Base class for polyhedra over \(\QQ\)

class sage.geometry.polyhedron.base_QQ.Polyhedron_QQ(parent, Vrep, Hrep, Vrep_minimal=None, Hrep_minimal=None, pref_rep=None, mutable=False, **kwds)

Bases: sage.geometry.polyhedron.base.Polyhedron_base

Base class for Polyhedra over \(\QQ\)

Hstar_function(acting_group=None, output=None)

Return \(H^*\) as a rational function in \(t\) with coefficients in the ring of class functions of the acting_group of this polytope.

Here, \(H^*(t) = \sum_{m} \chi_{m\text{self}}t^m \det(Id-\rho(t))\). The irreducible characters of acting_group form an orthonormal basis for the ring of class functions with values in \(\CC\). The coefficients of \(H^*(t)\) are expressed in this basis.

INPUT:

  • acting_group – (default=None) a permgroup object. A subgroup of the polytope’s restricted_automorphism_group. If None, it is set to the full restricted_automorphism_group of the polytope. The acting group should always use output=’permutation’.

  • output – string. an output option. The allowed values are:

    • None (default): returns the rational function \(H^*(t)\). \(H^*\) is a rational function in \(t\) with coefficients in the ring of class functions.

    • 'e_series_list': Returns a list of the ehrhart_series for the fixed_subpolytopes of each conjugacy class representative.

    • 'determinant_vec': Returns a list of the determinants of \(Id-\rho*t\) for each conjugacy class representative.

    • 'Hstar_as_lin_comb': Returns a vector of the coefficients of the irreducible representations in the expression of \(H^*\).

    • 'prod_det_es': Returns a vector of the product of determinants and the Ehrhart series.

    • 'complete': Returns a list with Hstar, Hstar_as_lin_comb, character table of the acting group, and whether Hstar is effective.

OUTPUT:

The default output is the rational function \(H^*\). \(H^*\) is a rational function in \(t\) with coefficients in the ring of class functions. There are several output options to see the intermediary outputs of the function.

EXAMPLES:

The \(H^*\)-polynomial of the standard (\(d-1\))-dimensional simplex \(S = conv(e_1, \dots, e_d)\) under its restricted_automorphism_group is equal to 1 = \(\chi_{trivial}\) (Prop 6.1 [Stap2011]). Here is the computation for the 3-dimensional standard simplex:

sage: S = polytopes.simplex(3, backend = 'normaliz'); S              # optional - pynormaliz
A 3-dimensional polyhedron in ZZ^4 defined as the convex hull of 4 vertices
sage: G = S.restricted_automorphism_group(output = 'permutation'); G # optional - pynormaliz
Permutation Group with generators [(2,3), (1,2), (0,1)]
sage: len(G)                                                         # optional - pynormaliz
24
sage: Hstar = S._Hstar_function_normaliz(G); Hstar                   # optional - pynormaliz
chi_4
sage: G.character_table()                                            # optional - pynormaliz
[ 1 -1  1  1 -1]
[ 3 -1  0 -1  1]
[ 2  0 -1  2  0]
[ 3  1  0 -1 -1]
[ 1  1  1  1  1]

The next example is Example 7.6 in [Stap2011], and shows that \(H^*\) is not always a polynomial. Let P be the polytope with vertices \(\pm(0,0,1),\pm(1,0,1), \pm(0,1,1), \pm(1,1,1)\) and let G = \(\Zmod{2}\) act on P as follows:

