Adding some more examples and doc.

src/sage/algebras/down_up_algebra.py

@@ -59,6 +59,17 @@ class DownUpAlgebra(CombinatorialFreeModule):
  where `y` covers `x` and `z` covers `y`. For `r`-differential posets
  we have `du - ud = r 1` and afford a representation of a
  :class:`Weyl algebra <sage.algebras.weyl_algebra.DifferentialWeylAlgebra>`.
+ This is obtained from DU(0, 1, 2r)`. For a `(q,r)`-differential poset,
+ we have the `d` and `u` operators satisfying
+
+ .. MATH::
+
+ \begin{aligned}
+ d^2u & = q(q+1) dud - q^3 ud^2 + r d,
+ \\ du^2 & = q(q+1) udu - q^3 u^2d + r u,
+ \end{aligned}
+
+ or `\alpha = q(q+1)`, `\beta = -q^3`, and `\gamma = r`.
 
  EXAMPLES:
 
@@ -108,6 +119,32 @@ class DownUpAlgebra(CombinatorialFreeModule):
  sage: d*u^2 - 2*u*d*u + u^2*d == g*u
  True
 
+ Young's lattice is known to be a differential poset. Thus we can
+ construct a representation of `DU(0, 1, 2)` on this poset (which
+ gives a proof that Fomin's :class:`growth diagrams <GrowthDiagram>`
+ are equivalent to edge local rules or shadow lines construction
+ for :func:`RSK`)::
+
+ sage: DU = algebras.DownUp(0, 1, 2)
+ sage: d, u = DU.gens()
+ sage: d^2*u == 0*d*u*d + 1*u*d*d + 2*d
+ True
+ sage: d*u^2 == 0*u*d*u + 1*u*u*d + 2*u
+ True
+
+ sage: YL = CombinatorialFreeModule(DU.base_ring(), Partitions())
+ sage: def d_action(la):
+ ....: return YL.sum_of_monomials(la.remove_cell(*c) for c in la.removable_cells())
+ sage: def u_action(la):
+ ....: return YL.sum_of_monomials(la.add_cell(*c) for c in la.addable_cells())
+ sage: D = YL.module_morphism(on_basis=d_action, codomain=YL)
+ sage: U = YL.module_morphism(on_basis=u_action, codomain=YL)
+ sage: for la in PartitionsInBox(5, 5):
+ ....: b = YL.basis()[la]
+ ....: assert (D*D*U)(b) == 0*(D*U*D)(b) + 1*(U*D*D)(b) + 2*D(b)
+ ....: assert (D*U*U)(b) == 0*(U*D*U)(la) + 1*(U*U*D)(b) + 2*U(b)
+ ....: assert (D*U)(b) == (U*D)(b) + b # the Weyl algebra relation
+
  .. TODO::
 
  Implement the homogenized version.
@@ -497,6 +534,17 @@ class VermaModule(CombinatorialFreeModule):
  sage: B = crystals.Tableaux(['A',1], shape=[5])
  sage: [b.weight() for b in B]
  [(5, 0), (4, 1), (3, 2), (2, 3), (1, 4), (0, 5)]
+
+ An example with periodic weights (see Theorem 2.13 of [BR1998]_)::
+
+ sage: k.<z6> = CyclotomicField(6)
+ sage: al = z6 + 1
+ sage: (al - 1)^6 == 1
+ True
+ sage: DU = algebras.DownUp(al, 1-al, 0)
+ sage: V = DU.verma_module(5)
+ sage: list(V.weights()[:8])
+ [5, 5*z6 + 5, 10*z6, 10*z6 - 5, 5*z6 - 5, 0, 5, 5*z6 + 5]
  """
  @staticmethod
  def __classcall_private__(cls, DU, la):