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# SAGE’s To Do list

There is still some work to do :-) :

Warning

This list is currently very incomplete as most doctests do not use the `.. todo::` markup.

Todo

Rewrite the hand-written TODOs by using the correct `.. todo::` markup.

The combined to do list is only available in the html version of the reference manual.

Todo

Rewrite the hand-written TODOs by using the correct `.. todo::` markup.

(The original entry is located in /build/sage-mathematics-doc/src/sage-6.4.1/src/doc/en/reference/todolist.rst, line 13.)

Todo

Deprecate this in favor of a method called \(center()\) once subalgebras are properly implemented in Sage.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/clifford_algebra.py:docstring of sage.algebras.clifford_algebra.CliffordAlgebra.center_basis, line 12.)

Todo

Deprecate this in favor of a method called \(supercenter()\) once subalgebras are properly implemented in Sage.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/clifford_algebra.py:docstring of sage.algebras.clifford_algebra.CliffordAlgebra.supercenter_basis, line 12.)

Todo

Add a category for Hopf superalgebras (perhaps part of trac ticket #16513).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/clifford_algebra.py:docstring of sage.algebras.clifford_algebra.ExteriorAlgebra, line 42.)

Todo

Define a similar method for general Clifford algebras once the morphism to exterior algebras is implemented.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/clifford_algebra.py:docstring of sage.algebras.clifford_algebra.ExteriorAlgebra.Element.constant_coefficient, line 3.)

Todo

Define a similar method for general Clifford algebras once the morphism to exterior algebras is implemented.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/clifford_algebra.py:docstring of sage.algebras.clifford_algebra.ExteriorAlgebra.Element.scalar, line 10.)

Todo

Implement a class for bilinear forms and rewrite this method to use that class.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/clifford_algebra.py:docstring of sage.algebras.clifford_algebra.ExteriorAlgebra.lifted_bilinear_form, line 53.)

Todo

Another way to compute this bilinear form seems to be to map \(x\) and \(y\) to the appropriate Clifford algebra and there compute \(x^t y\), then send the result back to the exterior algebra and return its constant coefficient. Or something like this. Once the maps to the Clifford and back are implemented, check if this is faster.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/clifford_algebra.py:docstring of sage.algebras.clifford_algebra.ExteriorAlgebra.lifted_bilinear_form, line 108.)

Todo

An example illustrating unitary flag.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/finite_dimensional_algebras/finite_dimensional_algebra_morphism.py:docstring of sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_morphism.FiniteDimensionalAlgebraMorphism, line 35.)

Todo

- Coercion doesn’t work yet, there is some cheating about assumptions
- The optional argument `check` controls checking the degeneracy conditions. Furthermore, the default values interfere with non-degeneracy conditions.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/free_algebra.py:docstring of sage.algebras.free_algebra.FreeAlgebra_generic.g_algebra, line 4.)

Todo

Implement multi-parameter Iwahori-Hecke algebras together with their Kazhdan-Lusztig bases. That is, Iwahori-Hecke algebras with (possibly) different parameters for each conjugacy class of simple reflections in the underlying Coxeter group.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/iwahori_hecke_algebra.py:docstring of sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra, line 305.)

Todo

When given “generic parameters” we should return the generic Iwahori-Hecke algebra with these parameters and allow the user to work inside this algebra rather than doing calculations behind the scenes in a copy of the generic Iwahori-Hecke algebra. The main problem is that it is not clear how to recognise when the parameters are “generic”.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/iwahori_hecke_algebra.py:docstring of sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra, line 312.)

Todo

- “filtered” DFTs
- more idfts
- more examples for probability, stats, theory of FTs

(The original entry is located in /usr/lib/python2.7/site-packages/sage/gsl/dft.py:docstring of sage.gsl.dft, line 52.)

Todo

Read the parent of the elements of S; if \(\QQ\) or \(\CC\) leave as is; if AbelianGroup, use abelian_group_dual; if some other implemented Group (permutation, matrix), call .characters() and test if the index list is the set of conjugacy classes.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/gsl/dft.py:docstring of sage.gsl.dft.IndexedSequence.dft, line 41.)

Todo

Add an example.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/additive_magmas.py:docstring of sage.categories.additive_magmas.AdditiveMagmas.ParentMethods.summation, line 34.)

Todo

Add an example.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/additive_magmas.py:docstring of sage.categories.additive_magmas.AdditiveMagmas.ParentMethods.summation_from_element_class_add, line 34.)

Todo

This could be deduced from `AdditiveSemigroups.Homsets.extra_super_categories()`. See comment in `Objects.SubcategoryMethods.Homsets()`.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/additive_monoids.py:docstring of sage.categories.additive_monoids.AdditiveMonoids.Homsets.extra_super_categories, line 12.)

Todo

add a description of this category

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/affine_weyl_groups.py:docstring of sage.categories.affine_weyl_groups.AffineWeylGroups, line 3.)

Todo

should return an enumerated set, with iterator, ...

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/affine_weyl_groups.py:docstring of sage.categories.affine_weyl_groups.AffineWeylGroups.ParentMethods.affine_grassmannian_elements_of_given_length, line 13.)

Todo

Should this category be a `CategoryWithAxiom`?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/affine_weyl_groups.py:docstring of sage.categories.affine_weyl_groups.AffineWeylGroups.additional_structure, line 9.)

Todo

- Add support for non commutative rings (this is currently not supported by the subcategory `AlgebraModules`).
- Make `AlgebraIdeals(R)`, return `CommutativeAlgebraIdeals(R)` when `R` is commutative.
- If useful, implement `AlgebraLeftIdeals` and `AlgebraRightIdeals` of which `AlgebraIdeals` would be a subcategory.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/algebra_ideals.py:docstring of sage.categories.algebra_ideals.AlgebraIdeals, line 8.)

Todo

Should \(R\) be a commutative ring?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/algebras.py:docstring of sage.categories.algebras.Algebras, line 14.)

Todo

This category should be a `CategoryWithAxiom`.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/bialgebras.py:docstring of sage.categories.bialgebras.Bialgebras.additional_structure, line 9.)

Todo

Should this category be a `CategoryWithAxiom`?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/bimodules.py:docstring of sage.categories.bimodules.Bimodules.additional_structure, line 9.)

Todo

Improve this explanation.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category.py:docstring of sage.categories.category.Category._without_axioms, line 8.)

Todo

Add an optional argument to allow for:

    sage: Realizations(A, category = Blahs()) # todo: not implemented

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category.py:docstring of sage.categories.category.Category.Realizations, line 39.)

Todo

The latter is a consequence of `EuclideanDomains` currently being a structure category. Is this what we want?

    sage: EuclideanDomains().is_full_subcategory(Rings())
    False

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category.py:docstring of sage.categories.category.Category.is_full_subcategory, line 47.)

Todo

Further remove the base ring (see also trac ticket #15801).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category.py:docstring of sage.categories.category.category_graph, line 15.)

Todo

Specify whether or not one should systematically use @cached_method in the definition of the axiom. And make sure all the definition of axioms in Sage are consistent in this respect!

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category_with_axiom.py:docstring of sage.categories.category_with_axiom, line 376.)

Todo

We could possibly define an @axiom decorator? This could hide two little implementation details: whether or not to make the method a cached method, and the call to _with_axiom(...) under the hood. It could do possibly do some more magic. The gain is not obvious though.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category_with_axiom.py:docstring of sage.categories.category_with_axiom, line 382.)

Todo

Explore ways to get rid of this global `all_axioms` tuple, and/or have automatic registration there, and/or having a register_axiom(...) method.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category_with_axiom.py:docstring of sage.categories.category_with_axiom, line 421.)

Todo

Other categories that would be better implemented via an axiom depending on a join category include:

- `Algebras`: defining an associative unital algebra as a ring and a module satisfying the suitable compatibility axiom between inner multiplication and multiplication by scalars (bilinearity). Of course this should be implemented at the level of `MagmaticAlgebras`, if not higher.
- `Bialgebras`: defining an bialgebra as an algebra and coalgebra where the coproduct is a morphism for the product.
- `Bimodules`: defining a bimodule as a left and right module where the two actions commute.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category_with_axiom.py:docstring of sage.categories.category_with_axiom, line 488.)

Todo

- Design and implement an idiom for the definition of an axiom by a join category.
- Or support more advanced joins, through some hook or registration process to specify that a given category is the intersection of two (or more) categories.
- Or at least improve the above workaround to avoid the last issue; this possibly could be achieved using a class `Magmas.Distributive` with a bit of `__classcall__` magic.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category_with_axiom.py:docstring of sage.categories.category_with_axiom, line 505.)