sage: P = Polyhedron(vertices=[[0,0,1],[0,0,-1],[1,0,1],[-1,0,-1],[0,1,1],   # optional - pynormaliz
....: [0,-1,-1],[1,1,1],[-1,-1,-1]],backend='normaliz')                      # optional - pynormaliz
sage: K = P.restricted_automorphism_group(output = 'permutation')            # optional - pynormaliz
sage: G = K.subgroup(gens = [K[6]]); G                                       # optional - pynormaliz
Subgroup generated by [(0,2)(1,3)(4,6)(5,7)] of (Permutation Group with generators [(2,4)(3,5), (1,2)(5,6), (0,1)(2,3)(4,5)(6,7), (0,7)(1,3)(2,5)(4,6)])
sage: conj_reps = G.conjugacy_classes_representatives()                      # optional - pynormaliz
sage: Dict = P.permutations_to_matrices(conj_reps, acting_group = G)   # optional - pynormaliz
sage: list(Dict.keys())[0]                                                   # optional - pynormaliz
(0,2)(1,3)(4,6)(5,7)
sage: list(Dict.values())[0]                                                 # optional - pynormaliz
[-1  0  1  0]
[ 0  1  0  0]
[ 0  0  1  0]
[ 0  0  0  1]
sage: len(G)                                                                 # optional - pynormaliz
2
sage: G.character_table()                                                    # optional - pynormaliz
[ 1  1]
[ 1 -1]

Then we calculate the rational function \(H^*(t)\):

sage: Hst = P._Hstar_function_normaliz(G); Hst     # optional - pynormaliz
(chi_0*t^4 + (3*chi_0 + 3*chi_1)*t^3 + (8*chi_0 + 2*chi_1)*t^2 + (3*chi_0 + 3*chi_1)*t + chi_0)/(t + 1)

To see the exact as written in [Stap2011], we can format it as 'Hstar_as_lin_comb'. The first coordinate is the coefficient of the trivial character; the second is the coefficient of the sign character:

sage: lin = P._Hstar_function_normaliz(G,output = 'Hstar_as_lin_comb'); lin  # optional - pynormaliz
((t^4 + 3*t^3 + 8*t^2 + 3*t + 1)/(t + 1), (3*t^3 + 2*t^2 + 3*t)/(t + 1))
ehrhart_polynomial(engine=None, variable='t', verbose=False, dual=None, irrational_primal=None, irrational_all_primal=None, maxdet=None, no_decomposition=None, compute_vertex_cones=None, smith_form=None, dualization=None, triangulation=None, triangulation_max_height=None, **kwds)

Return the Ehrhart polynomial of this polyhedron.

The polyhedron must be a lattice polytope. Let \(P\) be a lattice polytope in \(\RR^d\) and define \(L(P,t) = \# (tP\cap \ZZ^d)\). Then E. Ehrhart proved in 1962 that \(L\) coincides with a rational polynomial of degree \(d\) for integer \(t\). \(L\) is called the Ehrhart polynomial of \(P\). For more information see the Wikipedia article Ehrhart_polynomial. The Ehrhart polynomial may be computed using either LattE Integrale or Normaliz by setting engine to ‘latte’ or ‘normaliz’ respectively.

INPUT:

  • engine – string; The backend to use. Allowed values are:

    • None (default); When no input is given the Ehrhart polynomial is computed using LattE Integrale (optional)

    • 'latte'; use LattE integrale program (optional)

    • 'normaliz'; use Normaliz program (optional package pynormaliz). The backend of self must be set to ‘normaliz’.

  • variable – string (default: ‘t’); The variable in which the Ehrhart polynomial should be expressed.

  • When the engine is ‘latte’, the additional input values are:

    • verbose - boolean (default: False); If True, print the whole output of the LattE command.

    The following options are passed to the LattE command, for details consult the LattE documentation:

    • dual - boolean; triangulate and signed-decompose in the dual space

    • irrational_primal - boolean; triangulate in the dual space, signed-decompose in the primal space using irrationalization.

    • irrational_all_primal - boolean; triangulate and signed-decompose in the primal space using irrationalization.

    • maxdet – integer; decompose down to an index (determinant) of maxdet instead of index 1 (unimodular cones).

    • no_decomposition – boolean; do not signed-decompose simplicial cones.

