Not all Verma module morphisms are found #36793
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…ensional Lie algebra representation
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The goal of this PR is to implement the BGG resolution of a finite
dimensional simple Lie algebra representation $L_{\lambda}$. In order to
implement this, we need to implement a number of features:
- Simple modules (this does not assume finite dimensional).
- Dual modules in BGG category $\mathcal{O}$.
- Fix embeddings of Verma modules sagemath#36793.
- Improve construction of elements of free abelian monoids when strings
are indices.
- Move `to_root_vector()` to a method of fundamental weights and cache
the inverse Cartan matrix.
- Add `is_dominant_weight()` method to weight lattice realization
elements.
- Add `is_verma_dominant()` to define a dominant weight in the sense of
the Humphreys 2008 reference, which means the Verma module is projective
(that is, maximal in its dot action orbit).
- Add the category of `FiniteDimensionalKacMoodyAlgebras`.
- `@cache_in_parent_method` marked `weak_le` for Coxeter groups.
- Subclass of PBW bases for semisimple Lie algebras.
We also add methods to compute the contravariant form of a Verma module
and add tester methods for them being simple or projective.
It is possible to split this up into smaller parts, but it is natural
for them to go together given the final goal.
One major TODO is to speed up the computation of the bases of the simple
modules. The main place to optimize this would be to improve the
multiplication of the PBW basis code, which for finite dimensional Lie
algebras (not necessarily semisimple), this could be improved by using
the fixed basis order and using the monomials as vectors to avoid lots
of calls for getting the sort order and comparing. In particular, the
triangular sorting order for the Chevalley basis implementation has
error messages being raised (and caught) for comparing the roots and
coroots. This will be done at some point on a separate PR.
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### ⌛ Dependencies
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URL: sagemath#37297
Reported by: Travis Scrimshaw
Reviewer(s): Matthias Köppe, Travis Scrimshaw
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Sage claims that there is no module morphism between these two Verma modules:
However, there should be one when
w \leq vin Bruhat order. Indeed, we can find the singular vector by an explicit computation:This comes from the fact the current implementation for singular vectors only works with the weights are "strongly linked," or equivalently when they differ by left weak Bruhat order.
The more general algorithm from, e.g., de Graaf's work should be implemented for the singular vectors.
Expected Behavior
This should have dimension 1.
Environment
Checklist
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