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Merge branch 'species/rename' into lazy-species
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@mantepse
mantepse committed Feb 7, 2025
2 parents e781f39 + 15ed781 commit 8c3a64e
Showing 1 changed file with 112 additions and 55 deletions.
167 changes: 112 additions & 55 deletions src/sage/rings/species.py
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Original file line number Diff line number Diff line change
Expand Up @@ -33,6 +33,7 @@

from itertools import accumulate, chain, product

from sage.arith.misc import divisors
from sage.categories.graded_algebras_with_basis import GradedAlgebrasWithBasis
from sage.categories.modules import Modules
from sage.categories.monoids import Monoids
Expand All @@ -42,16 +43,18 @@
from sage.combinat.cyclic_sieving_phenomenon import orbit_decomposition
from sage.combinat.free_module import CombinatorialFreeModule
from sage.combinat.integer_vector import IntegerVectors
from sage.combinat.integer_vector_weighted import WeightedIntegerVectors
from sage.combinat.partition import Partitions, _Partitions
from sage.combinat.set_partition_ordered import OrderedSetPartitions
from sage.combinat.sf.sf import SymmetricFunctions
from sage.groups.perm_gps.constructor import PermutationGroupElement
from sage.groups.perm_gps.permgroup import PermutationGroup, PermutationGroup_generic
from sage.groups.perm_gps.permgroup_named import SymmetricGroup
from sage.libs.gap.libgap import libgap
from sage.misc.cachefunc import cached_method
from sage.misc.cachefunc import cached_method, cached_function
from sage.misc.fast_methods import WithEqualityById
from sage.misc.inherit_comparison import InheritComparisonClasscallMetaclass
from sage.misc.misc_c import prod
from sage.modules.free_module_element import vector
from sage.monoids.indexed_free_monoid import (IndexedFreeAbelianMonoid,
IndexedFreeAbelianMonoidElement)
Expand Down Expand Up @@ -320,13 +323,13 @@ def __lt__(self, other):
sage: len(P.cover_relations())
7
sage: sorted(P.cover_relations(), key=str)
[[C_4, {((1,2)(3,4),)}],
[[C_4, E_2(X^2)],
[E_2(E_2), C_4],
[E_2(E_2), Pb_4],
[E_4, E_2(E_2)],
[E_4, Eo_4],
[E_4, P_4],
[Eo_4, Pb_4],
[P_4, C_4],
[P_4, Pb_4],
[Pb_4, {((1,2)(3,4),)}]]
[Pb_4, E_2(X^2)]]
TESTS::
Expand Down Expand Up @@ -513,7 +516,7 @@ def _element_constructor_(self, G, pi=None, check=True):
sage: G = PermutationGroup([[(1,3),(5,7)]], domain=[1,3,5,7])
sage: A(G, ([1,3], [5,7]))
{((1,2)(3,4),): ({1, 2}, {3, 4})}
E_2(X*Y)
Test that errors are raised on some possible misuses::
Expand Down Expand Up @@ -594,7 +597,7 @@ def _element_constructor_(self, G, pi=None, check=True):

def _rename(self, n):
r"""
Names for common species.
Give common species of degree `n` their traditional names.
EXAMPLES::
Expand All @@ -609,9 +612,9 @@ def _rename(self, n):
sage: A(CyclicPermutationGroup(4), {1: range(1, 5)})
C_4(Y)
sage: A(DihedralGroup(4), {0: range(1, 5)})
P_4(X)
E_2(E_2(X))
sage: A(DihedralGroup(4), {1: range(1, 5)})
P_4(Y)
E_2(E_2(Y))
sage: A(AlternatingGroup(4), {0: range(1, 5)})
Eo_4(X)
sage: A(AlternatingGroup(4), {1: range(1, 5)})
Expand Down Expand Up @@ -652,6 +655,8 @@ def _rename(self, n):
[(i, i+1) for i in range(1, n, 2)]]
self(PermutationGroup(gens), pi, check=False).rename(f"Pb_{n}" + sort)

