def weight_in_root_lattice(self):
r"""
Return the weight of ``self`` as an element of the root lattice.
EXAMPLES::
sage: M = crystals.infinity.NakajimaMonomials(['F',4])
sage: m = M.module_generators[0].f_string([3,3,1,2,4])
sage: m.weight_in_root_lattice()
-alpha[1] - alpha[2] - 2*alpha[3] - alpha[4]
sage: M = crystals.infinity.NakajimaMonomials(['B',3,1])
sage: mg = M.module_generators[0]
sage: m = mg.f_string([1,3,2,0,1,2,3,0,0,1])
sage: m.weight_in_root_lattice()
-3*alpha[0] - 3*alpha[1] - 2*alpha[2] - 2*alpha[3]
sage: M = crystals.infinity.NakajimaMonomials(['C',3,1])
sage: m = M.module_generators[0].f_string([3,0,1,2,0])
sage: m.weight_in_root_lattice()
-2*alpha[0] - alpha[1] - alpha[2] - alpha[3]
"""
Q = RootSystem(self.parent().cartan_type()).root_lattice()
al = Q.simple_roots()
return Q.sum(e*al[k[0]] for k,e in six.iteritems(self._A))
def weight_in_root_lattice(self):
r"""
Return the weight of ``self`` as an element of the root lattice.
EXAMPLES::
sage: M = crystals.infinity.NakajimaMonomials(['F',4])
sage: m = M.module_generators[0].f_string([3,3,1,2,4])
sage: m.weight_in_root_lattice()
-alpha[1] - alpha[2] - 2*alpha[3] - alpha[4]
sage: M = crystals.infinity.NakajimaMonomials(['B',3,1])
sage: mg = M.module_generators[0]
sage: m = mg.f_string([1,3,2,0,1,2,3,0,0,1])
sage: m.weight_in_root_lattice()
-3*alpha[0] - 3*alpha[1] - 2*alpha[2] - 2*alpha[3]
sage: M = crystals.infinity.NakajimaMonomials(['C',3,1])
sage: m = M.module_generators[0].f_string([3,0,1,2,0])
sage: m.weight_in_root_lattice()
-2*alpha[0] - alpha[1] - alpha[2] - alpha[3]
"""
Q = RootSystem(self.parent().cartan_type()).root_lattice()
al = Q.simple_roots()
return Q.sum(e * al[k[0]] for k, e in self._A.items())
def weight_in_root_lattice(self):
r"""
Return the weight of ``self`` as an element of the root lattice.
EXAMPLES::
sage: M = crystals.infinity.NakajimaMonomials(['F',4])
sage: m = M.module_generators[0].f_string([3,3,1,2,4])
sage: m.weight_in_root_lattice()
-alpha[1] - alpha[2] - 2*alpha[3] - alpha[4]
sage: M = crystals.infinity.NakajimaMonomials(['B',3,1])
sage: mg = M.module_generators[0]
sage: m = mg.f_string([1,3,2,0,1,2,3,0,0,1])
sage: m.weight_in_root_lattice()
-3*alpha[0] - 3*alpha[1] - 2*alpha[2] - 2*alpha[3]
"""
Q = RootSystem(self.parent().cartan_type()).root_lattice()
alpha = Q.simple_roots()
path = self.to_highest_weight()
return Q(sum(-alpha[j] for j in path[1]))
def weight_in_root_lattice(self):
r"""
Return the weight of ``self`` as an element of the root lattice.
