Trac #24032: add noncrossing partition lattices to poset catalog

and make them work for Coxeter groups

URL: https://trac.sagemath.org/24032
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Christian Stump

src/sage/combinat/posets/poset_examples.py

@@ -35,6 +35,7 @@
     :meth:`~Posets.IntegerCompositions` | Return the poset of integer compositions of `n`.
     :meth:`~Posets.IntegerPartitions` | Return the poset of integer partitions of ``n``.
     :meth:`~Posets.IntegerPartitionsDominanceOrder` | Return the lattice of integer partitions on the integer `n` ordered by dominance.
+    :meth:`~Posets.NoncrossingPartitions` | Return the poset of noncrossing partitions of a finite Coxeter group ``W``.
     :meth:`~Posets.PentagonPoset` | Return the Pentagon poset.
     :meth:`~Posets.RandomLattice` | Return a random lattice on `n` elements.
     :meth:`~Posets.RandomPoset` | Return a random poset on `n` elements.
@@ -1157,6 +1158,27 @@ def CoxeterGroupAbsoluteOrderPoset(W, use_reduced_words=True):
             return Poset({s: s.absolute_covers() for s in W}, element_labels)
         return Poset({s: s.absolute_covers() for s in W})
 
+    @staticmethod
+    def NoncrossingPartitions(W):
+        """
+        Return the lattice of noncrossing partitions.
+
+        INPUT:
+
+        - ``W`` -- a finite Coxeter group or a Weyl group
+
+        EXAMPLES::
+
+            sage: W = CoxeterGroup(['A', 3])
+            sage: posets.NoncrossingPartitions(W)
+            Finite lattice containing 14 elements
+
+            sage: W = WeylGroup(['B', 2], prefix='s')
+            sage: posets.NoncrossingPartitions(W)
+            Finite lattice containing 6 elements
+        """
+        return W.noncrossing_partition_lattice()
+
     @staticmethod
     def SymmetricGroupAbsoluteOrderPoset(n, labels="permutations"):
         r"""

src/sage/combinat/root_system/coxeter_group.py

@@ -160,4 +160,3 @@ def CoxeterGroup(data, implementation="reflection", base_ring=None, index_set=No
 
 from sage.structure.sage_object import register_unpickle_override
 register_unpickle_override('sage.combinat.root_system.coxeter_group', 'CoxeterGroupAsPermutationGroup',  ReflectionGroup)
-

src/sage/combinat/root_system/coxeter_matrix.py

@@ -905,7 +905,7 @@ def is_simply_laced(self):
 
     def is_crystallographic(self):
         """
-        Return if ``self`` is crystallographic.
+        Return whether ``self`` is crystallographic.
 
         A Coxeter matrix is crystallographic if all non-diagonal entries
         are either 2, 4, or 6.
@@ -921,6 +921,21 @@ def is_crystallographic(self):
         L = [1, 2, 3, 4, 6]
         return all(x in L for row in self for x in row)
 
+    def is_irreducible(self):
+        """
+        Return whether ``self`` is irreducible.
+
+        A Coxeter matrix is irreducible if the Coxeter graph is connected.
+
+        EXAMPLES::
+
+            sage: CoxeterMatrix([['F',4],['A',1]]).is_irreducible()
+            False
+            sage: CoxeterMatrix(['H',3]).is_irreducible()
+            True
+        """
+        return self.coxeter_graph().is_connected()
+
     def is_finite(self):
         """
         Return if ``self`` is a finite type or ``False`` if unknown.

src/sage/groups/matrix_gps/coxeter_group.py

@@ -256,19 +256,19 @@ def __init__(self, coxeter_matrix, base_ring, index_set):
         We check that :trac:`16630` is fixed::
 
             sage: CoxeterGroup(['D',4], base_ring=QQ).category()
-            Category of finite coxeter groups
+            Category of finite irreducible coxeter groups
             sage: CoxeterGroup(['H',4], base_ring=QQbar).category()
-            Category of finite coxeter groups
+            Category of finite irreducible coxeter groups
             sage: F = CoxeterGroups().Finite()
             sage: all(CoxeterGroup([letter,i]) in F
             ....:     for i in range(2,5) for letter in ['A','B','D'])
             True
             sage: all(CoxeterGroup(['E',i]) in F for i in range(6,9))
             True
             sage: CoxeterGroup(['F',4]).category()
-            Category of finite coxeter groups
+            Category of finite irreducible coxeter groups
             sage: CoxeterGroup(['G',2]).category()
-            Category of finite coxeter groups
+            Category of finite irreducible coxeter groups
             sage: all(CoxeterGroup(['H',i]) in F for i in range(3,5))
             True
             sage: all(CoxeterGroup(['I',i]) in F for i in range(2,5))
@@ -318,6 +318,8 @@ def val(x):
             category = category.Finite()
         else:
             category = category.Infinite()
+        if self._matrix.is_irreducible():
+            category = category.Irreducible()
         self._index_set_inverse = {i: ii for ii,i in enumerate(self._matrix.index_set())}
         FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(n), base_ring,
                                                       gens, category=category)