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Adding Brauer characters, fixing doctest failures, other improvements.
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@tscrim
tscrim committed Nov 23, 2023
1 parent 6e43cb8 commit 2b2af14
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2 changes: 1 addition & 1 deletion src/sage/combinat/partition.py
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Expand Up @@ -5491,7 +5491,7 @@ def specht_module(self, base_ring=None):
EXAMPLES::
sage: SM = Partition([2,2,1]).specht_module(QQ); SM # optional - sage.modules
Specht module of [(0, 0), (0, 1), (1, 0), (1, 1), (2, 0)] over Rational Field
Specht module of [2, 2, 1] over Rational Field
sage: s = SymmetricFunctions(QQ).s() # optional - sage.modules
sage: s(SM.frobenius_image()) # optional - sage.modules
s[2, 2, 1]
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2 changes: 1 addition & 1 deletion src/sage/combinat/permutation.py
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Expand Up @@ -1262,7 +1262,7 @@ def __mul__(self, rp):
sage: p213 * SGA.an_element() # optional - sage.combinat sage.modules
3*[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + 2*[3, 1, 2]
sage: p213 * SM.an_element() # optional - sage.combinat sage.modules
2*B[0] - 4*B[1]
2*S[[1, 2], [3]] - 4*S[[1, 3], [2]]
"""
if not isinstance(rp, Permutation) and isinstance(rp, Element):
return get_coercion_model().bin_op(self, rp, operator.mul)
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274 changes: 220 additions & 54 deletions src/sage/combinat/specht_module.py
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Original file line number Diff line number Diff line change
Expand Up @@ -4,6 +4,13 @@
AUTHORS:
- Travis Scrimshaw (2023-1-22): initial version
- Travis Scrimshaw (2023-11-23): added simple modules based on code
from Sacha Goldman
.. TODO::
Integrate this with the implementations in
:mod:`sage.modules.with_basis.representation`.
"""

# ****************************************************************************
Expand All @@ -25,10 +32,10 @@
from sage.matrix.constructor import matrix
from sage.rings.rational_field import QQ
from sage.modules.with_basis.subquotient import SubmoduleWithBasis, QuotientModuleWithBasis
from sage.modules.free_module_element import vector
from sage.categories.modules_with_basis import ModulesWithBasis


class SymmetricGroupRepresentation():
class SymmetricGroupRepresentation:
"""
Mixin class for symmetric group (algebra) representations.
"""
Expand All @@ -41,48 +48,9 @@ def __init__(self, SGA):
sage: SM = Partition([3,1,1]).specht_module(GF(3))
sage: TestSuite(SM).run()
"""
self._semigroup = SGA.group()
self._SGA = SGA

def representation_matrix(self, elt):
r"""
Return the matrix corresponding to the left action of the symmetric
group (algebra) element ``elt`` on ``self``.
.. SEEALSO::
:class:`~sage.combinat.symmetric_group_representations.SpechtRepresentation`
EXAMPLES::
sage: SM = Partition([3,1,1]).specht_module(QQ)
sage: SM.representation_matrix(Permutation([2,1,3,5,4]))
[-1 0 0 0 0 0]
[ 0 0 0 -1 0 0]
[ 1 0 0 -1 1 0]
[ 0 -1 0 0 0 0]
[ 1 -1 1 0 0 0]
[ 0 -1 0 1 0 -1]
sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM = SGA.specht_module([(0,0), (0,1), (0,2), (1,0), (2,0)])
sage: SM.representation_matrix(Permutation([2,1,3,5,4]))
[-1 0 0 1 -1 0]
[ 0 0 1 0 -1 1]
[ 0 1 0 -1 0 1]
[ 0 0 0 0 -1 0]
[ 0 0 0 -1 0 0]
[ 0 0 0 0 0 -1]
sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM.representation_matrix(SGA([3,1,5,2,4]))
[ 0 -1 0 1 0 -1]
[ 0 0 0 0 0 -1]
[ 0 0 0 -1 0 0]
[ 0 0 -1 0 1 -1]
[ 1 0 0 -1 1 0]
[ 0 0 0 0 1 0]
"""
return matrix(self.base_ring(), [(elt * b).to_vector() for b in self.basis()])

@cached_method
def frobenius_image(self):
r"""
Expand Down Expand Up @@ -139,11 +107,142 @@ def frobenius_image(self):
from sage.combinat.sf.sf import SymmetricFunctions
BR = self.base_ring()
p = SymmetricFunctions(BR).p()
G = self._SGA.group()
G = self._semigroup
CCR = [(elt, elt.cycle_type()) for elt in G.conjugacy_classes_representatives()]
return p.sum(self.representation_matrix(elt).trace() / la.centralizer_size() * p[la]
for elt, la in CCR)