    • compute_vertex_cones – string; either ‘cdd’ or ‘lrs’ or ‘4ti2’

    • smith_form – string; either ‘ilio’ or ‘lidia’

    • dualization – string; either ‘cdd’ or ‘4ti2’

    • triangulation - string; ‘cddlib’, ‘4ti2’ or ‘topcom’

    • triangulation_max_height - integer; use a uniform distribution of height from 1 to this number

OUTPUT:

A univariate polynomial in variable over a rational field.

See also

latte the interface to LattE Integrale PyNormaliz

EXAMPLES:

To start, we find the Ehrhart polynomial of a three-dimensional simplex, first using engine='latte'. Leaving the engine unspecified sets the engine to ‘latte’ by default:

sage: simplex = Polyhedron(vertices=[(0,0,0),(3,3,3),(-3,2,1),(1,-1,-2)])
sage: simplex = simplex.change_ring(QQ)
sage: poly = simplex.ehrhart_polynomial(engine='latte')  # optional - latte_int
sage: poly                                               # optional - latte_int
7/2*t^3 + 2*t^2 - 1/2*t + 1
sage: poly(1)                                            # optional - latte_int
6
sage: len(simplex.integral_points())                     # optional - latte_int
6
sage: poly(2)                                            # optional - latte_int
36
sage: len((2*simplex).integral_points())                 # optional - latte_int
36

Now we find the same Ehrhart polynomial, this time using engine='normaliz'. To use the Normaliz engine, the simplex must be defined with backend='normaliz':

sage: simplex = Polyhedron(vertices=[(0,0,0),(3,3,3),(-3,2,1),(1,-1,-2)], backend='normaliz') # optional - pynormaliz
sage: simplex = simplex.change_ring(QQ)                                                       # optional - pynormaliz
sage: poly = simplex.ehrhart_polynomial(engine = 'normaliz')                                  # optional - pynormaliz
sage: poly                                                                                    # optional - pynormaliz
7/2*t^3 + 2*t^2 - 1/2*t + 1

If the engine='normaliz', the backend should be 'normaliz', otherwise it returns an error:

sage: simplex = Polyhedron(vertices=[(0,0,0),(3,3,3),(-3,2,1),(1,-1,-2)])
sage: simplex = simplex.change_ring(QQ)
sage: simplex.ehrhart_polynomial(engine='normaliz')  # optional - pynormaliz
Traceback (most recent call last):
...
TypeError: The backend of the polyhedron should be 'normaliz'

The polyhedron should be compact:

sage: C = Polyhedron(backend='normaliz',rays=[[1,2],[2,1]])  # optional - pynormaliz
sage: C = C.change_ring(QQ)                                  # optional - pynormaliz
sage: C.ehrhart_polynomial()                                 # optional - pynormaliz
Traceback (most recent call last):
...
ValueError: Ehrhart polynomial only defined for compact polyhedra

The polyhedron should have integral vertices:

sage: L = Polyhedron(vertices = [[0],[1/2]])
sage: L.ehrhart_polynomial()
Traceback (most recent call last):
...
TypeError: the polytope has nonintegral vertices, use ehrhart_quasipolynomial with backend 'normaliz'
ehrhart_quasipolynomial(variable='t', engine=None, verbose=False, dual=None, irrational_primal=None, irrational_all_primal=None, maxdet=None, no_decomposition=None, compute_vertex_cones=None, smith_form=None, dualization=None, triangulation=None, triangulation_max_height=None, **kwds)

Compute the Ehrhart quasipolynomial of this polyhedron with rational vertices.

If the polyhedron is a lattice polytope, returns the Ehrhart polynomial, a univariate polynomial in variable over a rational field. If the polyhedron has rational, nonintegral vertices, returns a tuple of polynomials in variable over a rational field. The Ehrhart counting function of a polytope \(P\) with rational vertices is given by a quasipolynomial. That is, there exists a positive integer \(l\) and \(l\) polynomials \(ehr_{P,i} \text{ for } i \in \{1,\dots,l \}\) such that if \(t\) is equivalent to \(i\) mod \(l\) then \(tP \cap \mathbb Z^d = ehr_{P,i}(t)\).

