''SimpleIndices'' represents indices of [[SimpleTensor]] and [[TensorField]]. It inherits all properties of [[Indices]] but does not sort indices., so indices in ''SimpleIndices''  appear exactly in that order as defined by the user:
<sxh groovy; gutter: false>
def indices = '_pqrs^sp'.si
println indices
</sxh>
<sxh plain; gutter: false>
   > _pqrs^sp
</sxh>
On the other hand, since indices of different [[IndexType|types]] have different mathematical nature, [[SimpleIndices]] are sorted according to the type (preserving relative ordering of indices of same type):
<sxh groovy; gutter: false>
println '_{mn}^{\\beta\\alpha}_{ba\\alpha}'.si
</sxh>
<sxh plain; gutter: false>
   > _{mnba}^{\beta\alpha}_{\alpha}
</sxh>

All properties and methods of ''SimpleIndices'' leaves the relative ordering of indices unchanged.


''SimpleIndices'' has ''.symmetries'' property which allows to define their permutational symmetries.

====Features====
Since ''SimpleIndices'' is a subtype of [[Indices]], it inherits all its properties. However,  in contrast to [[Indices]], ''SimpleIndices'' and all its properties and methods preserve the relative ordering of indices.

Consider examples. Parse ''SimpleIndices'' from string:
<sxh groovy; gutter: true>
def indices = '^an_ab^m_pq^b'.si
println indices
</sxh>
<sxh plain; gutter: false>
   > ^{an}_{ab}^{m}_{pq}^{b}
</sxh>
Get just free indices:
<sxh groovy; gutter: true; first-line: 3>
println indices.free
</sxh>
<sxh plain; gutter: false>
   > ^{nm}_{pq}
</sxh>
Invert indices:
<sxh groovy; gutter: true; first-line: 4>
println indices.inverted
</sxh>
<sxh plain; gutter: false>
   > _{an}^{ab}_{m}^{pq}_{b}
</sxh>
Get only contravariant indices:
<sxh groovy; gutter: true; first-line: 5>
println indices.upper
</sxh>
<sxh plain; gutter: false>
   > ^{anmb}
</sxh>
Get only covariant indices:
<sxh groovy; gutter: true; first-line: 6>
println indices.lower
</sxh>
<sxh plain; gutter: false>
   > _{abpq}
</sxh>

Parse indices with multiple types:
<sxh groovy; gutter: true>
def indices = '^AB_pqrs^C_A^sp'.si
println indices
</sxh>
<sxh plain; gutter: false>
   > _{pqrs}^{spABC}_{A}
</sxh>
Get just Latin upper case:
<sxh groovy; gutter: true; first-line: 3>
println indices.getOfType(LatinUpper)
</sxh>
<sxh plain; gutter: false>
   > ^{ABC}_{A}
</sxh>
Get first index:
<sxh groovy; gutter: true; first-line: 4>
println indices[0].toStringIndex()
</sxh>
<sxh plain; gutter: false>
   > _{p}
</sxh>
Get first Latin upper index:
<sxh groovy; gutter: true; first-line: 5>
println indices[LatinUpper, 0].toStringIndex()
</sxh>
<sxh plain; gutter: false>
   > ^{A}
</sxh>
Get several indices at a time:
<sxh groovy; gutter: true; first-line: 6>
println indices[1, 3, 4]
</sxh>
<sxh plain; gutter: false>
   > _{qs}^{s}
</sxh>
Get range:
<sxh groovy; gutter: true; first-line: 7>
println indices[1..4]
</sxh>
<sxh plain; gutter: false>
   > _{qrs}^{s}
</sxh>

As one can see, all methods in the above examples return indices which have same relative ordering as in the initial object.

====Symmetries====
Symmetries can be attached to a single ''SimpleIndices'' object by using ''.symmetry'' property:
<sxh groovy; gutter: false>
def indices = '_abcdef'.si
indices.symmetries.addSymmetry([[0, 1], [2, 3]].p)
indices.symmetries.addSymmetry([[0, 3, 5]].p)
println indices.symmetries.permutationGroup.order()
</sxh>
<sxh plain; gutter: false>
   > 60
</sxh>
Symmetries of simple indices can be used in transformations like [[Symmetrize]] or other methods like [[GenerateTensor]] or [[Reduce]].

The ''.symmetries'' property return a container of [[documentation:ref:permutation| permutations]] and a corresponding [[PermutationGroup]]. The latter can be accessed by taking ''.permutationGroup'' property.

It should be noted, that when specifying symmetries of simple tensors like
<sxh groovy; gutter: true>
'f_abc'.t.indices.symmetries.addSymmetry( [1, 0].p )
</sxh>
these symmetries will be automatically attached to indices of all instances of tensor ''f_abc'', i. e. ''f_pqr'' or ''f_a^a_y'' etc.:
<sxh groovy; gutter: true;first-line: 2>
println 'f_pq^i'.t.indices.symmetries.permutationGroup
</sxh>
<sxh plain; gutter: false>
   > Group( +[[0, 1]] )
 </sxh>

On the other hand, when specifying symmetries of "detached" indices:
<sxh groovy; gutter: false>
def ind = '_abc'.si
ind.si.indices.symmetries.addSymmetry( [1, 0].p )
</sxh>
they will be attached just to the particular object (''ind'' in the above example). 

====See also====
  * Related guides: [[documentation:guide:tensors_and_indices]], [[documentation:guide:symmetries_of_tensors]]
  * Reference material: [[documentation:ref:indices]], [[documentation:ref:indextype]], [[documentation:ref:simpletensor]], [[documentation:ref:tensorfield]], [[documentation:ref:permutation]], [[documentation:ref:symmetrize]], [[documentation:ref:generatetensor]], [[documentation:ref:reduce]], [[documentation:ref:permutationgroup]]
  * JavaDocs: [[http://api.redberry.cc/redberry/1.1.9/java-api/cc/redberry/core/indices/SimpleIndices.html| SimpleIndices]]
  * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/core/src/main/java/cc/redberry/core/indices/SimpleIndices.java|SimpleIndices.java]]