Skip to content

Commit

Permalink
Trac #34393: add method "tensor_factors" to tensor products
Browse files Browse the repository at this point in the history 
so that one can easily access them

URL: https://trac.sagemath.org/34393
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Matthias Koeppe, Travis Scrimshaw
  • Loading branch information
Release Manager committed Sep 20, 2022
2 parents 3063144 + b81dbf0 commit a6a72a9
Show file tree
Hide file tree
Showing 3 changed files with 120 additions and 37 deletions.
47 changes: 30 additions & 17 deletions src/sage/algebras/schur_algebra.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -454,6 +454,19 @@ def _repr_(self):
msg += " over {}"
return msg.format(self._r, self._n, self.base_ring())

def construction(self):
"""
Return ``None``.
There is no functorial construction for ``self``.
EXAMPLES::
sage: T = SchurTensorModule(QQ, 2, 3)
sage: T.construction()
"""
return None

def _monomial_product(self, xi, v):
"""
Result of acting by the basis element ``xi`` of the corresponding
Expand Down Expand Up @@ -581,11 +594,11 @@ def GL_irreducible_character(n, mu, KK):
A = M._schur
SGA = M._sga

#make ST the superstandard tableau of shape mu
# make ST the superstandard tableau of shape mu
from sage.combinat.tableau import from_shape_and_word
ST = from_shape_and_word(mu, list(range(1, r + 1)), convention='English')

#make ell the reading word of the highest weight tableau of shape mu
# make ell the reading word of the highest weight tableau of shape mu
ell = [i + 1 for i, l in enumerate(mu) for dummy in range(l)]

e = M.basis()[tuple(ell)] # the element e_l
Expand All @@ -607,17 +620,17 @@ def GL_irreducible_character(n, mu, KK):
y = A.basis()[schur_rep] * e # M.action_by_Schur_alg(A.basis()[schur_rep], e)
carter_lusztig.append(y.to_vector())

#Therefore, we now have carter_lusztig as a list giving the basis
#of `V_\mu`
# Therefore, we now have carter_lusztig as a list giving the basis
# of `V_\mu`

#We want to think of expressing this character as a sum of monomial
#symmetric functions.
# We want to think of expressing this character as a sum of monomial
# symmetric functions.

#We will determine a basis element for each m_\lambda in the
#character, and we want to keep track of them by \lambda.
# We will determine a basis element for each m_\lambda in the
# character, and we want to keep track of them by \lambda.

#That means that we only want to pick out the basis elements above for
#those semistandard words whose content is a partition.
# That means that we only want to pick out the basis elements above for
# those semistandard words whose content is a partition.

contents = Partitions(r, max_length=n).list()
# all partitions of r, length at most n
Expand Down Expand Up @@ -648,15 +661,15 @@ def GL_irreducible_character(n, mu, KK):
except ValueError:
pass

#There is an inner product on the Carter-Lusztig module V_\mu; its
#maximal submodule is exactly the kernel of the inner product.
# There is an inner product on the Carter-Lusztig module V_\mu; its
# maximal submodule is exactly the kernel of the inner product.

#Now, for each possible partition content, we look at the graded piece of
#that degree, and we record how these elements pair with each of the
#elements of carter_lusztig.
# Now, for each possible partition content, we look at the graded piece of
# that degree, and we record how these elements pair with each of the
# elements of carter_lusztig.

#The kernel of this pairing is the part of this graded piece which is
#not in the irreducible module for \mu.
# The kernel of this pairing is the part of this graded piece which is
# not in the irreducible module for \mu.

length = len(carter_lusztig)

Expand Down
56 changes: 55 additions & 1 deletion src/sage/categories/modules.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -12,14 +12,15 @@
# *****************************************************************************

from sage.misc.cachefunc import cached_method
from sage.misc.abstract_method import abstract_method
from sage.misc.lazy_import import LazyImport
from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
from sage.categories.morphism import SetMorphism
from sage.categories.homsets import HomsetsCategory
from sage.categories.homset import Hom
from .category import Category
from .category_types import Category_module
from sage.categories.tensor import TensorProductsCategory, tensor
from sage.categories.tensor import TensorProductsCategory, TensorProductFunctor, tensor
from .dual import DualObjectsCategory
from sage.categories.cartesian_product import CartesianProductsCategory
from sage.categories.sets_cat import Sets
Expand Down Expand Up @@ -891,3 +892,56 @@ def extra_super_categories(self):
[Category of modules over Integer Ring]
"""
return [self.base_category()]

class ParentMethods:
"""
Implement operations on tensor products of modules.
"""
def construction(self):
"""
Return the construction of ``self``.
EXAMPLES::
sage: A = algebras.Free(QQ,2)
sage: T = A.tensor(A)
sage: T.construction()
(The tensor functorial construction,
(Free Algebra on 2 generators (None0, None1) over Rational Field,
Free Algebra on 2 generators (None0, None1) over Rational Field))
"""
try:
factors = self.tensor_factors()
except (TypeError, NotImplementedError):
from sage.misc.superseded import deprecation
deprecation(34393, "implementations of Modules().TensorProducts() now must define the method tensor_factors")
return None
return (TensorProductFunctor(),
factors)

