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


Ignore:
Timestamp:
06/09/22 00:19:04 (2 months ago)
Author:
selia
Comment:

added description for equivariant Ehrhart ticket

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  • ReleaseTours/sage-9.6

    v46 v47  
    291291=== Equivariant Ehrhart theory ===
    292292
    293 #27637
    294 
     293Sage 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.
     294The computation is made with the method `Hstar_function` of a rational polytope.
     295The fixed subset of a polytope under the action of a group element may also be computed via the methods `fixed_subpolytope` or `fixed_subpolytopes`.
     296
     297As 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.
     298As shown by Ardila, Supina, and Vindas Meléndez, the corresponding H*-series is polynomial and effective:
     299{{{
     300sage: p3 = polytopes.permutahedron(3, backend = 'normaliz')     
     301sage: G = p3.restricted_automorphism_group(output='permutation')
     302sage: reflection12 = G([(0,2),(1,4),(3,5)])                     
     303sage: reflection23 = G([(0,1),(2,3),(4,5)])                     
     304sage: S3 = G.subgroup(gens=[reflection12, reflection23])         
     305sage: S3.is_isomorphic(SymmetricGroup(3))                       
     306True
     307sage: p3.Hstar_function(S3, output='complete')
     308{'Hstar': chi_0*t^2 + (chi_0 + chi_1 + chi_2)*t + chi_0,
     309 'Hstar_as_lin_comb': (t^2 + t + 1, t, t),
     310 'conjugacy_class_reps': [(), (0,1)(2,3)(4,5), (0,3,4)(1,5,2)],
     311 'character_table': [ 1  1  1]
     312 [ 1 -1  1]
     313 [ 2  0 -1],
     314 'is_effective': True}
     315sage: p3.fixed_subpolytope(reflection12)
     316A 1-dimensional polyhedron in QQ^3 defined as the convex hull of 2 vertices
     317sage: p3.fixed_subpolytope(reflection12).vertices()
     318(A vertex at (3/2, 3/2, 3), A vertex at (5/2, 5/2, 1))
     319}}}
    295320
    296321== Manifolds ==