INPUT:

  • variable – string (default: ‘t’); The variable in which the Ehrhart polynomial should be expressed.

  • engine – string; The backend to use. Allowed values are:

    • None (default); When no input is given the Ehrhart polynomial is computed using Normaliz (optional)

    • 'latte'; use LattE Integrale program (requires optional package ‘latte_int’)

    • 'normaliz'; use the Normaliz program (requires optional package ‘pynormaliz’). The backend of self must be set to ‘normaliz’.

  • When the engine is ‘latte’, the additional input values are:

    • verbose - boolean (default: False); If True, print the whole output of the LattE command.

    The following options are passed to the LattE command, for details consult the LattE documentation:

    • dual - boolean; triangulate and signed-decompose in the dual space

    • irrational_primal - boolean; triangulate in the dual space, signed-decompose in the primal space using irrationalization.

    • irrational_all_primal - boolean; triangulate and signed-decompose in the primal space using irrationalization.

    • maxdet – integer; decompose down to an index (determinant) of maxdet instead of index 1 (unimodular cones).

    • no_decomposition – boolean; do not signed-decompose simplicial cones.

    • compute_vertex_cones – string; either ‘cdd’ or ‘lrs’ or ‘4ti2’

    • smith_form – string; either ‘ilio’ or ‘lidia’

    • dualization – string; either ‘cdd’ or ‘4ti2’

    • triangulation - string; ‘cddlib’, ‘4ti2’ or ‘topcom’

    • triangulation_max_height - integer; use a uniform distribution of height from 1 to this number

OUTPUT:

A univariate polynomial over a rational field or a tuple of such polynomials.

See also

latte the interface to LattE Integrale PyNormaliz

Warning

If the polytope has rational, non integral vertices, it must have backend='normaliz'.

EXAMPLES:

As a first example, consider the line segment [0,1/2]. If we dilate this line segment by an even integral factor \(k\), then the dilated line segment will contain \(k/2 +1\) lattice points. If \(k\) is odd then there will be \(k/2+1/2\) lattice points in the dilated line segment. Note that it is necessary to set the backend of the polytope to ‘normaliz’:

sage: line_seg = Polyhedron(vertices=[[0],[1/2]],backend='normaliz') # optional - pynormaliz
sage: line_seg                                                       # optional - pynormaliz
A 1-dimensional polyhedron in QQ^1 defined as the convex hull of 2 vertices
sage: line_seg.ehrhart_quasipolynomial()                             # optional - pynormaliz
(1/2*t + 1, 1/2*t + 1/2)

For a more exciting example, let us look at the subpolytope of the 3 dimensional permutahedron fixed by the reflection across the hyperplane \(x_1 = x_4\):

sage: verts = [[3/2, 3, 4, 3/2],
....:  [3/2, 4, 3, 3/2],
....:  [5/2, 1, 4, 5/2],
....:  [5/2, 4, 1, 5/2],
....:  [7/2, 1, 2, 7/2],
....:  [7/2, 2, 1, 7/2]]
sage: subpoly = Polyhedron(vertices=verts, backend='normaliz') # optional - pynormaliz
sage: eq = subpoly.ehrhart_quasipolynomial()    # optional - pynormaliz
sage: eq                                        # optional - pynormaliz
(4*t^2 + 3*t + 1, 4*t^2 + 2*t)
sage: eq = subpoly.ehrhart_quasipolynomial()    # optional - pynormaliz
sage: eq                                        # optional - pynormaliz
(4*t^2 + 3*t + 1, 4*t^2 + 2*t)
sage: even_ep = eq[0]                           # optional - pynormaliz
sage: odd_ep  = eq[1]                           # optional - pynormaliz
sage: even_ep(2)                                # optional - pynormaliz
23
sage: ts = 2*subpoly                            # optional - pynormaliz
sage: ts.integral_points_count()                # optional - pynormaliz latte_int
23
sage: odd_ep(1)                                 # optional - pynormaliz
6
sage: subpoly.integral_points_count()           # optional - pynormaliz latte_int
6