EXAMPLES::
sage: M = InfinityCrystalOfNakajimaMonomials(['F',4])
sage: m = M.module_generators[0].f_string([3,3,1,2,4])
sage: m.weight_in_root_lattice()
-alpha[1] - alpha[2] - 2*alpha[3] - alpha[4]
sage: M = InfinityCrystalOfNakajimaMonomials(['B',3,1])
sage: mg = M.module_generators[0]
sage: m = mg.f_string([1,3,2,0,1,2,3,0,0,1])
sage: m.weight_in_root_lattice()
-3*alpha[0] - 3*alpha[1] - 2*alpha[2] - 2*alpha[3]
"""
Q = RootSystem(self.parent().cartan_type()).root_lattice()
alpha = Q.simple_roots()
path = self.to_highest_weight()
return Q(sum(-alpha[j] for j in path[1]))
def __init__(self, cartan_type, prefix, finite=True):
r"""
EXAMPLES::
sage: from sage.combinat.root_system.fundamental_group import FundamentalGroupOfExtendedAffineWeylGroup
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',3,1])
sage: F in Groups().Commutative().Finite()
True
sage: TestSuite(F).run()
"""
def leading_support(beta):
r"""
Given a dictionary with one key, return this key
"""
supp = beta.support()
assert len(supp) == 1
return supp[0]
self._cartan_type = cartan_type
self._prefix = prefix
special_node = cartan_type.special_node()
self._special_nodes = cartan_type.special_nodes()
# initialize dictionaries with the entries for the
# distinguished special node
# dictionary of inverse elements
inverse_dict = {}
inverse_dict[special_node] = special_node
# dictionary for the action of special automorphisms by
# permutations of the affine Dynkin nodes
auto_dict = {}
for i in cartan_type.index_set():
auto_dict[special_node, i] = i
# dictionary for the finite Weyl component of the special automorphisms
reduced_words_dict = {}
reduced_words_dict[0] = tuple([])
if cartan_type.dual().is_untwisted_affine():
# this combines the computations for an untwisted affine
# type and its affine dual
cartan_type = cartan_type.dual()
if cartan_type.is_untwisted_affine():
cartan_type_classical = cartan_type.classical()
I = [i for i in cartan_type_classical.index_set()]
Q = RootSystem(cartan_type_classical).root_lattice()
alpha = Q.simple_roots()
omega = RootSystem(
cartan_type_classical).weight_lattice().fundamental_weights()
W = Q.weyl_group(prefix="s")
for i in self._special_nodes:
if i == special_node:
continue
antidominant_weight, reduced_word = omega[
i].to_dominant_chamber(reduced_word=True, positive=False)
reduced_words_dict[i] = tuple(reduced_word)
w0i = W.from_reduced_word(reduced_word)
idual = leading_support(-antidominant_weight)
inverse_dict[i] = idual
auto_dict[i, special_node] = i
for j in I:
if j == idual:
auto_dict[i, j] = special_node
else:
auto_dict[i, j] = leading_support(w0i.action(alpha[j]))
self._action = Family(
self._special_nodes, lambda i: Family(cartan_type.index_set(),
lambda j: auto_dict[i, j]))
self._dual_node = Family(self._special_nodes, inverse_dict.__getitem__)
self._reduced_words = Family(self._special_nodes,
reduced_words_dict.__getitem__)
if finite:
cat = Category.join(
(Groups().Commutative().Finite(), EnumeratedSets()))
else:
cat = Groups().Commutative().Infinite()
Parent.__init__(self, category=cat)
def product_on_basis(self, left, right):
r"""
Return ``left`` multiplied by ``right`` in ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: a2 = R.algebra_generators()['a2']
sage: ac1 = R.algebra_generators()['ac1']
sage: a2 * ac1 # indirect doctest
a2*ac1
sage: ac1 * a2
-I + a2*ac1 - s1 - s2 + 1/2*s1*s2*s1
sage: x = R.an_element()
sage: [y * x for y in R.some_elements()]
[0,
3*ac1 + 2*s1 + a1,
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2,
3*a1*ac1 + 2*a1*s1 + a1^2,
3*a2*ac1 + 2*a2*s1 + a1*a2,
3*s1*ac1 + 2*I - a1*s1,
3*s2*ac1 + 2*s2*s1 + a1*s2 + a2*s2,
3*ac1^2 - 2*s1*ac1 + 2*I + a1*ac1 + 2*s1 + 1/2*s2 + 1/2*s1*s2*s1,
3*ac1*ac2 + 2*s1*ac1 + 2*s1*ac2 - I + a1*ac2 - s1 - s2 + 1/2*s1*s2*s1]
sage: [x * y for y in R.some_elements()]
[0,
3*ac1 + 2*s1 + a1,
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2,