Todo

The above example violates the specifications (a category should be modelled by at most one class), so it’s appropriate that it fails. Yet, the error message could be usefully complemented by some hint at what the source of the problem is (a category implemented in two distinct classes). Leaving a large enough piece of the backtrace would be useful though, so that one can explore where the issue comes from (e.g. with post mortem debugging).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category_with_axiom.py:docstring of sage.categories.category_with_axiom, line 1095.)

Todo

The following specifications would be desirable but are not yet implemented:

- A functorial construction category (Graded, CartesianProducts, ...) having a `Category_singleton` as base category should be a `CategoryWithAxiom_singleton`.

  Nothing difficult to implement, but this will need to rework the current “no subclass of a concrete class” assertion test of `Category_singleton.__classcall__()`.

- Similarly, a covariant functorial construction category having a `Category_over_base_ring` as base category should be a `Category_over_base_ring`.

The following specification might be desirable, or not:

- A join category involving a `Category_over_base_ring` should be a `Category_over_base_ring`. In the mean time, a `base_ring` method is automatically provided for most of those by `Modules.SubcategoryMethods.base_ring()`.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category_with_axiom.py:docstring of sage.categories.category_with_axiom, line 1257.)

Todo

Detail this a bit. What could typically go wrong is a situation where, for some category `C1`, `C1.A()` specifies a category `C2` as super category such that `C2.A()` specifies `C3` as super category such that ...; this would clearly cause an infinite execution. Note that this situation violates the specifications since `C1.A()` is supposed to be a subcategory of `C2.A()`, ... so we would have an infinite increasing chain of constructible categories.

It’s reasonnable to assume that there is a finite number of axioms defined in the code. There remains to use this assumption to argue that any infinite execution of the algorithm would give rise to such an infinite sequence.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category_with_axiom.py:docstring of sage.categories.category_with_axiom, line 1412.)

Todo

Decide whether we care about this feature. In such a situation, we are not really defining a new axiom, but just defining an axiom as an alias for a couple others, which might not be that useful.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category_with_axiom.py:docstring of sage.categories.category_with_axiom.Blahs.Blue_extra_super_categories, line 15.)

Todo

Improve the infrastructure to detect and report this violation of the specifications, if this is easy. Otherwise, it’s not so bad: when defining an axiom A in a category `Cs` the first thing one is supposed to doctest is that `Cs().A()` works. So the problem should not go unnoticed.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/category_with_axiom.py:docstring of sage.categories.category_with_axiom.Blahs.Blue_extra_super_categories, line 22.)

Todo

add a demo of usual computations on Coxeter groups.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/coxeter_groups.py:docstring of sage.categories.coxeter_groups.CoxeterGroups, line 40.)

Todo

- Use the symmetric group in the examples (for nicer output), and print the edges for a stronger test.
- The constructed poset should be lazy, in order to handle large / infinite Coxeter groups.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/finite_coxeter_groups.py:docstring of sage.categories.finite_coxeter_groups.FiniteCoxeterGroups.ParentMethods.bruhat_poset, line 44.)

Todo

- Use the symmetric group in the examples (for nicer output), and print the edges for a stronger test.
- The constructed poset should be lazy, in order to handle large / infinite Coxeter groups.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/finite_coxeter_groups.py:docstring of sage.categories.finite_coxeter_groups.FiniteCoxeterGroups.ParentMethods.weak_lattice, line 70.)

Todo

- Use the symmetric group in the examples (for nicer output), and print the edges for a stronger test.
- The constructed poset should be lazy, in order to handle large / infinite Coxeter groups.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/finite_coxeter_groups.py:docstring of sage.categories.finite_coxeter_groups.FiniteCoxeterGroups.ParentMethods.weak_poset, line 70.)

Todo

`sage.combinat.debruijn_sequence.DeBruijnSequences` should not inherit from this class. If that is solved, then `FiniteEnumeratedSets` shall be turned into a subclass of `Category_singleton`.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/finite_enumerated_sets.py:docstring of sage.categories.finite_enumerated_sets.FiniteEnumeratedSets, line 25.)

Todo

make this optional

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/finite_semigroups.py:docstring of sage.categories.finite_semigroups.FiniteSemigroups, line 17.)

Todo

Get rid of this workaround once there is a more systematic approach for the alias `Modules(QQ)` -> `VectorSpaces(QQ)`. Probably the later should be a category with axiom, and covariant constructions should play well with axioms.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/graded_modules.py:docstring of sage.categories.graded_modules.GradedModules.extra_super_categories, line 18.)

Todo

- Check which methods would be better located in `Monoid.Algebras` or `Groups.Finite.Algebras`.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/groups.py:docstring of sage.categories.groups.Groups.Algebras, line 24.)

Todo

Should this category be a `CategoryWithAxiom`?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/highest_weight_crystals.py:docstring of sage.categories.highest_weight_crystals.HighestWeightCrystals.additional_structure, line 9.)

Todo

- Design decision: how much of the homset comes from the category of `X` and `Y`, and how much from the specific `X` and `Y`. In particular, do we need several parent classes depending on `X` and `Y`, or does the difference only lie in the elements (i.e. the morphism), and of course how the parent calls their constructors.
- Specify the protocol for the `_Hom_` hook in case of ambiguity (e.g. if both a parent and some category thereof provide one).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/homset.py:docstring of sage.categories.homset.Hom, line 174.)

Todo

Refactor during the upcoming homset cleanup.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/homset.py:docstring of sage.categories.homset.Homset.element_class_set_morphism, line 8.)

Todo

This category should be a `CategoryWithAxiom`, the axiom specifying the compability between the magma and module structure.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/magmatic_algebras.py:docstring of sage.categories.magmatic_algebras.MagmaticAlgebras.additional_structure, line 9.)

Todo

- Clarify the distinction, if any, with `BiModules(R, R)`. In particular, if \(R\) is a commutative ring (e.g. a field), some pieces of the code possibly assume that \(M\) is a symmetric `R`-`R`-bimodule:

  \[r*x = x*r \qquad \forall r \in R \text{ and } x \in M\]

- Make sure that non symmetric modules are properly supported by all the code, and advertise it.

- Make sure that non commutative rings are properly supported by all the code, and advertise it.

- Add support for base semirings.

- Implement a `FreeModules(R)` category, when so prompted by a concrete use case: e.g. modeling a free module with several bases (using `Sets.SubcategoryMethods.Realizations()`) or with an atlas of local maps (see e.g. trac ticket #15916).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/modules.py:docstring of sage.categories.modules.Modules, line 57.)

Todo

Generalize this so that any homset category of a full subcategory of modules over a base ring is a category over this base ring.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/modules.py:docstring of sage.categories.modules.Modules.Homsets.base_ring, line 6.)

Todo

- Explain why this does not commute with `WithBasis()`
- Improve the support for covariant functorial constructions categories over a base ring so as to get rid of the `base_ring` argument.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/modules.py:docstring of sage.categories.modules.Modules.SubcategoryMethods.Graded, line 18.)

Todo

handle base being a category

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/modules.py:docstring of sage.categories.modules.Modules.SubcategoryMethods.base_ring, line 6.)

Todo

Should this category be a `CategoryWithAxiom`?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/modules.py:docstring of sage.categories.modules.Modules.additional_structure, line 8.)

Todo

- Implement an optimized `_call_()` function.
- Generalize to a mapcoeffs.
- Generalize to a mapterms.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/modules_with_basis.py:docstring of sage.categories.modules_with_basis.DiagonalModuleMorphism, line 23.)

Todo

`End(X)` is an algebra.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/modules_with_basis.py:docstring of sage.categories.modules_with_basis.ModulesWithBasis, line 85.)

Todo

Should codomain be `self` by default in the diagonal and triangular cases?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/modules_with_basis.py:docstring of sage.categories.modules_with_basis.ModulesWithBasis.ParentMethods.module_morphism, line 189.)

Todo

Extract a method to linearize a multilinear morphism, and delegate the work there.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/modules_with_basis.py:docstring of sage.categories.modules_with_basis.ModulesWithBasis.TensorProducts.ElementMethods.apply_multilinear_morphism, line 99.)

Todo

This has nothing to do here!!! Should there be a library for pointwise operations on functions somewhere in Sage?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/modules_with_basis.py:docstring of sage.categories.modules_with_basis.pointwise_inverse_function, line 21.)

Todo

- Design a mechanism to specify that an axiom is compatible with taking subsets. Examples: `Finite`, `Associative`, `Commutative` (when meaningful), but not `Infinite` nor `Unital`.
- Design a mechanism to specify that, when \(B\) is a subcategory of \(A\), a \(B\)-homset is a subset of the corresponding \(A\) homset. And use it to recover all the relevant axioms from homsets in super categories.
- For instances of redundant code due to this missing feature, see:
  - `AdditiveMonoids.Homsets.extra_super_categories()`
  - `HomsetsCategory.extra_super_categories()` (slightly different nature)
  - plus plenty of spots where this is not implemented.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/objects.py:docstring of sage.categories.objects.Objects.SubcategoryMethods.Homsets, line 46.)