_atomic_set_like_species(n, self._names)

def __contains__(self, x):
r"""
Return whether ``x`` is in ``self``.
Expand Down Expand Up @@ -780,11 +785,7 @@ def _an_element_(self):
sage: A = AtomicSpecies("X, Y")
sage: a = A.an_element(); a
{((1,2)(3,4),): ({1, 2}, {3, 4})}
sage: a.rename("E_2(XY)")
sage: a
E_2(XY)
E_2(X*Y)
"""
G = PermutationGroup([[(2 * s + 1, 2 * s + 2) for s in range(self._arity)]])
m = {s: [2 * s + 1, 2 * s + 2] for s in range(self._arity)}
Expand Down Expand Up @@ -908,7 +909,7 @@ class MolecularSpecies(IndexedFreeAbelianMonoid):
sage: M = MolecularSpecies("X,Y")
sage: G = PermutationGroup([[(1,2),(3,4)], [(5,6)]])
sage: M(G, {0: [5,6], 1: [1,2,3,4]})
E_2(X)*{((1,2)(3,4),): ({}, {1, 2, 3, 4})}
E_2(X)*E_2(Y^2)
"""
@staticmethod
def __classcall__(cls, names):
Expand Down Expand Up @@ -1003,11 +1004,11 @@ def _element_constructor_(self, G, pi=None, check=True):
sage: a = lambda g, x: SetPartition([[g(e) for e in b] for b in x])
sage: X = SetPartitions(4, [2, 2])
sage: M((X, a, 'left'), {0: X.base_set()})
P_4
E_2(E_2)
sage: X = SetPartitions(8, [4, 2, 2])
sage: M((X, a, 'left'), {0: X.base_set()}, check=False)
E_4*P_4
E_4*E_2(E_2)
TESTS::
Expand All @@ -1019,7 +1020,7 @@ def _element_constructor_(self, G, pi=None, check=True):
sage: G = PermutationGroup([[(2,3),(4,5)]], domain=[2,3,4,5])
sage: M(G, {0: [2, 3], 1: [4, 5]})
E_2(XY)
E_2(X*Y)
sage: X = SetPartitions(4, 2)
sage: a = lambda g, x: SetPartition([[g(e) for e in b] for b in x])
Expand All @@ -1030,7 +1031,7 @@ def _element_constructor_(self, G, pi=None, check=True):
sage: G = PermutationGroup([[(1,3),(5,7)]], domain=[1,3,5,7])
sage: M(G, ([1,3], [5,7]))
E_2(XY)
E_2(X*Y)
sage: G = PermutationGroup([[(1,2), (3,4,5,6)]])
sage: a = M(G, {0:[1,2], 1:[3,4,5,6]})
Expand Down Expand Up @@ -1239,23 +1240,23 @@ def _richcmp_(self, other, op):
sage: len(P.cover_relations())
17
sage: sorted(P.cover_relations(), key=str)
[[C_4, {((1,2)(3,4),)}],
[[C_4, E_2(X^2)],
[E_2(E_2), C_4],
[E_2(E_2), E_2^2],
[E_2(E_2), Pb_4],
[E_2(X^2), X^4],
[E_2^2, E_2(X^2)],
[E_2^2, X^2*E_2],
[E_2^2, {((1,2)(3,4),)}],
[E_4, E_2(E_2)],
[E_4, Eo_4],
[E_4, P_4],
[E_4, X*E_3],
[Eo_4, Pb_4],
[Eo_4, X*C_3],
[P_4, C_4],
[P_4, E_2^2],
[P_4, Pb_4],
[Pb_4, {((1,2)(3,4),)}],
[Pb_4, E_2(X^2)],
[X*C_3, X^4],
[X*E_3, X*C_3],
[X*E_3, X^2*E_2],
[X^2*E_2, X^4],
[{((1,2)(3,4),)}, X^4]]
[X^2*E_2, X^4]]
TESTS::
Expand Down Expand Up @@ -1312,7 +1313,7 @@ def permutation_group(self):
sage: M = MolecularSpecies("X,Y")
sage: G = PermutationGroup([[(1,2),(3,4)], [(5,6)]])
sage: A = M(G, {0: [5,6], 1: [1,2,3,4]}); A
E_2(X)*{((1,2)(3,4),): ({}, {1, 2, 3, 4})}
E_2(X)*E_2(Y^2)
sage: A.permutation_group()
(Permutation Group with generators [(3,4)(5,6), (1,2)],
(frozenset({1, 2}), frozenset({3, 4, 5, 6})))
Expand All @@ -1337,7 +1338,7 @@ def permutation_group(self):
sage: G = PermutationGroup([[(1,2),(3,4)], [(5,6)]])
sage: A = M(G, {0: [5,6], 1: [1,2,3,4]})
sage: A * B
E_2(X)*C_3(X)*{((1,2)(3,4),): ({}, {1, 2, 3, 4})}
E_2(X)*C_3(X)*E_2(Y^2)
sage: (A*B).permutation_group()
(Permutation Group with generators [(6,7)(8,9), (3,4,5), (1,2)],
(frozenset({1, 2, 3, 4, 5}), frozenset({6, 7, 8, 9})))
Expand Down Expand Up @@ -1502,7 +1503,7 @@ def __call__(self, *args):
sage: X(E2)
E_2
sage: E2(E2)
P_4
E_2(E_2)
sage: M = MolecularSpecies(["X","Y"])
sage: X = M(SymmetricGroup(1), {0: [1]})
Expand Down Expand Up @@ -1747,22 +1748,22 @@ def tilde(self):
sage: P = PolynomialSpecies(QQ, "X")
sage: sortkey = lambda x: (len(x[1]), sum(x[1].coefficients()), str(x[0]))
sage: n=4; table(sorted([(m, P.monomial(m).tilde()) for m in M.subset(n)], key=sortkey))
X^4 X^4
X^2*E_2 2*X^2*E_2
{((1,2)(3,4),)} 2*{((1,2)(3,4),)}
X*C_3 3*X*C_3
C_4 4*C_4
E_2^2 4*E_2^2
Pb_4 4*Pb_4