6*I + 3*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 - 2*a1*s1 + a1^2,
-3*I + 3*a2*ac1 - 3*s1 - 3*s2 + 3/2*s1*s2*s1 + 2*a1*s1 + 2*a2*s1 + a1*a2,
-3*s1*ac1 + 2*I + a1*s1,
3*s2*ac1 + 3*s2*ac2 + 2*s1*s2 + a1*s2,
3*ac1^2 + 2*s1*ac1 + a1*ac1,
3*ac1*ac2 + 2*s1*ac2 + a1*ac2]
"""
# Make copies of the internal dictionaries
dl = dict(left[2]._monomial)
dr = dict(right[0]._monomial)
# If there is nothing to commute
if not dl and not dr:
return self.monomial((left[0], left[1] * right[1], right[2]))
R = self.base_ring()
I = self._cartan_type.index_set()
P = PolynomialRing(R, 'x', len(I))
G = P.gens()
gens_dict = {a: G[i] for i, a in enumerate(I)}
Q = RootSystem(self._cartan_type).root_lattice()
alpha = Q.simple_roots()
alphacheck = Q.simple_coroots()
def commute_w_hd(w, al): # al is given as a dictionary
ret = P.one()
for k in al:
x = sum(c * gens_dict[i] for i, c in alpha[k].weyl_action(w))
ret *= x**al[k]
ret = ret.dict()
for k in ret:
yield (self._hd({I[i]: e
for i, e in enumerate(k) if e != 0}), ret[k])
# Do Lac Ra if they are both non-trivial
if dl and dr:
il = dl.keys()[0]
ir = dr.keys()[0]
# Compute the commutator
terms = self._product_coroot_root(il, ir)
# remove the generator from the elements
dl[il] -= 1
if dl[il] == 0:
del dl[il]
dr[ir] -= 1
if dr[ir] == 0:
del dr[ir]
# We now commute right roots past the left reflections: s Ra = Ra' s
cur = self._from_dict({(hd, s * right[1], right[2]): c * cc
for s, c in terms
for hd, cc in commute_w_hd(s, dr)})
cur = self.monomial((left[0], left[1], self._h(dl))) * cur
# Add back in the commuted h and hd elements
rem = self.monomial((left[0], left[1], self._h(dl)))
rem = rem * self.monomial(
(self._hd({ir: 1}), self._weyl.one(), self._h({il: 1})))
rem = rem * self.monomial((self._hd(dr), right[1], right[2]))
return cur + rem
if dl:
# We have La Ls Lac Rs Rac,
# so we must commute Lac Rs = Rs Lac'
# and obtain La (Ls Rs) (Lac' Rac)
ret = P.one()
for k in dl:
x = sum(c * gens_dict[i] for i, c in alphacheck[k].weyl_action(
right[1].reduced_word(), inverse=True))
ret *= x**dl[k]
ret = ret.dict()
w = left[1] * right[1]
return self._from_dict({
(left[0], w,
self._h({I[i]: e
for i, e in enumerate(k) if e != 0}) * right[2]):
ret[k]
for k in ret
})
# Otherwise dr is non-trivial and we have La Ls Ra Rs Rac,
# so we must commute Ls Ra = Ra' Ls
w = left[1] * right[1]
return self._from_dict({(left[0] * hd, w, right[2]): c
for hd, c in commute_w_hd(left[1], dr)})
def __init__(self, cartan_type, prefix, finite=True):
r"""
EXAMPLES::
sage: from sage.combinat.root_system.fundamental_group import FundamentalGroupOfExtendedAffineWeylGroup
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',3,1])
sage: F in Groups().Commutative().Finite()
True
sage: TestSuite(F).run()
"""
def leading_support(beta):
r"""
Given a dictionary with one key, return this key
"""
supp = beta.support()
assert len(supp) == 1
return supp[0]
self._cartan_type = cartan_type
self._prefix = prefix
special_node = cartan_type.special_node()
self._special_nodes = cartan_type.special_nodes()
# initialize dictionaries with the entries for the distinguished special node
# dictionary of inverse elements
inverse_dict = {}
inverse_dict[special_node] = special_node
# dictionary for the action of special automorphisms by permutations of the affine Dynkin nodes
auto_dict = {}
for i in cartan_type.index_set():
auto_dict[special_node,i] = i
# dictionary for the finite Weyl component of the special automorphisms
reduced_words_dict = {}
reduced_words_dict[0] = tuple([])
if cartan_type.dual().is_untwisted_affine():
# this combines the computations for an untwisted affine type and its affine dual
cartan_type = cartan_type.dual()
if cartan_type.is_untwisted_affine():
cartan_type_classical = cartan_type.classical()
I = [i for i in cartan_type_classical.index_set()]
Q = RootSystem(cartan_type_classical).root_lattice()