Todo

shall we accept only permutations with finite support or not?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/permutation_groups.py:docstring of sage.categories.permutation_groups.PermutationGroups, line 11.)

Todo

Give a concrete example, typically using `ElementWrapper`.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/primer.py:docstring of sage.categories.primer, line 278.)

Todo

Improve the printing of functorial constructions and joins to raise this potentially dangerous ambiguity.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/primer.py:docstring of sage.categories.primer, line 1458.)

Todo

Add an optional argument to allow for:

    sage: Realizations(A, category = Blahs()) # todo: not implemented

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/realizations.py:docstring of sage.categories.realizations.Realizations, line 39.)

Todo

Should this category be a `CategoryWithAxiom`?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/regular_crystals.py:docstring of sage.categories.regular_crystals.RegularCrystals.additional_structure, line 9.)

Todo

(see: http://trac.sagemath.org/sage_trac/wiki/CategoriesRoadMap)

- Make Rings() into a subcategory or alias of Algebras(ZZ);
- A parent P in the category `Rings()` should automatically be in the category `Algebras(P)`.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/rings.py:docstring of sage.categories.rings.Rings, line 27.)

Todo

Make `Schemes()` a singleton category (and remove `Schemes` from the workaround in `category_types.Category_over_base._test_category_over_bases()`).

This is currently incompatible with the dispatching below.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/schemes.py:docstring of sage.categories.schemes.Schemes, line 17.)

Todo

What category structure is there on Homsets of schemes?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/schemes.py:docstring of sage.categories.schemes.Schemes.Homsets.extra_super_categories, line 8.)

Todo

check that the result above is correct now

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/schemes.py:docstring of sage.categories.schemes.Schemes.Homsets.extra_super_categories, line 12.)

Todo

- Add more options for constructing subgraphs of the Cayley graph, handling the standard use cases when exploring large/infinite semigroups (a predicate, generators of an ideal, a maximal length in term of the generators)
- Specify good default layout/plot/latex options in the graph
- Generalize to combinatorial modules with module generators / operators

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/semigroups.py:docstring of sage.categories.semigroups.Semigroups.ParentMethods.cayley_graph, line 110.)

Todo

Draw the typical commutative diagram.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/sets_cat.py:docstring of sage.categories.sets_cat.Sets.SubcategoryMethods.Subquotients, line 27.)

Todo

use a more interesting example, like \(\ZZ/n\ZZ\).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/sets_cat.py:docstring of sage.categories.sets_cat.Sets.SubcategoryMethods.Subquotients, line 121.)

Todo

Fix the failing test by making `C` a singleton category. This will require some fiddling with the assertion in `Category_singleton.__classcall__()`

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/sets_cat.py:docstring of sage.categories.sets_cat.Sets.WithRealizations.ParentMethods.Realizations, line 14.)

Todo

- This should be moved to `Sets().WithGrading()`.
- Should the grading set be a parameter for this category?
- Does the enumeration need to be compatible with the grading? Be careful that the fact that graded components are allowed to be finite or infinite make the answer complicated.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/sets_with_grading.py:docstring of sage.categories.sets_with_grading.SetsWithGrading, line 72.)

Todo

Should this category be a `CategoryWithAxiom`?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/vector_spaces.py:docstring of sage.categories.vector_spaces.VectorSpaces.additional_structure, line 9.)

Todo

Should this category be a `CategoryWithAxiom`?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/categories/weyl_groups.py:docstring of sage.categories.weyl_groups.WeylGroups.additional_structure, line 8.)

Todo

Implement multi-parameter Iwahori-Hecke algebras together with their Kazhdan-Lusztig bases. That is, Iwahori-Hecke algebras with (possibly) different parameters for each conjugacy class of simple reflections in the underlying Coxeter group.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/iwahori_hecke_algebra.py:docstring of sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra, line 305.)

Todo

When given “generic parameters” we should return the generic Iwahori-Hecke algebra with these parameters and allow the user to work inside this algebra rather than doing calculations behind the scenes in a copy of the generic Iwahori-Hecke algebra. The main problem is that it is not clear how to recognise when the parameters are “generic”.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/algebras/iwahori_hecke_algebra.py:docstring of sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra, line 312.)

Todo

Do we want to implement the following syntactic sugar:

    with t.clone() as tt:
        tt.labels[1,2] = 3 ?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/abstract_tree.py:docstring of sage.combinat.abstract_tree.AbstractLabelledClonableTree.set_label, line 34.)

Todo

It is currently not possible to use `LabelledBinaryTree()` as a shorthand for `LabelledBinaryTree(None)` (in analogy to similar syntax in the `BinaryTree` class).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/binary_tree.py:docstring of sage.combinat.binary_tree.LabelledBinaryTree, line 50.)

Todo

GUAVA commands:
  - VandermondeMat
  - GrayMat returns a list of all different vectors of length n over the field F, using Gray ordering.
Not in GAP:
  - Rencontres numbers http://en.wikipedia.org/wiki/Rencontres_number

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/combinat.py:docstring of sage.combinat.combinat, line 93.)

Todo

Incorporate this method into the `_repr_` for finite Cartan type.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/crystals/alcove_path.py:docstring of sage.combinat.crystals.alcove_path.CrystalOfAlcovePathsElement.integer_sequence, line 4.)

Todo

Better doctest

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/crystals/alcove_path.py:docstring of sage.combinat.crystals.alcove_path.CrystalOfAlcovePathsElement.is_admissible, line 47.)

Todo

- Vocabulary and conventions:
  - For a classical crystal: connected / highest weight / irreducible
  - ...
- Layout instructions for plot() for rank 2 types
- RestrictionOfCrystal

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/crystals/crystals.py:docstring of sage.combinat.crystals.crystals, line 97.)

Todo

FIXME:

- Do we want to specify the columns increasingly or decreasingly? That is, should this be `Tab(columns = [[1,3],[2,4]])`?
- Make this fully consistent with `Tableau()`!

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/crystals/tensor_product.py:docstring of sage.combinat.crystals.tensor_product.CrystalOfTableaux, line 72.)

Todo

Implement finite non-Desarguesian plane as in [We07] and Wikipedia article Non-Desarguesian_plane.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/designs/block_design.py:docstring of sage.combinat.designs.block_design, line 28.)

Todo

Implement DerivedDesign and ComplementaryDesign.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/designs/design_catalog.py:docstring of sage.combinat.designs.design_catalog, line 58.)

Todo

Implement recursive constructions from Buratti “Recursive for difference matrices and relative difference families” (1998) and Jungnickel “Composition theorems for difference families and regular planes” (1978)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/designs/difference_family.py:docstring of sage.combinat.designs.difference_family.difference_family, line 170.)

Todo

The XML data from the designtheory.org database contains a wealth of information about things like automorphism groups, transitivity, cycle type representatives, etc, but none of this data is made available through the current implementation.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/designs/ext_rep.py:docstring of sage.combinat.designs.ext_rep, line 14.)

Todo

extend this to m-Dyck words

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/dyck_word.py:docstring of sage.combinat.dyck_word.CompleteDyckWords_size.random_element, line 11.)

Todo

At the moment, the letters of the alphabets need to be hashable.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/finite_state_machine.py:docstring of sage.combinat.finite_state_machine.FiniteStateMachine.determine_alphabets, line 17.)

Todo

Do the iteration in place to save on copying time

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/integer_list.py:docstring of sage.combinat.integer_list.IntegerListsLex.count, line 7.)

Todo

Placeholder. Implement a proper check.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/integer_list.py:docstring of sage.combinat.integer_list.IntegerListsLexElement.check, line 4.)

Todo

Move this into Cython.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/integer_list.py:docstring of sage.combinat.integer_list.first, line 8.)

Todo

should the order of the arguments `n` and `weight` be exchanged to simplify the logic ?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/integer_vector_weighted.py:docstring of sage.combinat.integer_vector_weighted.WeightedIntegerVectors, line 39.)

Todo

Integer vectors should accept `max_part` as a single argument, and the following should change:

    sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), max_part=12); S.ambient()
    Integer vectors

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/integer_vectors_mod_permgroup.py:docstring of sage.combinat.integer_vectors_mod_permgroup.IntegerVectorsModPermutationGroup_with_constraints.ambient, line 10.)

Todo

To study this, it would be more natural to define interval-posets on arbitrary ordered sets rather than just on \(\{1, 2, \ldots, n\}\).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/interval_posets.py:docstring of sage.combinat.interval_posets.TamariIntervalPoset.insertion, line 25.)

