Opened 10 years ago

Last modified 7 years ago

#12914 new task

Representation theory of finite semigroups

Reported by: nthiery Owned by: sage-combinat
Priority: major Milestone: sage-6.4
Component: combinatorics Keywords:
Cc: sage-combinat Merged in:
Authors: Reviewers:
Report Upstream: N/A Work issues:
Branch: Commit:
Dependencies: #11111,#12919 Stopgaps:

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Description (last modified by 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

Change History (9)

comment:1 Changed 10 years ago by nthiery

  • Cc sage-combinat added
  • Component changed from PLEASE CHANGE to combinatorics
  • Owner changed from tbd to sage-combinat
  • Type changed from PLEASE CHANGE to task

comment:2 Changed 10 years ago by nthiery

  • Dependencies set to #11111,#12919

comment:3 Changed 9 years ago by jdemeyer

  • Milestone changed from sage-5.11 to sage-5.12

comment:4 Changed 8 years ago by vbraun_spam

  • Milestone changed from sage-6.1 to sage-6.2

comment:5 Changed 8 years ago by vbraun_spam

  • Milestone changed from sage-6.2 to sage-6.3

comment:6 Changed 8 years ago by vbraun_spam

  • Milestone changed from sage-6.3 to sage-6.4

comment:7 Changed 7 years ago by nthiery

  • Description modified (diff)

comment:8 Changed 7 years ago by nthiery

  • Description modified (diff)

comment:9 Changed 7 years ago by nthiery

  • Description modified (diff)
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