A polytope with rational nonintegral vertices must have backend='normaliz':

sage: line_seg = Polyhedron(vertices=[[0],[1/2]])
sage: line_seg.ehrhart_quasipolynomial()
Traceback (most recent call last):
...
TypeError: The backend of the polyhedron should be 'normaliz'

The polyhedron should be compact:

sage: C = Polyhedron(backend='normaliz',rays=[[1/2,2],[2,1]])  # optional - pynormaliz
sage: C.ehrhart_quasipolynomial()                              # optional - pynormaliz
Traceback (most recent call last):
...
ValueError: Ehrhart quasipolynomial only defined for compact polyhedra

If the polytope happens to be a lattice polytope, the Ehrhart polynomial is returned:

sage: simplex = Polyhedron(vertices=[(0,0,0),(3,3,3),(-3,2,1),(1,-1,-2)], backend='normaliz') # optional - pynormaliz
sage: simplex = simplex.change_ring(QQ)                                                       # optional - pynormaliz
sage: poly = simplex.ehrhart_quasipolynomial(engine='normaliz')                               # optional - pynormaliz
sage: poly                                                                                    # optional - pynormaliz
7/2*t^3 + 2*t^2 - 1/2*t + 1
sage: simplex.ehrhart_polynomial()                                                            # optional - pynormaliz latte_int
7/2*t^3 + 2*t^2 - 1/2*t + 1
fixed_subpolytope(vertex_permutation)

Return the fixed subpolytope of this polytope by the cyclic action of vertex_permutation.

The fixed subpolytope of this polytope under the vertex_permutation is the subset of this polytope that is fixed pointwise.

INPUT:

  • vertex_permutation – permutation; a permutation of the vertices of self.

OUTPUT:

A subpolytope of self.

Note

The vertex_permutation is obtained as a permutation of the vertices represented as a permutation. For example, vertex_permutation = self.restricted_automorphism_group(output=’permutation’).

Requiring a lattice polytope as opposed to a rational polytope as input is purely conventional.

EXAMPLES:

The fixed subpolytopes of the cube can be obtained as follows:

sage: Cube = polytopes.cube(backend = 'normaliz')                   # optional - pynormaliz
sage: AG = Cube.restricted_automorphism_group(output='permutation') # optional - pynormaliz
sage: reprs = AG.conjugacy_classes_representatives()                # optional - pynormaliz

The fixed subpolytope of the identity element of the group is the entire cube:

sage: reprs[0]                                                      # optional - pynormaliz
()
sage: Cube.fixed_subpolytope(vertex_permutation = reprs[0])         # optional - pynormaliz
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 8
vertices
sage: _.vertices()                                                  # optional - pynormaliz
(A vertex at (-1, -1, -1),
A vertex at (-1, -1, 1),
A vertex at (-1, 1, -1),
A vertex at (-1, 1, 1),
A vertex at (1, -1, -1),
A vertex at (1, -1, 1),
A vertex at (1, 1, -1),
A vertex at (1, 1, 1))

You can obtain non-trivial examples:

sage: fsp1 = Cube.fixed_subpolytope(reprs[8]);fsp1                  # optional - pynormaliz
A 0-dimensional polyhedron in QQ^3 defined as the convex hull of 1 vertex
sage: fsp1.vertices()                                               # optional - pynormaliz
(A vertex at (0, 0, 0),)
sage: fsp2 = Cube.fixed_subpolytope(reprs[3]);fsp2                  # optional - pynormaliz
A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 4 vertices
sage: fsp2.vertices()                                               # optional - pynormaliz
(A vertex at (-1, -1, 0),
A vertex at (-1, 1, 0),
A vertex at (1, -1, 0),
A vertex at (1, 1, 0))

