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

Depends on #11111 Depends on #12919

CC: @sagetrac-sage-combinat

Component: combinatorics

Issue created by migration from https://trac.sagemath.org/ticket/12914

[nthiery]

Dependencies: #11111,#12919

[nthiery]

Description changed:

--- 
+++ 
@@ -1,10 +1,11 @@
 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:
-  * What to merge now ; what to merge later
   * How to specify an indexing of the J-classes
-  * Should JTrivial / ... be adjectives?
   * 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()?

[nthiery]

Description changed:

--- 
+++ 
@@ -9,7 +9,7 @@
   * 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
+  * 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)?
@@ -29,4 +29,16 @@
   * 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
+

[nthiery]

Description changed:

--- 
+++ 
@@ -18,6 +18,9 @@
     * 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