#12914: Representation theory of finite semigroups
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Reporter: nthiery | Owner: sage-combinat
Type: task | Status: new
Priority: major | Milestone: sage-6.4
Component: combinatorics | Resolution:
Keywords: | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: #11111,#12919 | Stopgaps:
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Description changed by nthiery:
Old description:
> Add support for representation theory of finite semigroups. Quite some
> stuff is available in the sage-combinat queue.
>
> * #18230: basic hierarchy of categories for representations of monoids,
> lie algebras, ...
> * #18001: implement categories for H, L, R, J-trivial monoids
> * #16659: decomposition of finite dimensional associative algebras
> * Required discussions about the current features:
> * How to specify an indexing of the J-classes
> * Should representation theory questions be asked to the semigroup or
> its algebra?
> * S.character_ring(QQ, ZZ) or S.algebra(QQ).character_ring(ZZ) ?
> * S.simple_modules(QQ) or S.algebra(QQ).simple_modules()?
> * Character rings (code by Nicolas in the Sage-Combinat queue)
> * Should this be called Character ring?
> * How to specify the two base rings (for the representations / for the
> character ring)?
> * Should left and right characters live in the same space (with
> realizations)?
> e.g.:
> * Should there be coercions or conversions between the basis of left-
> class modules and right-class modules?
> * Should the basis of simple modules on the left and on the right be
> identified?
> * How to handle subspaces (like for projective modules when the Cartan
> matrix is not invertible)
> * If we discover that a semigroup is J-trivial, how to propagate this
> information to its algebra, character ring, ...?
>
> * Features that remain to be implemented:
> * is_r_trivial + _test_r_trivial and friends
> * Group of a regular J-class
> * Character table for any monoid
> * Cartan matrix for any monoid
> * Group of a non regular J-class
> * Cartan matrix by J-classes
> * Radical filtration of a module
> * Recursive construction of a triangular basis of the radical
>
> Related features:
>
> - Toy implementation of Specht modules as quotient of the space
> spanned by tabloids by the span of XXX.
>
> Code by Franco available. Dependencies: 11111=None!
>
> - LRegularBand code by Franco
>
> - Interface to the Monoids GAP package
>
> - Representation theory of monoids
New description:
Add support for representation theory of finite semigroups. Quite some
stuff is available in the sage-combinat queue.
* #18230: basic hierarchy of categories for representations of monoids,
lie algebras, ...
* #18001: implement categories for H, L, R, J-trivial monoids
* #16659: decomposition of finite dimensional associative algebras
* Required discussions about the current features:
* How to specify an indexing of the J-classes
* Should representation theory questions be asked to the semigroup or its
algebra?
* S.character_ring(QQ, ZZ) or S.algebra(QQ).character_ring(ZZ) ?
* S.simple_modules(QQ) or S.algebra(QQ).simple_modules()?
* Character rings (code by Nicolas in the Sage-Combinat queue)
* Should this be called Character ring?
* How to specify the two base rings (for the representations / for the
character ring)?
* Should left and right characters live in the same space (with
realizations)?
e.g.:
* Should there be coercions or conversions between the basis of left-
class modules and right-class modules?
* Should the basis of simple modules on the left and on the right be
identified?
* How to handle subspaces (like for projective modules when the Cartan
matrix is not invertible)
* If we discover that a semigroup is J-trivial, how to propagate this
information to its algebra, character ring, ...?
* how to handle bimodules: do we want to see as two (facade?)
modules, one on the left, and one on the right
* Features that remain to be implemented:
* is_r_trivial + _test_r_trivial and friends
* Group of a regular J-class
* Character table for any monoid
* Cartan matrix for any monoid
* Group of a non regular J-class
* Cartan matrix by J-classes
* Radical filtration of a module
* Recursive construction of a triangular basis of the radical
Related features:
- Toy implementation of Specht modules as quotient of the space
spanned by tabloids by the span of XXX.
Code by Franco available. Dependencies: 11111=None!
- LRegularBand code by Franco
- Interface to the Monoids GAP package
- Representation theory of monoids
--
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Ticket URL: <http://trac.sagemath.org/ticket/12914#comment:9>
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