The next example shows that fixed_subpolytope works for rational polytopes:

sage: P = Polyhedron(vertices = [[0,0],[3/2,0],[3/2,3/2],[0,3/2]], backend ='normaliz') # optional - pynormaliz
sage: P.vertices()                                                   # optional - pynormaliz
(A vertex at (0, 0),
 A vertex at (0, 3/2),
 A vertex at (3/2, 0),
 A vertex at (3/2, 3/2))
sage: G = P.restricted_automorphism_group(output = 'permutation');G  # optional - pynormaliz
Permutation Group with generators [(1,2), (0,1)(2,3), (0,3)]
sage: len(G)                                                         # optional - pynormaliz
8
sage: G[2]                                                           # optional - pynormaliz
(0,1)(2,3)
sage: fixed_set = P.fixed_subpolytope(G[2]); fixed_set               # optional - pynormaliz
A 1-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices
sage: fixed_set.vertices()                                           # optional - pynormaliz
(A vertex at (0, 3/4), A vertex at (3/2, 3/4))
fixed_subpolytopes(conj_class_reps)

Return the fixed subpolytopes of this polytope under the actions of the given conjugacy class representatives.

The conj_class_reps are representatives of the conjugacy classes of a subgroup of the automorphism group of this polytope. For an element of the automorphism group, the fixed subpolytope is the subset of this polytope that is fixed pointwise.

INPUT:

  • conj_class_reps – a list of representatives of the conjugacy classes of the subgroup of the restricted_automorphism_group of the polytope. Each element is written as a permutation of the vertices of the polytope.

OUTPUT:

A dictionary where the elements of conj_class_reps are keys and the fixed subpolytopes are values.

Note

Two elements in the same conjugacy class fix lattice-isomorphic subpolytopes.

EXAMPLES:

Here is an example for the square:

sage: p = polytopes.hypercube(2, backend = 'normaliz'); p               # optional - pynormaliz
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices
sage: aut_p = p.restricted_automorphism_group(output = 'permutation')   # optional - pynormaliz
sage: aut_p.order()                                                     # optional - pynormaliz
8
sage: conj_list = aut_p.conjugacy_classes_representatives(); conj_list  # optional - pynormaliz
[(), (1,2), (0,1)(2,3), (0,1,3,2), (0,3)(1,2)]
sage: p.fixed_subpolytopes(conj_list)                                   # optional - pynormaliz
{(): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices,
(1,2): A 1-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices,
(0,1)(2,3): A 1-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices,
(0,1,3,2): A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex,
(0,3)(1,2): A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex}
integral_points_count(verbose=False, use_Hrepresentation=False, explicit_enumeration_threshold=1000, preprocess=True, **kwds)

Return the number of integral points in the polyhedron.

This method uses the optional package latte_int if an estimate for lattice points based on bounding boxes exceeds explicit_enumeration_threshold.

INPUT:

  • verbose (boolean; False by default) – whether to display verbose output.

  • use_Hrepresentation - (boolean; False by default) – whether to send the H or V representation to LattE

  • preprocess - (boolean; True by default) – whether, if the integral hull is known to lie in a coordinate hyperplane, to tighten bounds to reduce dimension

See also

latte the interface to LattE interfaces

EXAMPLES:

sage: P = polytopes.cube()
sage: P.integral_points_count()
27
sage: P.integral_points_count(explicit_enumeration_threshold=0) # optional - latte_int
27

We enlarge the polyhedron to force the use of the generating function methods implemented in LattE integrale, rather than explicit enumeration.

sage: (1000000000*P).integral_points_count(verbose=True) # optional - latte_int This is LattE integrale… … Total time:… 8000000012000000006000000001

