cutgeneratingfunctionology/multirow/equivariant-iii-remark-2-13.sage:…

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cutgeneratingfunctionology/multirow/equivariant-iii-remark-2-13.sage

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+## Example from Equivariant III, Remark 2.13
+
+def remark_2_13_vertices():
+    """
+    The additivity domain from Remark 2.13.
+    
+    The following doctests verify some claims of this remark.
+
+    TESTS::
+
+        sage: vertices = remark_2_13_vertices()
+        sage: F = Polyhedron(vertices)
+        sage: len(vertices) == len(F.vertices())
+        True
+        sage: p1F, p2F, p3F = projections(F)
+        sage: sorted(p1F.vertices_list())
+        [[0, 3], [0, 4], [3, 4]]
+        sage: sorted(p2F.vertices_list())
+        [[4, 1], [5, 0], [5, 2]]
+        sage: sorted(p3F.vertices_list())
+        [[5, 5], [5, 6], [8, 4]]
+        sage: F == FIJK(p1F, p2F, p3F)
+        True
+
+    EXAMPLES::
+
+        Reproduce a part of Figure 4.
+
+        sage: plot(p1F,polygon='red') + plot(p2F,polygon='blue') + plot(p3F,polygon='green') #not tested
+
+    """
+    vertices = [ vector (v)
+                 for v in [ [ 0, 4, 5, 1 ], [ 0, 3, 5, 2 ], [ 2, 11/3, 4, 1],
+                            [ 1, 4, 4, 1 ], [ 8/3, 35/9, 4, 1 ], [ 5/2, 4, 4, 1 ],
+                            [ 1, 10/3, 5, 2 ], [ 0, 4, 5, 2 ], [ 3, 4, 5, 0 ] ] ]
+    return vertices
+
+def projections(F):
+    """
+    Return the projections p1, p2, p3 of a face F.
+
+    Currently only implemented for the two-dimensional case.
+    """
+    vertices = F.vertices()
+    p1 = matrix(QQ, [[1, 0, 0, 0], [0, 1, 0, 0]])
+    p1F = Polyhedron([ p1 * v.vector() for v in vertices ])
+    p2 = matrix(QQ, [[0, 0, 1, 0], [0, 0, 0, 1]])
+    p2F = Polyhedron([ p2 * v.vector() for v in vertices ])
+    p3 = p1 + p2
+    p3F = Polyhedron([ p3 * v.vector() for v in vertices ])
+    return p1F, p2F, p3F
+
+def FIJK(I, J, K):
+    """
+    The face (polyhedron) F(I, J, K), where I, J, K are polyhedra.
+    """
+    zero = [0] * I.ambient_dim()
+    ieqs = []
+    eqns = []
+    ieqs.extend( [ [i[0]] + list(i[1:]) + zero
+                   for i in I.inequality_generator() ] )
+    eqns.extend( [ [i[0]] + list(i[1:]) + zero
+                   for i in I.equation_generator() ] )
+    ieqs.extend( [ [i[0]] + zero + list(i[1:])
+                   for i in J.inequality_generator() ] )
+    eqns.extend( [ [i[0]] + zero + list(i[1:])
+                   for i in J.equation_generator() ] )
+    ieqs.extend( [ [i[0]] + list(i[1:]) + list(i[1:])
+                   for i in K.inequality_generator() ] )
+    eqns.extend( [ [i[0]] + list(i[1:]) + list(i[1:])
+                   for i in K.equation_generator() ] )
+    return Polyhedron(ieqs=ieqs, eqns=eqns)