Todo

Functionality to add:

- plotter will not plot edge labels higher than 2; e.g. in BK puzzles, the labels are 1,..., n and so in 3-step examples, none of the edge labels with 3 appear
- we should also have a 3-step puzzle pieces constructor, taken from p22 of Arxiv math/0610538
- implement the bijection from puzzles to tableaux; see for example R. Vakil, A geometric Littlewood-Richardson rule, Arxiv math/0302294 or K. Purbhoo, Puzzles, Tableaux and Mosaics, Arxiv 0705.1184.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/knutson_tao_puzzles.py:docstring of sage.combinat.knutson_tao_puzzles, line 12.)

Todo

This construction holds more generally for prime powers \(q\) congruent to \(3 \mod{4}\). We should implement these but we first need to implement Quadratic character for \(GF(q)\).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/matrices/hadamard_matrix.py:docstring of sage.combinat.matrices.hadamard_matrix.H1, line 5.)

Todo

This construction holds more generally for prime powers \(q\) congruent to \(1 \mod{4}\). We should implement these but we first need to implement Quadratic character for \(GF(q)\).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/matrices/hadamard_matrix.py:docstring of sage.combinat.matrices.hadamard_matrix.H2, line 6.)

Todo

Fix the failing test by making `C` a singleton category. This will require some fiddling with the assertion in `Category_singleton.__classcall__()`

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/generic_basis_code.py:docstring of sage.combinat.ncsf_qsym.generic_basis_code.BasesOfQSymOrNCSF, line 14.)

Todo

Generalize this to all graded vector spaces?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/generic_basis_code.py:docstring of sage.combinat.ncsf_qsym.generic_basis_code.BasesOfQSymOrNCSF.ElementMethods.degree_negation, line 33.)

Todo

Generalize this to all graded vector spaces?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/generic_basis_code.py:docstring of sage.combinat.ncsf_qsym.generic_basis_code.BasesOfQSymOrNCSF.ParentMethods.degree_negation, line 47.)

Todo

Despite the `__repr__`, this is NOT an endomorphism!

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/generic_basis_code.py:docstring of sage.combinat.ncsf_qsym.generic_basis_code.GradedModulesWithInternalProduct.ParentMethods.internal_product, line 34.)

Todo

Despite the `__repr__`, this is NOT an endomorphism!

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/generic_basis_code.py:docstring of sage.combinat.ncsf_qsym.generic_basis_code.GradedModulesWithInternalProduct.ParentMethods.itensor, line 34.)

Todo

Despite the `__repr__`, this is NOT an endomorphism!

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/generic_basis_code.py:docstring of sage.combinat.ncsf_qsym.generic_basis_code.GradedModulesWithInternalProduct.ParentMethods.kronecker_product, line 34.)

Todo

demonstrate how to customize the basis names

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/ncsf.py:docstring of sage.combinat.ncsf_qsym.ncsf.NonCommutativeSymmetricFunctions, line 162.)

Todo

explain the other changes of bases!

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/ncsf.py:docstring of sage.combinat.ncsf_qsym.ncsf.NonCommutativeSymmetricFunctions, line 213.)

Todo

- Bases: monomial, fundamental, forgotten, quasi_schur_dual simple() ? (<=> simple modules of HS_n; to be discussed with Florent)
- Multiplication in:
  - fundamental and monomial (cf. Lenny’s code)
  - ribbon (from Mike’s code)
- Coproducts (most done by coercions)
- some_elements in all bases
- Systematic coercion checks (in AlgebrasWithBasis().Abstract())

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/ncsf.py:docstring of sage.combinat.ncsf_qsym.ncsf.NonCommutativeSymmetricFunctions, line 259.)

Todo

this could be generalized to any free algebra.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/ncsf.py:docstring of sage.combinat.ncsf_qsym.ncsf.NonCommutativeSymmetricFunctions.MultiplicativeBasesOnPrimitiveElements, line 14.)

Todo

Implement this directly on the monomial basis maybe? The \((0, I)\)-matrices are a pain to generate from their definition, but maybe there is a good algorithm. If so, the above “further examples” should be moved to the M-method.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/qsym.py:docstring of sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions.Bases.ElementMethods.internal_coproduct, line 184.)

Todo

Implement this directly on the monomial basis maybe? The \((0, I)\)-matrices are a pain to generate from their definition, but maybe there is a good algorithm. If so, the above “further examples” should be moved to the M-method.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/qsym.py:docstring of sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions.Bases.ElementMethods.kronecker_coproduct, line 184.)

Todo

The conversion from the M basis to the HWL basis is currently implemented in the naive way (inverting the base-change matrix in the other direction). This matrix is not triangular (not even after any permutations of the bases), and there could very well be a faster method (the one given by Hazewinkel?).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/qsym.py:docstring of sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions.HazewinkelLambda, line 39.)

Todo

accept an alphabet as input

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ncsf_qsym/qsym.py:docstring of sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions.Monomial.Element.expand, line 14.)

Todo

Reimplement like `remove_horizontal_border_strip` using `IntegerListsLex`

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/partition.py:docstring of sage.combinat.partition.Partition.add_horizontal_border_strip, line 15.)

Todo

Check in Knuth AOCP4.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/partition.py:docstring of sage.combinat.partition.Partitions_n.random_element_uniform, line 21.)

Todo

This docstring needs to be fixed. First, the definition does not match the implementation (or the examples). Second, this doesn’t seem to be defined in [GarStan1984] (the descent monomial in their (7.23) is different).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/permutation.py:docstring of sage.combinat.permutation.Permutation.descent_polynomial, line 25.)

Todo

- generalize this feature by accepting a family of operators as input
- move up in some appropriate category

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/posets/linear_extensions.py:docstring of sage.combinat.posets.linear_extensions.LinearExtensionsOfPoset.markov_chain_digraph, line 8.)

Todo

Should the vertices of the diagram have the poset as parent?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/posets/posets.py:docstring of sage.combinat.posets.posets.FinitePoset.hasse_diagram, line 5.)

Todo

- Is it acceptable to have those two features for a single method?
- In particular, we miss a short idiom to get the default linear extension

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/posets/posets.py:docstring of sage.combinat.posets.posets.FinitePoset.linear_extension, line 49.)

Todo

add tests as in combinat::rankers

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/ranker.py:docstring of sage.combinat.ranker.on_fly, line 27.)

Todo

Implement a direct action of \(e_0\) without moving to KR crystals.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/rigged_configurations/kr_tableaux.py:docstring of sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableauxElement.e, line 3.)

Todo

Implement a direct action of \(\epsilon_0\) without moving to KR crystals.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/rigged_configurations/kr_tableaux.py:docstring of sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableauxElement.epsilon, line 3.)

Todo

Implement a direct action of \(f_0\) without moving to KR crystals.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/rigged_configurations/kr_tableaux.py:docstring of sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableauxElement.f, line 3.)

Todo

Compute \(\phi_0\) without moving to KR crystals.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/rigged_configurations/kr_tableaux.py:docstring of sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableauxElement.phi, line 3.)

Todo

Implement \(e_0\) without appealing to tensor product of KR tableaux.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/rigged_configurations/rigged_configuration_element.py:docstring of sage.combinat.rigged_configurations.rigged_configuration_element.KRRiggedConfigurationElement.e, line 8.)

Todo

Implement \(f_0\) without appealing to tensor product of KR tableaux.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/rigged_configurations/rigged_configuration_element.py:docstring of sage.combinat.rigged_configurations.rigged_configuration_element.KRRiggedConfigurationElement.f, line 8.)

Todo

Convert this to using multiplicities \(m_i\) (perhaps with a dictionary?)?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/rigged_configurations/rigged_partition.py:docstring of sage.combinat.rigged_configurations.rigged_partition, line 18.)

Todo

add a method `set_mutable()` as, say, for matrices

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/cartan_type.py:docstring of sage.combinat.root_system.cartan_type, line 205.)

Todo

add a method `set_mutable()` as, say, for matrices

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/cartan_type.py:docstring of sage.combinat.root_system.cartan_type.CartanType, line 207.)

Todo

add some reducible Cartan types (suggestions?)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/cartan_type.py:docstring of sage.combinat.root_system.cartan_type.CartanTypeFactory.samples, line 50.)

Todo

Add the picture here, once root system plots in the weight lattice will be implemented. In the mean time, the reader may look up the dual picture on Figure 2 of [HST09] which was produced by MuPAD-Combinat.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/cartan_type.py:docstring of sage.combinat.root_system.cartan_type.CartanType_affine.translation_factors, line 131.)

Todo

Add tests for the above assumptions, and also that the classical operators \(T_1, \ldots, T_n\) from \(T\) and \(T_Y\) coincide.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/hecke_algebra_representation.py:docstring of sage.combinat.root_system.hecke_algebra_representation.CherednikOperatorsEigenvectors, line 39.)

