sage.tensor.modules: Add SymPowerFreeModule

[mkoeppe]

FiniteRankFreeModule defines a method exterior_power, creating an instance of ExtPowerFreeModule.

We propose to add a method symmetric_power and class SymPowerFreeModule. (More generally, we could have submodules of tensors with prescribed symmetries; this is currently only available on the level of elements, not parents.)

They will be submodules of TensorFreeModules. (#30169 makes ExtPowerFreeModules quotients.)

The existing method FiniteRankFreeModule.sym_bilinear_form might be changed to produce elements of it for p=2.

Depends on #30229

CC: @egourgoulhon @dimpase @yuan-zhou @tscrim @mjungmath

Component: linear algebra

Reviewer: Dima Pasechnik

Issue created by migration from https://trac.sagemath.org/ticket/30095

[mkoeppe]

Description changed:

--- 
+++ 
@@ -2,6 +2,6 @@
 
 We propose to add a method `symmetric_power` and class `SymPowerFreeModule`.
 
-The existing method `ExtPowerFreeModule.sym_bilinear_form` might be changed to produce elements of it for p=2.
+The existing method `FiniteRankFreeModule.sym_bilinear_form` might be changed to produce elements of it for p=2.
 
 

[mkoeppe]

Description changed:

--- 
+++ 
@@ -2,6 +2,8 @@
 
 We propose to add a method `symmetric_power` and class `SymPowerFreeModule`.
 
+They will be submodules of `TensorFreeModule`s. (#30169 makes `ExtPowerFreeModule`s quotients.)
+
 The existing method `FiniteRankFreeModule.sym_bilinear_form` might be changed to produce elements of it for p=2.
 
 

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1,6 +1,6 @@
 `FiniteRankFreeModule` defines a method `exterior_power`, creating an instance of `ExtPowerFreeModule`.
 
-We propose to add a method `symmetric_power` and class `SymPowerFreeModule`.
+We propose to add a method `symmetric_power` and class `SymPowerFreeModule`. (More generally, we could have submodules of tensors with prescribed symmetries; this is currently only available on the level of elements, not parents.)
 
 They will be submodules of `TensorFreeModule`s. (#30169 makes `ExtPowerFreeModule`s quotients.)
 

[mkoeppe]

Dependencies: #30229

[mjungmath]

comment:7

I think one has to be very careful here, especially with the last proposal. If the base field has not characteristic zero, as described also on Wikipedia, a symmetric bilinear form cannot be identified with an element of the symmetric algebra.

Addendum: I just noticed, we are not even in the manifold setting.

[mkoeppe]

comment:8

Thanks. Yes, we'll have to be careful

[mkoeppe]

comment:10

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[mkoeppe]

comment:14

Done as part of #30229

[dimpase]

Reviewer: Dima Pasechnik

[fchapoton]

comment:16

closing as duplicate, to test the new trac machine