Action of a sympy TensorSymmetry

[Matthias Köppe]

SymPy's TensorSymmetry (​https://docs.sympy.org/latest/modules/tensor/tensor.html) uses a trick to represent (monoterm) tensor symmetries and antisymmetries as permutation groups: Two extra elements (in the example below, 3 and 4) keep track of the sign

sage: TensorSymmetry.fully_symmetric(3)                                                                                               
TensorSymmetry((0, 1), ((4)(0 1), (4)(1 2)))
sage: TensorSymmetry.fully_symmetric(-3)    # fully antisymmetric 3 indices                                                                                      
TensorSymmetry((0, 1), ((0 1)(3 4), (1 2)(3 4)))

The 2-cycle (3 4) represents the antisymmetry. What is displayed there is the "base and strong generating system".

This trick can of course be generalized to Mathematica's "phased permutation groups" (#30276) by using longer cycles.

We define a class TensorSymmetryGroup

• which can be initialized from sage.tensor sym and antisym lists
• can convert to/from sympy.tensor.tensor.TensorSymmetry
• defines _get_action_ for acting on tensor modules / components

[Matthias Köppe]

I'm thinking of putting this in sage.groups.tensor.monoterm_symmetry

[Dima Pasechnik]

the symmetry group there is an arbitrary permutation group, no? So sympy has an algorithm to find the autmorphism group of a weighted hypergraph?

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New commits:

​ad4f3f1

src/sage/groups/tensor/monoterm_symmetry.py: New

[Matthias Köppe]

Replying to dimpase:

the symmetry group there is an arbitrary permutation group, no? So sympy has an algorithm to find the autmorphism group of a weighted hypergraph?

The automorphism group is the input, not the output. This code is for defining spaces of tensors with prescribed symmetries.

[Dima Pasechnik]

Your input is a bunch of permutations, not a group. Why do you force the user to figure them out?

I'd rather see it as a group, and yes, it's perfectly possible in Sage to compute a strong generating set and a base, given a permutation group. E.g. one can use the corresponding GAP ​functionality via libgap.

[Matthias Köppe]

Replying to dimpase:

Your input is a bunch of permutations, not a group. Why do you force the user to figure them out?

Sorry, what are you referring to?

The example lines on the ticket description using SymPy show the output of SymPy's TensorSymmetry constructor, which happens to show the BSGS.

In the branch attached here, the input is a symmetry specification in the style of sage.tensor.

[Dima Pasechnik]

All I can see is an example with sym=(1, 2) - it's totally unclear what kind of input is expected, due to lack of any documentation.

[Matthias Köppe]

OK. I will adapt documentation from sage.tensor.modules.comp, where this input is documented.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​3629386

TensorSymmetryGroup: Adapt some documentation from sage.tensor.modules.comp

[Dima Pasechnik]

+        * ``sym = [(0,2), (1,3,4)]`` for a symmetry between the 1st and 3rd
+          indices and a symmetry between the 2nd, 4th and 5th indices.

is ambiguous. E.g. sym = [(0,2), (1,3)] might either mean one order 2 symmetry (i.e. group of symmetries of order 2), or a list of generators for a group of order 4. In the former can one should say something like a symmetry which acts as a permutation (13)(24). In the latter, say they generate a group...