Representation theory of finite semigroups

[nthiery]

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