# Standard bases of free submodules of tensor modules defined by some monoterm symmetries#

AUTHORS:

• Matthias Koeppe (2020-2022): initial version

class sage.tensor.modules.tensor_free_submodule_basis.TensorFreeSubmoduleBasis_sym(tensor_module, symbol, latex_symbol=None, indices=None, latex_indices=None, symbol_dual=None, latex_symbol_dual=None)[source]#

Bases: `Basis_abstract`

Standard basis of a free submodule of a tensor module with prescribed monoterm symmetries.

EXAMPLES:

```sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: T11 = M.tensor_module(1,1)
sage: e11 = T11.basis('e')
sage: for a in e11: a.display()
e_0⊗e^0
e_0⊗e^1
e_0⊗e^2
e_1⊗e^0
e_1⊗e^1
e_1⊗e^2
e_2⊗e^0
e_2⊗e^1
e_2⊗e^2
```
```>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> T11 = M.tensor_module(Integer(1),Integer(1))
>>> e11 = T11.basis('e')
>>> for a in e11: a.display()
e_0⊗e^0
e_0⊗e^1
e_0⊗e^2
e_1⊗e^0
e_1⊗e^1
e_1⊗e^2
e_2⊗e^0
e_2⊗e^1
e_2⊗e^2
```
keys()[source]#

Return an iterator for the keys (indices) of the family.

EXAMPLES:

```sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: T11 = M.tensor_module(1,1)
sage: e11 = T11.basis('e')
sage: list(e11.keys())
[(0, 0), (0, 1), (0, 2),
(1, 0), (1, 1), (1, 2),
(2, 0), (2, 1), (2, 2)]
```
```>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> T11 = M.tensor_module(Integer(1),Integer(1))
>>> e11 = T11.basis('e')
>>> list(e11.keys())
[(0, 0), (0, 1), (0, 2),
(1, 0), (1, 1), (1, 2),
(2, 0), (2, 1), (2, 2)]
```
values()[source]#

Return an iterator for the elements of the family.

EXAMPLES:

```sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: T11 = M.tensor_module(1,1)
sage: e11 = T11.basis('e')
sage: [b.disp() for b in e11.values()]
[e_0⊗e^0, e_0⊗e^1, e_0⊗e^2,
e_1⊗e^0, e_1⊗e^1, e_1⊗e^2,
e_2⊗e^0, e_2⊗e^1, e_2⊗e^2]
```
```>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> T11 = M.tensor_module(Integer(1),Integer(1))
>>> e11 = T11.basis('e')
>>> [b.disp() for b in e11.values()]
[e_0⊗e^0, e_0⊗e^1, e_0⊗e^2,
e_1⊗e^0, e_1⊗e^1, e_1⊗e^2,
e_2⊗e^0, e_2⊗e^1, e_2⊗e^2]
```