Representation theory of finite semigroups #12914
Description
Add support for representation theory of finite semigroups. Quite some
stuff is available in the sage-combinat queue.
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Basic hierarchy of categories for representations of monoids, lie algebras, ... #18230: basic hierarchy of categories for representations of monoids, lie algebras, ...
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Implement category for H, L, R, J trivials monoids #18001: implement categories for H, L, R, J-trivial monoids
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Representation theory of finite dimensional associative algebras #16659: decomposition of finite dimensional associative algebras
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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
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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:
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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!
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LRegularBand code by Franco
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Interface to the Monoids GAP package
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Representation theory of monoids
Depends on #11111
Depends on #12919
CC: @sagetrac-sage-combinat
Component: combinatorics
Issue created by migration from https://trac.sagemath.org/ticket/12914