X*E_3 X*E_3 + X^2*E_2 + X*C_3
Eo_4 Eo_4 + 2*X*C_3 + Pb_4
P_4 2*P_4 + E_2^2 + Pb_4 + C_4
E_4 E_4 + E_2^2 + X*C_3 + P_4 + C_4
X^4 X^4
E_2(X^2) 2*E_2(X^2)
X^2*E_2 2*X^2*E_2
X*C_3 3*X*C_3
C_4 4*C_4
E_2^2 4*E_2^2
Pb_4 4*Pb_4
X*E_3 X*E_3 + X^2*E_2 + X*C_3
Eo_4 Eo_4 + 2*X*C_3 + Pb_4
E_2(E_2) 2*E_2(E_2) + E_2^2 + Pb_4 + C_4
E_4 E_4 + E_2^2 + X*C_3 + E_2(E_2) + C_4
sage: P.<X,Y> = PolynomialSpecies(QQ)
sage: E2 = PolynomialSpecies(QQ, "X")(SymmetricGroup(2))
sage: E2(X*Y).tilde()
2*E_2(XY)
2*E_2(X*Y)
"""
P = self.parent()
M = P._indices
Expand Down Expand Up @@ -1886,7 +1887,7 @@ def _compose_with_singletons(self, names, args):
sage: P = PolynomialSpecies(ZZ, "X")
sage: C4 = P(CyclicPermutationGroup(4))
sage: C4._compose_with_singletons("X, Y", [[2, 2]])
E_2(XY) + X^2*Y^2
E_2(X*Y) + X^2*Y^2
sage: P = PolynomialSpecies(ZZ, ["X", "Y"])
sage: F = P(PermutationGroup([[(1,2,3), (4,5,6)]]), {0: [1,2,3], 1: [4,5,6]})
Expand Down Expand Up @@ -1974,12 +1975,12 @@ def _compose_with_weighted_singletons(self, names, multiplicities, degrees):
sage: C4 = P(CyclicPermutationGroup(4))
sage: C4._compose_with_weighted_singletons(["X"], [-1], [[4]])
-C_4 + {((1,2)(3,4),)}
-C_4 + E_2(X^2)
Exercise (2.5.17) in [BLL1998]_::
sage: C4._compose_with_weighted_singletons(["X", "Y"], [1, 1], [[2, 2]])
E_2(XY) + X^2*Y^2
E_2(X*Y) + X^2*Y^2
sage: C4._compose_with_weighted_singletons(["X", "Y"], [1, 1], [[3, 1]])
X^3*Y
sage: C4._compose_with_weighted_singletons(["X", "Y"], [1, 1], [[4, 0]])
Expand All @@ -1988,7 +1989,7 @@ def _compose_with_weighted_singletons(self, names, multiplicities, degrees):
Equation (4.60) in [ALL2002]_::
sage: C4._compose_with_weighted_singletons(["X", "Y"], [1, -1], [[2, 2]])
-E_2(XY) + 2*X^2*Y^2
-E_2(X*Y) + 2*X^2*Y^2
A bivariate example::
Expand All @@ -2004,7 +2005,7 @@ def _compose_with_weighted_singletons(self, names, multiplicities, degrees):
TESTS::
sage: (C4+E2^2)._compose_with_weighted_singletons(["X"], [-1], [[4]])
-C_4 + E_2^2 + {((1,2)(3,4),)} - 2*X^2*E_2 + X^4
-C_4 + E_2^2 + E_2(X^2) - 2*X^2*E_2 + X^4
sage: C4._compose_with_weighted_singletons(["X"], [-1, 0], [[4]])
Traceback (most recent call last):
Expand Down Expand Up @@ -2077,10 +2078,10 @@ def __call__(self, *args):
sage: E2(-X)
-E_2 + X^2
sage: E2(X^2)
{((1,2)(3,4),)}
E_2(X^2)
sage: E2(X + X^2)
E_2 + X^3 + {((1,2)(3,4),)}
E_2 + X^3 + E_2(X^2)
sage: P2 = PolynomialSpecies(QQ, ["X", "Y"])
sage: X = P2(SymmetricGroup(1), {0: [1]})
Expand All @@ -2089,7 +2090,7 @@ def __call__(self, *args):
E_2(X) + X*Y + E_2(Y)
sage: E2(X*Y)(E2(X), E2(Y))
{((7,8), (5,6), (3,4), (1,2), (1,3)(2,4)(5,7)(6,8)): ({1, 2, 3, 4}, {5, 6, 7, 8})}
E_2(E_2(X)*E_2(Y))
sage: R.<q> = QQ[]
sage: P = PolynomialSpecies(R, ["X"])
Expand Down Expand Up @@ -2298,13 +2299,13 @@ def _element_constructor_(self, G, pi=None, check=True):
sage: X = SetPartitions(4, 2)
sage: a = lambda g, x: SetPartition([[g(e) for e in b] for b in x])
sage: P((X, a, 'left'), {0: [1,2], 1: [3,4]})
E_2(X)*E_2(Y) + X^2*E_2(Y) + E_2(XY) + Y^2*E_2(X)
E_2(X)*E_2(Y) + X^2*E_2(Y) + E_2(X*Y) + Y^2*E_2(X)
sage: P = PolynomialSpecies(ZZ, ["X"])
sage: X = SetPartitions(4, 2)
sage: a = lambda g, x: SetPartition([[g(e) for e in b] for b in x])
sage: P((X, a, 'left'), {0: [1,2,3,4]})
X*E_3 + P_4
X*E_3 + E_2(E_2)
The species of permutation groups::
Expand All @@ -2316,7 +2317,7 @@ def _element_constructor_(self, G, pi=None, check=True):
....: H = S.subgroup(G.conjugate(pi).gens())
....: return next(K for K in X if K == H)
sage: P((X, act), {0: range(1, n+1)}, check=False)
4*E_4 + 4*P_4 + E_2^2 + 2*X*E_3
4*E_4 + 4*E_2(E_2) + E_2^2 + 2*X*E_3
Create a multisort species given an action::
Expand Down Expand Up @@ -2635,3 +2636,59 @@ def factor(s, c, d):
for s in range(self._arity))