Todo

Add an example where \(T_i\ne T_i^{-1}\)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/hecke_algebra_representation.py:docstring of sage.combinat.root_system.hecke_algebra_representation.HeckeAlgebraRepresentation.Tw_inverse, line 5.)

Todo

Add more tests

Add tests in type BC affine where the null coroot \(\delta^\vee\) can have non trivial coefficient in term of \(\alpha_0\)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/hecke_algebra_representation.py:docstring of sage.combinat.root_system.hecke_algebra_representation.HeckeAlgebraRepresentation.Y_lambdacheck, line 66.)

Todo

At this point, this method is constant. It’s meant as a starting point for implementing parameters depending on the node \(i\) of the Dynkin diagram.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/hecke_algebra_representation.py:docstring of sage.combinat.root_system.hecke_algebra_representation.HeckeAlgebraRepresentation.parameters, line 16.)

Todo

- Non-reduced case (Koornwinder polynomials).
- Non-equal parameters for the affine Hecke algebra.
- Choice of convention (dominant/anti-dominant, ...).
- More uniform implementation of the \(T_0^\vee\) operator.
- Optimizations, in particular in the calculation of the eigenvalues for the recursion.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/non_symmetric_macdonald_polynomials.py:docstring of sage.combinat.root_system.non_symmetric_macdonald_polynomials.NonSymmetricMacdonaldPolynomials, line 15.)

Todo

add his notes in latex

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/non_symmetric_macdonald_polynomials.py:docstring of sage.combinat.root_system.non_symmetric_macdonald_polynomials.NonSymmetricMacdonaldPolynomials, line 735.)

Todo

should this just return \(L\) in the simply laced case?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/non_symmetric_macdonald_polynomials.py:docstring of sage.combinat.root_system.non_symmetric_macdonald_polynomials.NonSymmetricMacdonaldPolynomials.L_check, line 3.)

Todo

- Use proposition 6.9 of [Haiman06] to check the action of the \(Y\) s on monomials.
- Generalize to any \(q_1\), \(q_2\).
- Check claim by Mark: all scalar products should occur in the finite weight lattice, with alpha 0 being the appropriate projection of the affine alpha 0. Question: can this be emulated by being at level 0?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/non_symmetric_macdonald_polynomials.py:docstring of sage.combinat.root_system.non_symmetric_macdonald_polynomials.NonSymmetricMacdonaldPolynomials.eigenvalue_experimental, line 87.)

Todo

Could we have nice \(\LaTeX\) labels in this graph?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/plot.py:docstring of sage.combinat.root_system.plot, line 477.)

Todo

Display the periodic orientation by adding a \(+\) and a \(-\) sign close to the label. Typically by using the associated root to shift a bit from the vertex upon which the hyperplane label is attached.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/plot.py:docstring of sage.combinat.root_system.plot.PlotOptions.reflection_hyperplane, line 38.)

Todo

- Optimize the code to only do the embedding/projection for T_0
- Add an option to specify at which level one wants to work. Currently this is level 0.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/root_lattice_realization_algebras.py:docstring of sage.combinat.root_system.root_lattice_realization_algebras.Algebras.ParentMethods.demazure_lusztig_operator_on_classical_on_basis, line 12.)

Todo

type free definition (Viviane’s definition uses that we are in the ambient space)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/root_lattice_realization_algebras.py:docstring of sage.combinat.root_system.root_lattice_realization_algebras.Algebras.ParentMethods.divided_difference_on_basis, line 8.)

Todo

make this work for Laurent polynomials too

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/root_lattice_realization_algebras.py:docstring of sage.combinat.root_system.root_lattice_realization_algebras.Algebras.ParentMethods.from_polynomial, line 26.)

Todo

Choose a good set of Cartan Type to run on. Rank >4 is too big. But \(C_1\) and \(B_1\) are boring.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/root_lattice_realization_algebras.py:docstring of sage.combinat.root_system.root_lattice_realization_algebras.Algebras.ParentMethods.twisted_demazure_lusztig_operators, line 75.)

Todo

Investigate why \(T_0^\vee\) currently does not satisfy the quadratic relation in type \(BC\). This should hopefuly be fixed when \(T_0^\vee\) will have a more uniform implementation:

    sage: cartan_type = CartanType(["BC",1,2])
    sage: KL = RootSystem(cartan_type).weight_lattice().algebra(K)
    sage: T = KL.twisted_demazure_lusztig_operators(q1,q2, convention="dominant")
    sage: T._test_relations()
    Traceback (most recent call last):
    ... tester.assert_(Ti(Ti(x,i,-q2),i,-q1).is_zero()) ...
    AssertionError: False is not true

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/root_lattice_realization_algebras.py:docstring of sage.combinat.root_system.root_lattice_realization_algebras.Algebras.ParentMethods.twisted_demazure_lusztig_operators, line 92.)

Todo

This implementation is only valid in the root or weight lattice

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/root_lattice_realizations.py:docstring of sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ElementMethods.is_parabolic_root, line 7.)

Todo

add a non simply laced example

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/root_lattice_realizations.py:docstring of sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ParentMethods.alphacheck, line 17.)

Todo

Provide an option for transparency?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/root_lattice_realizations.py:docstring of sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ParentMethods.plot_reflection_hyperplanes, line 65.)

Todo

The result should be an enumerated set, and handle infinite root systems.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/root_lattice_realizations.py:docstring of sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ParentMethods.roots, line 27.)

Todo

Rename to is_quantum_root

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/root_space.py:docstring of sage.combinat.root_system.root_space.RootSpaceElement.quantum_root, line 13.)

Todo

Lift to CombinatorialFreeModule.Element as canonical_inner_product

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/type_affine.py:docstring of sage.combinat.root_system.type_affine.AmbientSpace.Element.inner_product, line 23.)

Todo

Lift to CombinatorialFreeModule.Element as canonical_inner_product

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/type_affine.py:docstring of sage.combinat.root_system.type_affine.AmbientSpace.Element.scalar, line 23.)

Todo

Factor out this code with the classical ambient space.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/type_affine.py:docstring of sage.combinat.root_system.type_affine.AmbientSpace.coroot_lattice, line 6.)

Todo

Factor out this code with the classical ambient space.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/type_affine.py:docstring of sage.combinat.root_system.type_affine.AmbientSpace.simple_coroot, line 16.)

Todo

Currently `subdivide` is currently ignored.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/type_reducible.py:docstring of sage.combinat.root_system.type_reducible.CartanType.cartan_matrix, line 6.)

Todo

- merge with_apply_multi_module_morphism
- allow for any root space / lattice
- define properly the return type (depends on the base rings of the two spaces)
- make this robust for extended weight lattices (\(i\) might be “delta”)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/weight_space.py:docstring of sage.combinat.root_system.weight_space.WeightSpaceElement.scalar, line 4.)

Todo

Try to compute this directly without actually calculating the full symmetric and exterior squares.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/weyl_characters.py:docstring of sage.combinat.root_system.weyl_characters.WeylCharacterRing.Element.frobenius_schur_indicator, line 17.)

Todo

implement:

- Parabolic subrootsystems
- Parabolic subgroups with a set of nodes as argument

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/root_system/weyl_group.py:docstring of sage.combinat.root_system.weyl_group.ClassicalWeylSubgroup, line 26.)

Todo

delete this class once all coercions will be handled by Sage’s coercion model

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/classical.py:docstring of sage.combinat.sf.classical.SymmetricFunctionAlgebra_classical, line 3.)

Todo

Is there a not too difficult way to get the power-sum computations to work over any ring, not just one with coercion from \(\QQ\)?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/monomial.py:docstring of sage.combinat.sf.monomial.SymmetricFunctionAlgebra_monomial.antipode_by_coercion, line 25.)

Todo

Get rid of said technical “reasons”.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/new_kschur.py:docstring of sage.combinat.sf.new_kschur.kSchur, line 93.)

Todo

to be described

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/sf.py:docstring of sage.combinat.sf.sf.SymmetricFunctions, line 644.)

Todo

- Introduce fields with degree 1 elements as in MuPAD-Combinat, to get proper plethysm.
- Use UniqueRepresentation to get rid of all the manual cache handling for the bases
- Devise a mechanism so that pickling bases of symmetric functions pickles the coercions which have a cache.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/sf.py:docstring of sage.combinat.sf.sf.SymmetricFunctions, line 694.)

Todo

Most of the methods in this class are generic (manipulations of morphisms, ...) and should be generalized (or removed)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/sfa.py:docstring of sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic, line 3.)