@abstract_method(optional=True)
def tensor_factors(self):
"""
Return the tensor factors of this tensor product.
EXAMPLES::
sage: F = CombinatorialFreeModule(ZZ, [1,2])
sage: F.rename("F")
sage: G = CombinatorialFreeModule(ZZ, [3,4])
sage: G.rename("G")
sage: T = tensor([F, G]); T
F # G
sage: T.tensor_factors()
(F, G)
TESTS::
sage: M = CombinatorialFreeModule(ZZ, ((1, 1), (1, 2), (2, 1), (2, 2)),
....: category=ModulesWithBasis(ZZ).FiniteDimensional().TensorProducts())
sage: M.construction()
doctest:warning...
DeprecationWarning: implementations of Modules().TensorProducts() now must define the method tensor_factors
See https://trac.sagemath.org/34393 for details.
(VectorFunctor, Integer Ring)
"""
54 changes: 35 additions & 19 deletions src/sage/combinat/free_module.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -1183,8 +1183,8 @@ class CombinatorialFreeModule_Tensor(CombinatorialFreeModule):
We construct two free modules, assign them short names, and construct their tensor product::
sage: F = CombinatorialFreeModule(ZZ, [1,2]); F.__custom_name = "F"
sage: G = CombinatorialFreeModule(ZZ, [3,4]); G.__custom_name = "G"
sage: F = CombinatorialFreeModule(ZZ, [1,2]); F.rename("F")
sage: G = CombinatorialFreeModule(ZZ, [3,4]); G.rename("G")
sage: T = tensor([F, G]); T
F # G
Expand Down Expand Up @@ -1330,6 +1330,23 @@ def _repr_(self):
return symb.join("%s" % module for module in self._sets)
# TODO: make this overridable by setting _name

def tensor_factors(self):
"""
Return the tensor factors of this tensor product.
EXAMPLES::
sage: F = CombinatorialFreeModule(ZZ, [1,2])
sage: F.rename("F")
sage: G = CombinatorialFreeModule(ZZ, [3,4])
sage: G.rename("G")
sage: T = tensor([F, G]); T
F # G
sage: T.tensor_factors()
(F, G)
"""
return self._sets

def _ascii_art_(self, term):
"""
TESTS::
Expand Down Expand Up @@ -1454,8 +1471,8 @@ def tensor_constructor(self, modules):
EXAMPLES::
sage: F = CombinatorialFreeModule(ZZ, [1,2]); F.__custom_name = "F"
sage: G = CombinatorialFreeModule(ZZ, [3,4]); G.__custom_name = "G"
sage: F = CombinatorialFreeModule(ZZ, [1,2]); F.rename("F")
sage: G = CombinatorialFreeModule(ZZ, [3,4]); G.rename("G")
sage: H = CombinatorialFreeModule(ZZ, [5,6]); H.rename("H")
sage: f = F.monomial(1) + 2 * F.monomial(2)
Expand Down Expand Up @@ -1494,8 +1511,8 @@ def _tensor_of_elements(self, elements):
EXAMPLES::
sage: F = CombinatorialFreeModule(ZZ, [1,2]); F.__custom_name = "F"
sage: G = CombinatorialFreeModule(ZZ, [3,4]); G.__custom_name = "G"
sage: F = CombinatorialFreeModule(ZZ, [1,2]); F.rename("F")
sage: G = CombinatorialFreeModule(ZZ, [3,4]); G.rename("G")
sage: H = CombinatorialFreeModule(ZZ, [5,6]); H.rename("H")
sage: f = F.monomial(1) + 2 * F.monomial(2)
Expand Down Expand Up @@ -1629,9 +1646,9 @@ class CombinatorialFreeModule_CartesianProduct(CombinatorialFreeModule):
We construct two free modules, assign them short names, and construct their Cartesian product::
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.__custom_name = "F"
sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.__custom_name = "G"
sage: H = CombinatorialFreeModule(ZZ, [4,7]); H.__custom_name = "H"
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename("F")
sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename("G")
sage: H = CombinatorialFreeModule(ZZ, [4,7]); H.rename("H")
sage: S = cartesian_product([F, G])
sage: S
F (+) G
Expand Down Expand Up @@ -1703,7 +1720,7 @@ def _repr_(self):
sage: F = CombinatorialFreeModule(ZZ, [2,4,5])
sage: CP = cartesian_product([F, F]); CP # indirect doctest
Free module generated by {2, 4, 5} over Integer Ring (+) Free module generated by {2, 4, 5} over Integer Ring
sage: F.__custom_name = "F"; CP
sage: F.rename("F"); CP
F (+) F
"""
from sage.categories.cartesian_product import cartesian_product
Expand All @@ -1723,8 +1740,8 @@ def cartesian_embedding(self, i):
EXAMPLES::
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.__custom_name = "F"
sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.__custom_name = "G"
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename("F")
sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename("G")
sage: S = cartesian_product([F, G])
sage: phi = S.cartesian_embedding(0)
sage: phi(F.monomial(4) + 2 * F.monomial(5))
Expand Down Expand Up @@ -1757,8 +1774,8 @@ def cartesian_projection(self, i):
EXAMPLES::
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.__custom_name = "F"
sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.__custom_name = "G"
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename("F")
sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename("G")
sage: S = cartesian_product([F, G])
sage: x = S.monomial((0,4)) + 2 * S.monomial((0,5)) + 3 * S.monomial((1,6))
sage: S.cartesian_projection(0)(x)
Expand Down Expand Up @@ -1787,8 +1804,8 @@ def _cartesian_product_of_elements(self, elements):
EXAMPLES::
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.__custom_name = "F"
sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.__custom_name = "G"
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename("F")
sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename("G")
sage: S = cartesian_product([F, G])
sage: f = F.monomial(4) + 2 * F.monomial(5)
sage: g = 2*G.monomial(4) + G.monomial(6)
Expand Down Expand Up @@ -1826,12 +1843,11 @@ def cartesian_factors(self):
EXAMPLES::
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.__custom_name = "F"
sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.__custom_name = "G"
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename("F")
sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename("G")
sage: S = cartesian_product([F, G])
sage: S.cartesian_factors()
(F, G)
"""
return self._sets

Expand Down

0 comments on commit a6a72a9

Please sign in to comment.