Changes between Version 46 and Version 47 of ReleaseTours/sage-9.6

Timestamp:

06/09/22 00:19:04 (2 months ago)

Author:

selia

Comment:

added description for equivariant Ehrhart ticket

ReleaseTours/sage-9.6

@@ -291 +291 @@
 === Equivariant Ehrhart theory ===
 
-#27637
-
+Sage 9.6 allows for the calculation of the H*-series (also known as the φ-series) of a rational polytope which is invariant under the linear action of a finite group, as introduced by Alan Stapledon as part of equivariant Ehrhart theory #27637.
+The computation is made with the method `Hstar_function` of a rational polytope.
+The fixed subset of a polytope under the action of a group element may also be computed via the methods `fixed_subpolytope` or `fixed_subpolytopes`.
+
+As an example, consider the action of the symmetric group S3 on the 2-dimensional permutahedron in 3-dimensional space, given by permuting the three basis vectors.
+As shown by Ardila, Supina, and Vindas Meléndez, the corresponding H*-series is polynomial and effective:
+{{{
+sage: p3 = polytopes.permutahedron(3, backend = 'normaliz')      
+sage: G = p3.restricted_automorphism_group(output='permutation') 
+sage: reflection12 = G([(0,2),(1,4),(3,5)])                      
+sage: reflection23 = G([(0,1),(2,3),(4,5)])                      
+sage: S3 = G.subgroup(gens=[reflection12, reflection23])         
+sage: S3.is_isomorphic(SymmetricGroup(3))                        
+True
+sage: p3.Hstar_function(S3, output='complete')
+{'Hstar': chi_0*t^2 + (chi_0 + chi_1 + chi_2)*t + chi_0,
+ 'Hstar_as_lin_comb': (t^2 + t + 1, t, t),
+ 'conjugacy_class_reps': [(), (0,1)(2,3)(4,5), (0,3,4)(1,5,2)],
+ 'character_table': [ 1  1  1]
+ [ 1 -1  1]
+ [ 2  0 -1],
+ 'is_effective': True}
+sage: p3.fixed_subpolytope(reflection12)
+A 1-dimensional polyhedron in QQ^3 defined as the convex hull of 2 vertices
+sage: p3.fixed_subpolytope(reflection12).vertices()
+(A vertex at (3/2, 3/2, 3), A vertex at (5/2, 5/2, 1))
+}}}
 
 == Manifolds ==