Todo

This method is fast on the monomial and the powersum bases, while all other bases get converted to the monomial basis. For most bases, this is probably the quickest way to do, but at least the Schur basis should have a better option. (Quoting from Stanley’s EC2 [STA]: “D. G. Duncan, J. London Math. Soc. 27 (1952), 235-236, or Y. M. Chen, A. M. Garsia, and J. B. Remmel, Contemp. Math. 34 (1984), 109-153”.)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/sfa.py:docstring of sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.adams_operation, line 120.)

Todo

This method is fast on the monomial and the powersum bases, while all other bases get converted to the monomial basis. For most bases, this is probably the quickest way to do, but at least the Schur basis should have a better option. (Quoting from Stanley’s EC2 [STA]: “D. G. Duncan, J. London Math. Soc. 27 (1952), 235-236, or Y. M. Chen, A. M. Garsia, and J. B. Remmel, Contemp. Math. 34 (1984), 109-153”.)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/sfa.py:docstring of sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.frobenius, line 120.)

Todo

This implementation of the reduced Kronecker product is painfully slow.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/sfa.py:docstring of sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.reduced_kronecker_product, line 155.)

Todo

This function is an ugly hack using strings. It should be rewritten as soon as the bases of `SymmetricFunctions` are put on a more robust and systematic footing.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/sfa.py:docstring of sage.combinat.sf.sfa.SymmetricFunctionsBases.ParentMethods.corresponding_basis_over, line 72.)

Todo

generalize to Modules.Graded.Connected.ParentMethods

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/sf/sfa.py:docstring of sage.combinat.sf.sfa.SymmetricFunctionsBases.ParentMethods.one_basis, line 17.)

Todo

As is, this set is essentially the composition of `Compositions(n)` (which give the row lengths) and `SkewPartition(n, row_lengths=...)`, and one would want to “inherit” list and cardinality from this composition.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/skew_partition.py:docstring of sage.combinat.skew_partition.SkewPartitions_n, line 15.)

Todo

- implement subwords with repetitions
- implement the category of EnumeratedMultiset and inheritate from when needed (i.e. the initial word has repeated letters)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/subword.py:docstring of sage.combinat.subword, line 27.)

Todo

- construct the product of two irreducible representations.
- implement Induction/Restriction of representations.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/symmetric_group_representations.py:docstring of sage.combinat.symmetric_group_representations, line 1.)

Todo

Implement semistandard tableau tuples as defined in [DJM].

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/tableau_tuple.py:docstring of sage.combinat.tableau_tuple, line 180.)

Todo

Add link to some thematic tutorial on graphs

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/tutorial.py:docstring of sage.combinat.tutorial, line 17.)

Todo

add link to some tutorial on quotient rings

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/tutorial.py:docstring of sage.combinat.tutorial, line 468.)

Todo

hide the results by default

(The original entry is located in /usr/lib/python2.7/site-packages/sage/combinat/tutorial.py:docstring of sage.combinat.tutorial, line 1158.)

Todo

- in case of flood, suggest the user to install the off-line database instead.
- interface with the off-line database (or reimplement it).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/databases/oeis.py:docstring of sage.databases.oeis, line 123.)

Todo

Ask OEIS for a keyword ensuring that a sequence is infinite.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/databases/oeis.py:docstring of sage.databases.oeis.OEISSequence.is_finite, line 12.)

Todo

- ask OEIS to add a keyword telling whether the sequence comes from a power series, e.g. for http://oeis.org/A000182
- discover other possible conversions.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/databases/oeis.py:docstring of sage.databases.oeis.OEISSequence.natural_object, line 18.)

Todo

ask OEIS to add a “Sage program” field in the database ;)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/databases/oeis.py:docstring of sage.databases.oeis.OEISSequence.programs, line 12.)

Todo

- in case of flood, suggest the user to install the off-line database instead.
- interface with the off-line database (or reimplement it).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/databases/oeis.py:docstring of sage.databases.oeis, line 123.)

Todo

Ask OEIS for a keyword ensuring that a sequence is infinite.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/databases/oeis.py:docstring of sage.databases.oeis.OEISSequence.is_finite, line 12.)

Todo

- ask OEIS to add a keyword telling whether the sequence comes from a power series, e.g. for http://oeis.org/A000182
- discover other possible conversions.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/databases/oeis.py:docstring of sage.databases.oeis.OEISSequence.natural_object, line 18.)

Todo

ask OEIS to add a “Sage program” field in the database ;)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/databases/oeis.py:docstring of sage.databases.oeis.OEISSequence.programs, line 12.)

Todo

- construct as options different string representations for a permutation
  - the two intervals: str
  - the two intervals on one line: str_one_line
  - the separatrix diagram: str_separatrix_diagram
  - twin[0] and twin[1] for reduced permutation
  - nothing (useful for Rauzy diagram)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/dynamics/interval_exchanges/template.py:docstring of sage.dynamics.interval_exchanges.template, line 9.)

Todo

This function could probably be made faster.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/algebraic_closure_finite_field.py:docstring of sage.rings.algebraic_closure_finite_field.AlgebraicClosureFiniteFieldElement.nth_root, line 12.)

Todo

When trac ticket #10963 is merged we should remove that method and set the category to infinite fields (i.e. `Fields().Infinite()`).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/algebraic_closure_finite_field.py:docstring of sage.rings.algebraic_closure_finite_field.AlgebraicClosureFiniteField_generic.cardinality, line 5.)

Todo

When trac ticket #10963 is merged we should remove that method and set the category to infinite fields (i.e. `Fields().Infinite()`).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/algebraic_closure_finite_field.py:docstring of sage.rings.algebraic_closure_finite_field.AlgebraicClosureFiniteField_generic.is_finite, line 3.)

Todo

Implement associated Legendre polynomials and Zernike polynomials. (Neither is in Maxima.) Wikipedia article Associated_Legendre_polynomials Wikipedia article Zernike_polynomials

(The original entry is located in /usr/lib/python2.7/site-packages/sage/functions/orthogonal_polys.py:docstring of sage.functions.orthogonal_polys, line 261.)

Todo

Eventually, category should be `Sets` by default.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/geometry/lattice_polytope.py:docstring of sage.geometry.lattice_polytope.SetOfAllLatticePolytopesClass, line 50.)

Todo

Make it possible to draw Schlegel diagram for 4-polytopes.

    sage: P=Polyhedron(vertices=[[1,1,0,0],[1,2,0,0],[2,1,0,0],[0,0,1,0],[0,0,0,1]])
    sage: P
    A 4-dimensional polyhedron in ZZ^4 defined as the convex hull of 5 vertices
    sage: P.projection().tikz()
    Traceback (most recent call last):
    ...
    NotImplementedError: The polytope has to live in 2 or 3 dimensions.

Make it possible to draw 3-polytopes living in higher dimension.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/geometry/polyhedron/plot.py:docstring of sage.geometry.polyhedron.plot.Projection.tikz, line 88.)

Todo

This method sequentially tests each of the forbidden subgraphs in order to know whether the graph is a line graph, which is a very slow method. It could eventually be replaced by `root_graph()` when this method will not require an exponential time to run on general graphs anymore (see its documentation for more information on this problem)... and if it can be improved to return negative certificates !

(The original entry is located in /usr/lib/python2.7/site-packages/sage/graphs/graph.py:docstring of sage.graphs.graph.Graph.is_line_graph, line 17.)

Todo

Find a beautiful layout for this beautiful graph.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/graphs/graph_generators.py:docstring of sage.graphs.graph_generators.GraphGenerators.SchlaefliGraph, line 14.)

Todo

- Add tooltip like in http://bl.ocks.org/bentwonk/2514276.
- Add a zoom through scrolling (http://bl.ocks.org/mbostock/3681006).

(The original entry is located in /usr/lib/python2.7/site-packages/sage/graphs/graph_plot_js.py:docstring of sage.graphs.graph_plot_js, line 54.)

Todo

Technical things:

- Query the database for non-inclusion results so that comparisons can return `False`, and implement strict inclusions.
- Implement a proper search method for the classes not listed in `graph_classes`
- Some of the graph classes appearing in `graph_classes` already have a recognition algorithm implemented in Sage. It would be so nice to be able to write `g in Trees`, `g in Perfect`, `g in Chordal`, ... :-)

Long-term stuff:

- Implement simple accessors for all the information in the ISGCI database (as can be done from the website)
- Implement intersection of graph classes
- Write generic recognition algorithms for specific classes (when a graph class is defined by the exclusion of subgraphs, one can write a generic algorithm checking the existence of each of the graphs, and this method already exists in Sage).
- Improve the performance of Sage’s graph library by letting it take advantage of the properties of graph classes. For example, `Graph.independent_set()` could use the library to detect that a given graph is, say, a tree or a planar graph, and use a specialized algorithm for finding an independent set.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/graphs/isgci.py:docstring of sage.graphs.isgci, line 282.)