# TODO: Move these methods up to methods of general representations

def representation_matrix(self, elt):
r"""
Return the matrix corresponding to the left action of the symmetric
group (algebra) element ``elt`` on ``self``.
EXAMPLES::
sage: SM = Partition([3,1,1]).specht_module(QQ)
sage: SM.representation_matrix(Permutation([2,1,3,5,4]))
[-1 0 0 0 0 0]
[ 0 0 0 -1 0 0]
[ 1 0 0 -1 1 0]
[ 0 -1 0 0 0 0]
[ 1 -1 1 0 0 0]
[ 0 -1 0 1 0 -1]
sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM = SGA.specht_module([(0,0), (0,1), (0,2), (1,0), (2,0)])
sage: SM.representation_matrix(Permutation([2,1,3,5,4]))
[-1 0 0 1 -1 0]
[ 0 0 1 0 -1 1]
[ 0 1 0 -1 0 1]
[ 0 0 0 0 -1 0]
[ 0 0 0 -1 0 0]
[ 0 0 0 0 0 -1]
sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM.representation_matrix(SGA([3,1,5,2,4]))
[ 0 -1 0 1 0 -1]
[ 0 0 0 0 0 -1]
[ 0 0 0 -1 0 0]
[ 0 0 -1 0 1 -1]
[ 1 0 0 -1 1 0]
[ 0 0 0 0 1 0]
"""
return matrix(self.base_ring(), [(elt * b).to_vector() for b in self.basis()])

@cached_method
def character(self):
"""
Return the character of ``self``.
EXAMPLES::
sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM = SGA.specht_module([3,2])
sage: SM.character()
(5, 1, 1, -1, 1, -1, 0)
sage: matrix(SGA.specht_module(la).character() for la in Partitions(5))
[ 1 1 1 1 1 1 1]
[ 4 2 0 1 -1 0 -1]
[ 5 1 1 -1 1 -1 0]
[ 6 0 -2 0 0 0 1]
[ 5 -1 1 -1 -1 1 0]
[ 4 -2 0 1 1 0 -1]
[ 1 -1 1 1 -1 -1 1]
sage: SGA = SymmetricGroupAlgebra(QQ, SymmetricGroup(5))
sage: SM = SGA.specht_module([3,2])
sage: SM.character()
Character of Symmetric group of order 5! as a permutation group
sage: SM.character().values()
[5, 1, 1, -1, 1, -1, 0]
sage: matrix(SGA.specht_module(la).character().values() for la in reversed(Partitions(5)))
[ 1 -1 1 1 -1 -1 1]
[ 4 -2 0 1 1 0 -1]
[ 5 -1 1 -1 -1 1 0]
[ 6 0 -2 0 0 0 1]
[ 5 1 1 -1 1 -1 0]
[ 4 2 0 1 -1 0 -1]
[ 1 1 1 1 1 1 1]
sage: SGA.group().character_table()
[ 1 -1 1 1 -1 -1 1]
[ 4 -2 0 1 1 0 -1]
[ 5 -1 1 -1 -1 1 0]
[ 6 0 -2 0 0 0 1]
[ 5 1 1 -1 1 -1 0]
[ 4 2 0 1 -1 0 -1]
[ 1 1 1 1 1 1 1]
"""
G = self._semigroup
chi = [self.representation_matrix(g).trace()
for g in G.conjugacy_classes_representatives()]
try:
return G.character(chi)
except AttributeError:
return vector(chi, immutable=True)

@cached_method
def brauer_character(self):
"""
Return the Brauer character of ``self``.
EXAMPLES::
sage: SGA = SymmetricGroupAlgebra(GF(2), 5)
sage: SM = SGA.specht_module([3,2])
sage: SM.brauer_character()
(5, -1, 0)
sage: SM.simple_module().brauer_character()
(4, -2, -1)
"""
from sage.rings.number_field.number_field import CyclotomicField
from sage.arith.functions import lcm
G = self._semigroup
p = self.base_ring().characteristic()
# We manually compute the order since a Permutation does not implement order()
chi = []
for g in G.conjugacy_classes_representatives():
if p.divides(lcm(g.cycle_type())):
# ignore the non-p-regular elements
continue
evals = self.representation_matrix(g).eigenvalues()
K = evals[0].parent()
val = 0
for la in evals:
if la == K.one():
val += 1
continue
o = la.multiplicative_order()
zeta = CyclotomicField(o).gen()
prim = K.zeta(o)
for deg in range(o):
if prim**deg == la:
val += zeta ** deg
break
chi.append(val)

return vector(chi, immutable=True)


class SpechtModule(SubmoduleWithBasis, SymmetricGroupRepresentation):
r"""
Expand Down Expand Up @@ -542,6 +641,13 @@ class SpechtModuleTableauxBasis(SpechtModule):
This is constructed as a `S_n`-submodule of the :class:`TabloidModule`
(also referred to as the standard module).
.. SEEALSO::
- :class:`SpechtModule` for the generic diagram implementation
constructed as a left ideal of the group algebra
- :class:`~sage.combinat.symmetric_group_representations.SpechtRepresentation`
for an implementation of the representation by matrices.
"""
def __init__(self, ambient):
r"""
Expand Down Expand Up @@ -697,14 +803,25 @@ def simple_module(self):
r"""
Return the simple (or irreducible) `S_n`-submodule of ``self``.
.. SEEALSO::
:class:`~sage.combinat.specht_module.SimpleModule`
EXAMPLES::
sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: SM = SGA.specht_module([3,2])
sage: L = SM.simple_module()
sage: L.dimension()
1
sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM = SGA.specht_module([3,2])
sage: SM.simple_module() is SM
True
"""
if self.base_ring().characteristic() == 0:
return self
return SimpleModule(self)


