Trac #17560: Implement (quantum) Mobius algebras

Based on ''The Kazhdan-Lusztig polynomial of a matroid'' by Ben Elias,
Nicholas Proudfoot, and Max Wakefield by recently posted to the arXiv
(1412.7408), this implements their results for general graded lattices.
In particular, this implements the Mobius algebra, and it's
q-deformation (which I've coined as the quantum Mobius algebra). This
also implements KL polynomials for general graded posets.

In particular, you can use #14786 and recover the KL polynomials.
However the code in its current state is quite slow (most of the time is
spent constructing the digraphs for the posets), but faster
implementations can be done on followup tickets.

URL: http://trac.sagemath.org/17560
Reported by: tscrim
Ticket author(s): Travis Scrimshaw
Reviewer(s): Kevin Dilks

src/doc/en/reference/combinat/module_list.rst

@@ -171,6 +171,7 @@ Comprehensive Module list
     sage/combinat/posets/incidence_algebras
     sage/combinat/posets/lattices
     sage/combinat/posets/linear_extensions
+    sage/combinat/posets/moebius_algebra
     sage/combinat/posets/poset_examples
     sage/combinat/posets/posets
     sage/combinat/q_analogues

src/sage/algebras/catalog.py

@@ -25,6 +25,7 @@
 - :class:`algebras.Incidence <sage.combinat.posets.incidence_algebras.IncidenceAlgebra>`
 - :class:`algebras.IwahoriHecke
   <sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra>`
+- :class:`algebras.Mobius <sage.combinat.posets.mobius_algebra.MobiusAlgebra>`
 - :class:`algebras.Jordan
   <sage.algebras.jordan_algebra.JordanAlgebra>`
 - :class:`algebras.NilCoxeter
@@ -56,6 +57,7 @@
 lazy_import('sage.algebras.schur_algebra', 'SchurAlgebra', 'Schur')
 lazy_import('sage.algebras.commutative_dga', 'GradedCommutativeAlgebra', 'GradedCommutative')
 lazy_import('sage.combinat.posets.incidence_algebras', 'IncidenceAlgebra', 'Incidence')
+lazy_import('sage.combinat.posets.mobius_algebra', 'MobiusAlgebra', 'Mobius')
 lazy_import('sage.combinat.free_prelie_algebra', 'FreePreLieAlgebra', 'FreePreLie')
 del lazy_import # We remove the object from here so it doesn't appear under tab completion
 

src/sage/combinat/posets/__init__.py

@@ -17,6 +17,8 @@
 
 - :ref:`sage.combinat.posets.cartesian_product`
 
+- :ref:`sage.combinat.posets.moebius_algebra`
+
 - :ref:`sage.combinat.tamari_lattices`
 - :ref:`sage.combinat.interval_posets`
 - :ref:`sage.combinat.shard_order`

src/sage/combinat/posets/lattices.py

@@ -55,6 +55,7 @@
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+
 from sage.categories.finite_lattice_posets import FiniteLatticePosets
 from sage.combinat.posets.posets import Poset, FinitePoset
 from sage.combinat.posets.elements import (LatticePosetElement,
@@ -1147,6 +1148,37 @@ def frattini_sublattice(self):
         return LatticePoset(self.subposet([self[x] for x in
                 self._hasse_diagram.frattini_sublattice()]))
 
+    def moebius_algebra(self, R):
+        """
+        Return the Mobius algebra of ``self`` over ``R``.
+
+        EXAMPLES::
+
+            sage: L = posets.BooleanLattice(4)
+            sage: L.moebius_algebra(QQ)
+            Moebius algebra of Finite lattice containing 16 elements over Rational Field
+        """
+        from sage.combinat.posets.moebius_algebra import MoebiusAlgebra
+        return MoebiusAlgebra(R, self)
+
+    def quantum_moebius_algebra(self, q=None):
+        """
+        Return the quantum Mobius algebra of ``self`` with parameter ``q``.
+
+        INPUT:
+
+        - ``q`` -- (optional) the deformation parameter `q`
+
+        EXAMPLES::
+
+            sage: L = posets.BooleanLattice(4)
+            sage: L.quantum_moebius_algebra()
+            Quantum Moebius algebra of Finite lattice containing 16 elements
+             with q=q over Univariate Laurent Polynomial Ring in q over Integer Ring
+        """
+        from sage.combinat.posets.moebius_algebra import QuantumMoebiusAlgebra
+        return QuantumMoebiusAlgebra(self, q)
+
 ############################################################################
 
 FiniteMeetSemilattice._dual_class = FiniteJoinSemilattice