Todo

This method sequentially tests each of the forbidden subgraphs in order to know whether the graph is a line graph, which is a very slow method. It could eventually be replaced by `root_graph()` when this method will not require an exponential time to run on general graphs anymore (see its documentation for more information on this problem)... and if it can be improved to return negative certificates !

(The original entry is located in /usr/lib/python2.7/site-packages/sage/graphs/line_graph.py:docstring of sage.graphs.line_graph.is_line_graph, line 17.)

Todo

This code could probably be made more efficient by using FLINT polynomials and being written in Cython, using an array of fmpz_poly_t pointers or something... Right now just about the whole complement optimization is written in Python, and could be easily sped up.

(The original entry is located in docstring of sage.graphs.matchpoly.complete_poly, line 7.)

Todo

- Implement a non-naive fallback method for computing all the elements of the conjugacy class when the group is not defined in GAP, as the one in Butler’s paper.
- Define a sage method for gap matrices so that groups of matrices can use the quicker GAP algorithm rather than the naive one.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/groups/conjugacy_classes.py:docstring of sage.groups.conjugacy_classes, line 9.)

Todo

Implement a non-naive algorithm, cf. for instance G. Butler: “An Inductive Schema for Computing Conjugacy Classes in Permutation Groups”, Math. of Comp. Vol. 62, No. 205 (1994)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/groups/conjugacy_classes.py:docstring of sage.groups.conjugacy_classes.ConjugacyClass.set, line 4.)

Todo

- Include support for different orderings (currently only shortlex is used).
- Include the GAP package kbmag for more functionalities, including automatic structures and faster compiled functions.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/groups/finitely_presented.py:docstring of sage.groups.finitely_presented.RewritingSystem, line 46.)

Todo

Currently the label \(\infty\) is implemented as \(-1\) in the Coxeter matrix.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/groups/matrix_gps/coxeter_group.py:docstring of sage.groups.matrix_gps.coxeter_group.CoxeterMatrixGroup, line 31.)

Todo

Fix the broken hash.

    sage: G = SymmetricGroup(6)
    sage: G3 = G.subgroup([G((1,2,3,4,5,6)),G((1,2))])
    sage: hash(G) == hash(G3)  # todo: Should be True!
    False

(The original entry is located in /usr/lib/python2.7/site-packages/sage/groups/perm_gps/permgroup_named.py:docstring of sage.groups.perm_gps.permgroup_named.PermutationGroup_unique, line 1.)

Todo

Up to now, this group is only implemented for finite fields because of the limited support of automorphisms for arbitrary rings.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/groups/semimonomial_transformations/semimonomial_transformation_group.py:docstring of sage.groups.semimonomial_transformations.semimonomial_transformation_group, line 29.)

Todo

Up to now, this group is only implemented for finite fields because of the limited support of automorphisms for arbitrary rings.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/groups/semimonomial_transformations/semimonomial_transformation_group.py:docstring of sage.groups.semimonomial_transformations.semimonomial_transformation_group.SemimonomialTransformationGroup, line 31.)

Todo

Make subdivision work correctly on multiple factors.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/homology/chain_complex.py:docstring of sage.homology.chain_complex.ChainComplex_class.tensor, line 32.)

Todo

Create an animated image file (GIF) if spin is on and put data extracted from a file into a variable/string/structure to return

(The original entry is located in /usr/lib/python2.7/site-packages/sage/interfaces/jmoldata.py:docstring of sage.interfaces.jmoldata.JmolData, line 1.)

Todo

use this library in the `SymmetricFunctions` code, to make it easy to apply it to linear combinations of Schur functions.

(The original entry is located in docstring of sage.libs.lrcalc.lrcalc, line 95.)

Todo

Implement this method.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/logic/logic.py:docstring of sage.logic.logic.SymbolicLogic.prove, line 4.)

Todo

Implement this method.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/logic/logic.py:docstring of sage.logic.logic.SymbolicLogic.simplify, line 4.)

Todo

Implement faster algorithms, including a division-free one. Does [Rote2001], section 3.3 give one?

Check the implementation of the matchings used here for performance?

(The original entry is located in docstring of sage.matrix.matrix2.Matrix.pfaffian, line 82.)

Todo

Write abstract `RelabeledMatroid` class, and add `relabel()` method to the main `Matroid` class, together with `_relabel()` method that can be replaced by subclasses. Use the code from `is_isomorphism()` in `relabel()` to deal with a variety of input methods for the relabeling.

(The original entry is located in docstring of sage.matroids.basis_matroid.BasisMatroid.relabel, line 15.)

Todo

Add optional argument `groundset` to each method so users can customize the groundset of the matroid. We probably want some means of relabeling to accomplish that.

Add option to specify the field for represented matroids.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/matroids/catalog.py:docstring of sage.matroids.catalog, line 10.)

Todo

Optional arguments `ring` and `x`, such that the resulting matroid is represented over `ring` by a reduced matrix like `[-1 0 x]` `[ 1 -1 0]` `[ 0 1 -1]`

(The original entry is located in /usr/lib/python2.7/site-packages/sage/matroids/catalog.py:docstring of sage.matroids.catalog.Whirl, line 28.)

Todo

This important method can (and should) be optimized considerably. See [Hlineny] p.1219 for hints to that end.

(The original entry is located in docstring of sage.matroids.linear_matroid.LinearMatroid.has_field_minor, line 16.)

Todo

This important method can (and should) be optimized considerably. See [Hlineny] p.1219 for hints to that end.

(The original entry is located in docstring of sage.matroids.matroid.Matroid.has_minor, line 16.)

Todo

Implement this using the efficient algorithm from [BC79].

(The original entry is located in docstring of sage.matroids.matroid.Matroid.is_3connected, line 11.)

Todo

Make implementation more efficient, e.g. generalizing the approach from trac ticket #1314 from graphs to matroids.

(The original entry is located in docstring of sage.matroids.matroid.Matroid.tutte_polynomial, line 31.)

Todo

- Handle nicely (covariant) functorial constructions and axioms

(The original entry is located in docstring of sage.misc.c3_controlled.CmpKey, line 45.)

Todo

Make the following work nicely:

    sage: b.x?                # todo: not implemented
    sage: b.x??               # todo: not implemented

(The original entry is located in docstring of sage.misc.lazy_attribute.lazy_attribute, line 175.)

Todo

Improve the error message:

    sage: B().unimplemented_A # todo: not implemented
    Traceback (most recent call last):
    ...
    AttributeError: 'super' object has no attribute 'unimplemented_A'

(The original entry is located in docstring of sage.misc.lazy_attribute.lazy_attribute, line 387.)

Todo

- Add Pyrex source code inspection (I assume it doesn’t currently do this)
- Add ability to sort output by time
- Add option to constructor to print timing immediately when checkpoint is reached
- Migrate to Pyrex?
- Add ability to return timings in a more machine-friendly format

(The original entry is located in /usr/lib/python2.7/site-packages/sage/misc/profiler.py:docstring of sage.misc.profiler.Profiler, line 40.)

Todo

Make square root in the underlying rational field work.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/modular/modform_hecketriangle/graded_ring_element.py:docstring of sage.modular.modform_hecketriangle.graded_ring_element.FormsRingElement.sqrt, line 17.)

Todo

This should be moved to `sage.matrix.matrix_modn_dense` at some point.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/modular/overconvergent/hecke_series.py:docstring of sage.modular.overconvergent.hecke_series.ech_form, line 4.)

Todo

Refactor modules such that it only counts what category the base ring belongs to, but not what is its Python class.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/modules/free_module.py:docstring of sage.modules.free_module.FreeModuleFactory, line 122.)

Todo

Should implement a version of the algorithm that guarantees correct output. See Algorithm 4 in [Doyle-Krumm] for details of an implementation that takes precision issues into account.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/number_field/number_field.py:docstring of sage.rings.number_field.number_field.NumberField_absolute.elements_of_bounded_height, line 34.)

Todo

The `_new()` method should be overridden in this class to copy the `D` and `standard_embedding` attributes

(The original entry is located in docstring of sage.rings.number_field.number_field_element_quadratic, line 14.)

Todo

doctests for converting from other types of \(p\)-adic rings

(The original entry is located in docstring of sage.rings.padics.padic_fixed_mod_element.pAdicFixedModElement, line 82.)

Todo

(The original entry is located in docstring of sage.rings.padics.padic_generic_element.pAdicGenericElement.log, line 46.)

Todo

See comments at trac ticket #4805. Currently the absolute precision of the result may be less than the given value of absprec, and error-handling is imperfect.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/elliptic_curves/ell_point.py:docstring of sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_number_field.padic_elliptic_logarithm, line 26.)

Todo

make `GaloisAutomorphism` derive from GroupElement, so that one gets powers for free, etc.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/elliptic_curves/heegner.py:docstring of sage.schemes.elliptic_curves.heegner.GaloisAutomorphism, line 3.)