Element = PolynomialSpeciesElement


@cached_function
def _atomic_set_like_species(n, names):
r"""
Return a list of the atomic set like species of degree `n`,
and provide their traditional names.
INPUT:
- ``n`` -- positive integer, the degree
- ``names`` -- an iterable of strings for the sorts of the
species
EXAMPLES::
sage: from sage.rings.species import _atomic_set_like_species
sage: _atomic_set_like_species(6, "X")
[E_2(E_3), E_2(X*E_2), E_2(X^3), E_3(E_2), E_3(X^2), E_6]
sage: l = [len(_atomic_set_like_species(n, "X")) for n in range(12)]
sage: l
[0, 1, 1, 1, 3, 1, 6, 1, 10, 4, 12, 1]
sage: oeis(l) # optional - internet
0: A007650: Number of set-like atomic species of degree n.
sage: _atomic_set_like_species(4, "U, V")
[E_2(E_2(V)), E_2(E_2(U)), E_2(V^2), E_2(U*V), E_2(U^2), E_4(U), E_4(V)]
"""
if not n:
return []
M1 = MolecularSpecies("X")
M = MolecularSpecies(names)
if n == 1:
return [M(SymmetricGroup(1), {s: [1]}) for s in range(M._arity)]
result = []
for d in divisors(n):
if d == 1:
continue
if d == n:
result.extend(M(SymmetricGroup(n), {s: range(1, n+1)})
for s in range(M._arity))
continue
E_d = M1(SymmetricGroup(d))
l = []
w = []
for degree in range(1, n // d + 1):
a_degree = _atomic_set_like_species(degree, names)
l.extend(a_degree)
w.extend([degree]*len(a_degree))
for a in WeightedIntegerVectors(n // d, w):
G = prod(F ** e for F, e in zip(l, a))
F = E_d(G)
F.support()[0].rename(f"E_{d}({G})")
result.append(F)
return result

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