Expand Down Expand Up @@ -741,13 +858,22 @@ def __init__(self, specht_module):
sage: SM = SGA.specht_module([3,2])
sage: U = SM.maximal_submodule()
sage: TestSuite(U).run()
sage: SM = SGA.specht_module([2,1,1,1])
sage: SM.maximal_submodule()
Traceback (most recent call last):
...
NotImplementedError: only implemented for 3-regular partitions
"""
SymmetricGroupRepresentation.__init__(self, specht_module._SGA)

if specht_module.base_ring().characteristic() == 0:
p = specht_module.base_ring().characteristic()
if p == 0:
basis = []
else:
TM = specht_module._ambient
if not TM._shape.is_regular(p):
raise NotImplementedError(f"only implemented for {p}-regular partitions")
TV = TM._dense_free_module()
SV = TV.submodule(specht_module.lift.matrix().columns())
basis = (SV & SV.complement()).basis()
Expand Down Expand Up @@ -781,38 +907,78 @@ class SimpleModule(QuotientModuleWithBasis, SymmetricGroupRepresentation):
r"""
The simgle `S_n`-module associated with a partition `\lambda`.
The simple module `L^{\lambda}` is the quotient of the Specht module
The simple module `D^{\lambda}` is the quotient of the Specht module
`S^{\lambda}` by its :class:`maximal submodule <MaximalSpechtSubmodule>`
`U^{\lambda}`.
EXAMPLES::
sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: SM = SGA.specht_module([3,1,1])
sage: L = SM.simple_module()
sage: v = L.an_element(); v
2*L[[[1, 3, 5], [2], [4]]] + 2*L[[[1, 4, 5], [2], [3]]]
sage: D = SM.simple_module()
sage: v = D.an_element(); v
2*D[[[1, 3, 5], [2], [4]]] + 2*D[[[1, 4, 5], [2], [3]]]
sage: SGA.an_element() * v
2*L[[[1, 2, 4], [3], [5]]] + 2*L[[[1, 3, 5], [2], [4]]]
2*D[[[1, 2, 4], [3], [5]]] + 2*D[[[1, 3, 5], [2], [4]]]
We give an example on how to construct the decomposition matrix
(the Specht modules are a complete set of irreducible projective
modules) and the Cartan matrix of a symmetric group algebra::
sage: SGA = SymmetricGroupAlgebra(GF(3), 4)
sage: BM = matrix(SGA.simple_module(la).brauer_character()
....: for la in Partitions(4, regular=3))
sage: SBT = matrix(SGA.specht_module(la).brauer_character()
....: for la in Partitions(4))
sage: D = SBT * ~BM; D
[1 0 0 0]
[0 1 0 0]
[1 0 1 0]
[0 0 0 1]
[0 0 1 0]
sage: D.transpose() * D
[2 0 1 0]
[0 1 0 0]
[1 0 2 0]
[0 0 0 1]
We verify this against the direct computation (up to reindexing the
rows and columns)::
sage: SGA.cartan_invariants_matrix() # long time
[1 0 0 0]
[0 1 0 0]
[0 0 2 1]
[0 0 1 2]
"""
def __init__(self, specht_module):
"""
r"""
Initialize ``self``.
EXAMPLES::
sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: SM = SGA.specht_module([3,1,1])
sage: L = SM.simple_module()
sage: TestSuite(L).run()
sage: D = SM.simple_module()
sage: TestSuite(D).run()
sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: SM = SGA.specht_module([2,1,1,1])
sage: SM.simple_module()
Traceback (most recent call last):
...
ValueError: the partition must be 3-regular
"""
self._diagram = specht_module._diagram
p = specht_module.base_ring().characteristic()
if not self._diagram.is_regular(p):
raise ValueError(f"the partition must be {p}-regular")
SymmetricGroupRepresentation.__init__(self, specht_module._SGA)
cat = specht_module.category()
QuotientModuleWithBasis.__init__(self, specht_module.maximal_submodule(), cat, prefix='L')
QuotientModuleWithBasis.__init__(self, specht_module.maximal_submodule(), cat, prefix='D')

def _repr_(self):
"""
r"""
Return a string representation of ``self``.
EXAMPLES::
Expand Down
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