Todo

Unify `UnionOfIntervals` with the class `RealSet` introduced by trac ticket #13125; see trac ticket #16063.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/elliptic_curves/height.py:docstring of sage.schemes.elliptic_curves.height.UnionOfIntervals, line 19.)

Todo

Eventually we will want to run this in characteristic 3, so we need to: (a) Allow \(Q(x)\) to contain an \(x^2\) term, and (b) Remove the requirement that 3 be invertible. Currently this is used in the Toom-Cook algorithm to speed multiplication.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/monsky_washnitzer.py:docstring of sage.schemes.hyperelliptic_curves.monsky_washnitzer.SpecialCubicQuotientRing, line 17.)

Todo

write an example checking multiplication of these polynomials against Sage’s ordinary quotient ring arithmetic. I can’t seem to get the quotient ring stuff happening right now...

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/monsky_washnitzer.py:docstring of sage.schemes.hyperelliptic_curves.monsky_washnitzer.SpecialCubicQuotientRing, line 66.)

Todo

finish integrating tachyon – good default lights, camera

(The original entry is located in docstring of sage.plot.plot3d.base, line 11.)

Todo

Working with sparse matrices should usually give faster results, but with the current implementation it actually works slower. There should be a way to improve performance with regards to this.

(The original entry is located in docstring of sage.rings.polynomial.multi_polynomial_ring_generic.MPolynomialRing_generic.macaulay_resultant, line 39.)

Todo

What should we do about this method? Is nilpotency of a power series even decidable (assuming a nilpotency oracle in the base ring)? And I am not sure that returning `True` just because the series has finite precision and zero constant term is a good idea.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/multi_power_series_ring_element.py:docstring of sage.rings.multi_power_series_ring_element.MPowerSeries.is_nilpotent, line 13.)

Todo

This currently does not work for quivers with cycles, even if there are only finitely many paths from `start` to `end`.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/quivers/path_semigroup.py:docstring of sage.quivers.path_semigroup.PathSemigroup.all_paths, line 16.)

Todo

Change the wording `Reverse of ()` into something more meaningful.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/quivers/representation.py:docstring of sage.quivers.representation, line 389.)

Todo

Implement this method.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/ideal.py:docstring of sage.rings.ideal.Ideal_generic.absolute_norm, line 9.)

Todo

This is not implemented for many rings. Implement it!

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/ideal.py:docstring of sage.rings.ideal.Ideal_generic.is_maximal, line 4.)

Todo

Code is naive. Only keeps track of ideal generators as set during initialization of the ideal. (Can the base ring change? See example below.)

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/ideal.py:docstring of sage.rings.ideal.Ideal_generic.is_principal, line 4.)

Todo

The following skipped tests should be removed once trac ticket #13999 is fixed:

    sage: TestSuite(S).run(skip=['_test_nonzero_equal', '_test_elements', '_test_zero'])

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/quotient_ring.py:docstring of sage.rings.quotient_ring, line 15.)

Todo

Not yet implemented!

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/quotient_ring.py:docstring of sage.rings.quotient_ring.QuotientRing_nc.characteristic, line 3.)

Todo

Note that `ngens` counts 0 as a generator. Does this make sense? That is, since 0 only generates itself and the fact that this is true for all rings, is there a way to “knock it off” of the generators list if a generator of some original ring is modded out?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/quotient_ring.py:docstring of sage.rings.quotient_ring.QuotientRing_nc.ngens, line 3.)

Todo

Implement `ComplexIntervalFieldElement` multiplicative order similar to `ComplexNumber` multiplicative order with `_set_multiplicative_order(n)` and `ComplexNumber.multiplicative_order()` methods.

(The original entry is located in docstring of sage.rings.complex_interval, line 23.)

Todo

Implement `ComplexIntervalFieldElement` multiplicative order and set this output to have multiplicative order `n`.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/complex_interval_field.py:docstring of sage.rings.complex_interval_field.ComplexIntervalField_class.zeta, line 3.)

Todo

- implementation of matrices over the universal cyclotomic field.
- speed improvements of the cythonized methods.
- speed improvements for scalar multiples.
- Remove the inheritance from Field and FieldElement as soon as the methods `is_field(proof=True)` is implemented in the Fields category.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/rings/universal_cyclotomic_field/universal_cyclotomic_field.py:docstring of sage.rings.universal_cyclotomic_field.universal_cyclotomic_field, line 23.)

Todo

add heights to integer.pyx and remove special case

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/affine/affine_morphism.py:docstring of sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space.global_height, line 30.)

Todo

This could be improved.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/affine/affine_morphism.py:docstring of sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space.nth_iterate_map, line 7.)

Todo

p-adic heights

add heights to integer.pyx and remove special case

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/affine/affine_point.py:docstring of sage.schemes.affine.affine_point.SchemeMorphism_point_affine.global_height, line 34.)

Todo

Currently, `SchemeMorphism` copies code from `Map` rather than inheriting from it. This is to work around a bug in Cython: We want to create a common sub-class of `ModuleElement` and `SchemeMorphism`, but Cython would currently confuse cpdef attributes of the two base classes. Proper inheritance should be used as soon as this bug is fixed.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/generic/morphism.py:docstring of sage.schemes.generic.morphism.SchemeMorphism, line 7.)

Todo

Do the division when the base ring is p-adic or a function field so that the output is a polynomial.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/projective/projective_morphism.py:docstring of sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space.dynatomic_polynomial, line 41.)

Todo

add heights to integer.pyx and remove special case

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/projective/projective_morphism.py:docstring of sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space.global_height, line 42.)

Todo

Is there a more efficient way to do this?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/projective/projective_morphism.py:docstring of sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space.nth_iterate, line 7.)

Todo

This could be improved.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/projective/projective_morphism.py:docstring of sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space.nth_iterate_map, line 9.)

Todo

- move some of this to Cython so that it is faster especially the possible periods mod \(p\).
- have the last prime of good redution used also return the list of points instead of getting the information again for all_points.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/projective/projective_morphism.py:docstring of sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_field.rational_periodic_points, line 66.)

Todo

- do not return duplicate points
- improve hash to reduce memory of pointtable

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/projective/projective_morphism.py:docstring of sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_finite_field.possible_periods, line 41.)

Todo

p-adic heights

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/projective/projective_point.py:docstring of sage.schemes.projective.projective_point.SchemeMorphism_point_projective_ring.global_height, line 34.)

Todo

error bounds for dimension > 1

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/projective/projective_point.py:docstring of sage.schemes.projective.projective_point.SchemeMorphism_point_projective_ring.green_function, line 46.)

Todo

Is there a more efficient way to do this?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/schemes/projective/projective_point.py:docstring of sage.schemes.projective.projective_point.SchemeMorphism_point_projective_ring.nth_iterate, line 52.)

Todo

good name?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/sets/family.py:docstring of sage.sets.family.AbstractFamily.map, line 4.)

Todo

generalize to any number of families and merge with map?

(The original entry is located in /usr/lib/python2.7/site-packages/sage/sets/family.py:docstring of sage.sets.family.AbstractFamily.zip, line 5.)

Todo

FIXME: What should be the order of the result? That of `self.object()`? Or the order given by `set(self.object())`? Note that `__getitem__()` is currently implemented in term of this list method, which is really inefficient ...

(The original entry is located in /usr/lib/python2.7/site-packages/sage/sets/set.py:docstring of sage.sets.set.Set_object_enumerated.list, line 13.)

Todo

It is not yet possible to use `set_from_method` in conjunction with `cached_method`.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/sets/set_from_iterator.py:docstring of sage.sets.set_from_iterator.EnumeratedSetFromIterator_method_decorator, line 66.)

Todo

Move this method elsewhere (typically in the Modules category) so as not to pollute the namespace of all category objects.

(The original entry is located in docstring of sage.structure.category_object.CategoryObject.base_ring, line 33.)

Todo

title

(The original entry is located in /usr/lib/python2.7/site-packages/sage/structure/global_options.py:docstring of sage.structure.global_options.GlobalOptions.dispatch, line 1.)

Todo

Eventually, category should be `Sets` by default.

(The original entry is located in docstring of sage.structure.parent.Parent, line 50.)

Todo

Create a custom-made `SourPickle` for the last example.

(The original entry is located in docstring of sage.structure.sage_object.unpickle_all, line 68.)

Todo

Illustrate how this can be fixed on a case by case basis.

(The original entry is located in /usr/lib/python2.7/site-packages/sage/structure/unique_representation.py:docstring of sage.structure.unique_representation.CachedRepresentation, line 361.)

Todo

should reuse something preexisting ...

(The original entry is located in /usr/lib/python2.7/site-packages/sage/structure/unique_representation.py:docstring of sage.structure.unique_representation.unreduce, line 6.)

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