We shrink the polyhedron a little bit:

sage: Q = P*(8/9)
sage: Q.integral_points_count()
1
sage: Q.integral_points_count(explicit_enumeration_threshold=0) # optional - latte_int
1

Unbounded polyhedra (with or without lattice points) are not supported:

sage: P = Polyhedron(vertices=[[1/2, 1/3]], rays=[[1, 1]])
sage: P.integral_points_count()
Traceback (most recent call last):
...
NotImplementedError: ...
sage: P = Polyhedron(vertices=[[1, 1]], rays=[[1, 1]])
sage: P.integral_points_count()
Traceback (most recent call last):
...
NotImplementedError: ...

“Fibonacci” knapsacks (preprocessing helps a lot):

sage: def fibonacci_knapsack(d, b, backend=None):
....:     lp = MixedIntegerLinearProgram(base_ring=QQ)
....:     x = lp.new_variable(nonnegative=True)
....:     lp.add_constraint(lp.sum(fibonacci(i+3)*x[i] for i in range(d)) <= b)
....:     return lp.polyhedron(backend=backend)
sage: fibonacci_knapsack(20, 12).integral_points_count() # does not finish with preprocess=False
33
is_effective(Hstar, Hstar_as_lin_comb)

Test for the effectiveness of the Hstar series of this polytope.

The Hstar series of the polytope is determined by the action of a subgroup of the polytope’s restricted_automorphism_group. The Hstar series is effective if it is a polynomial in \(t\) and the coefficient of each \(t^i\) is an effective character in the ring of class functions of the acting group. A character \(\rho\) is effective if the coefficients of the irreducible representations in the expression of \(\rho\) are non-negative integers.

INPUT:

  • Hstar – a rational function in \(t\) with coefficients in the ring of class functions.

  • Hstar_as_lin_comb – vector. The coefficients of the irreducible representations of the acting group in the expression of Hstar as a linear combination of irreducible representations with coefficients in the field of rational functions in \(t\).

OUTPUT:

Boolean. Whether the Hstar series is effective.

See also

Hstar_function()

EXAMPLES:

The \(H^*\) series of the two-dimensional permutahedron under the action of the symmetric group is effective:

sage: p2 = polytopes.permutahedron(3, backend = 'normaliz')      # optional - pynormaliz
sage: G = p2.restricted_automorphism_group(output='permutation') # optional - pynormaliz
sage: H = G.subgroup(gens=[G.gens()[1],G.gens()[2]])             # optional - pynormaliz
sage: H.order()                                                  # optional - pynormaliz
6
sage: [Hstar, Hlin] = [p2.Hstar_function(H), p2.Hstar_function(H, output = 'Hstar_as_lin_comb')] # optional - pynormaliz
sage: p2.is_effective(Hstar,Hlin)   # optional - pynormaliz
True

If the \(H^*\)-series is not polynomial, then it is not effective:

sage: P = Polyhedron(vertices=[[0,0,1],[0,0,-1],[1,0,1],[-1,0,-1],[0,1,1], # optional - pynormaliz
....: [0,-1,-1],[1,1,1],[-1,-1,-1]],backend='normaliz')                    # optional - pynormaliz
sage: G = P.restricted_automorphism_group(output = 'permutation')          # optional - pynormaliz
sage: H = G.subgroup(gens = [G[6]])                                        # optional - pynormaliz
sage: Hstar = P.Hstar_function(H); Hstar                                   # optional - pynormaliz
(chi_0*t^4 + (3*chi_0 + 3*chi_1)*t^3 + (8*chi_0 + 2*chi_1)*t^2 + (3*chi_0 + 3*chi_1)*t + chi_0)/(t + 1)
sage: Hstar_lin = P.Hstar_function(H, output = 'Hstar_as_lin_comb')        # optional - pynormaliz
sage: P.is_effective(Hstar, Hstar_lin)                               # optional - pynormaliz
False