alpha = Q.simple_roots()
omega = RootSystem(cartan_type_classical).weight_lattice().fundamental_weights()
W = Q.weyl_group(prefix="s")
for i in self._special_nodes:
if i == special_node:
continue
antidominant_weight, reduced_word = omega[i].to_dominant_chamber(reduced_word=True, positive=False)
reduced_words_dict[i] = tuple(reduced_word)
w0i = W.from_reduced_word(reduced_word)
idual = leading_support(-antidominant_weight)
inverse_dict[i] = idual
auto_dict[i,special_node] = i
for j in I:
if j == idual:
auto_dict[i,j] = special_node
else:
auto_dict[i,j] = leading_support(w0i.action(alpha[j]))
self._action = Family(self._special_nodes, lambda i: Family(cartan_type.index_set(), lambda j: auto_dict[i,j]))
self._dual_node = Family(self._special_nodes, inverse_dict.__getitem__)
self._reduced_words = Family(self._special_nodes, reduced_words_dict.__getitem__)
if finite:
cat = Category.join((Groups().Commutative().Finite(),EnumeratedSets()))
else:
cat = Groups().Commutative().Infinite()
Parent.__init__(self, category = cat)
def product_on_basis(self, left, right):
r"""
Return ``left`` multiplied by ``right`` in ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: a2 = R.algebra_generators()['a2']
sage: ac1 = R.algebra_generators()['ac1']
sage: a2 * ac1 # indirect doctest
a2*ac1
sage: ac1 * a2
-I + a2*ac1 - s1 - s2 + 1/2*s1*s2*s1
sage: x = R.an_element()
sage: [y * x for y in R.some_elements()]
[0,
3*ac1 + 2*s1 + a1,
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2,
3*a1*ac1 + 2*a1*s1 + a1^2,
3*a2*ac1 + 2*a2*s1 + a1*a2,
3*s1*ac1 + 2*I - a1*s1,
3*s2*ac1 + 2*s2*s1 + a1*s2 + a2*s2,
3*ac1^2 - 2*s1*ac1 + 2*I + a1*ac1 + 2*s1 + 1/2*s2 + 1/2*s1*s2*s1,
3*ac1*ac2 + 2*s1*ac1 + 2*s1*ac2 - I + a1*ac2 - s1 - s2 + 1/2*s1*s2*s1]
sage: [x * y for y in R.some_elements()]
[0,
3*ac1 + 2*s1 + a1,
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2,
6*I + 3*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 - 2*a1*s1 + a1^2,
-3*I + 3*a2*ac1 - 3*s1 - 3*s2 + 3/2*s1*s2*s1 + 2*a1*s1 + 2*a2*s1 + a1*a2,
-3*s1*ac1 + 2*I + a1*s1,
3*s2*ac1 + 3*s2*ac2 + 2*s1*s2 + a1*s2,
3*ac1^2 + 2*s1*ac1 + a1*ac1,
3*ac1*ac2 + 2*s1*ac2 + a1*ac2]
"""
# Make copies of the internal dictionaries
dl = dict(left[2]._monomial)
dr = dict(right[0]._monomial)
# If there is nothing to commute
if not dl and not dr:
return self.monomial((left[0], left[1] * right[1], right[2]))
R = self.base_ring()
I = self._cartan_type.index_set()
P = PolynomialRing(R, 'x', len(I))
G = P.gens()
gens_dict = {a:G[i] for i,a in enumerate(I)}
Q = RootSystem(self._cartan_type).root_lattice()
alpha = Q.simple_roots()
alphacheck = Q.simple_coroots()
def commute_w_hd(w, al): # al is given as a dictionary
ret = P.one()
for k in al:
x = sum(c * gens_dict[i] for i,c in alpha[k].weyl_action(w))
ret *= x**al[k]
ret = ret.dict()
for k in ret:
yield (self._hd({I[i]: e for i,e in enumerate(k) if e != 0}), ret[k])
# Do Lac Ra if they are both non-trivial
if dl and dr:
il = dl.keys()[0]
ir = dr.keys()[0]
# Compute the commutator
terms = self._product_coroot_root(il, ir)
# remove the generator from the elements
dl[il] -= 1
if dl[il] == 0:
del dl[il]
dr[ir] -= 1
if dr[ir] == 0:
del dr[ir]
# We now commute right roots past the left reflections: s Ra = Ra' s
cur = self._from_dict({ (hd, s*right[1], right[2]): c * cc
for s,c in terms
for hd, cc in commute_w_hd(s, dr) })
cur = self.monomial( (left[0], left[1], self._h(dl)) ) * cur
# Add back in the commuted h and hd elements
rem = self.monomial( (left[0], left[1], self._h(dl)) )
rem = rem * self.monomial( (self._hd({ir:1}), self._weyl.one(),
self._h({il:1})) )
rem = rem * self.monomial( (self._hd(dr), right[1], right[2]) )
return cur + rem
if dl:
# We have La Ls Lac Rs Rac,
# so we must commute Lac Rs = Rs Lac'
# and obtain La (Ls Rs) (Lac' Rac)
ret = P.one()
for k in dl:
x = sum(c * gens_dict[i]
for i,c in alphacheck[k].weyl_action(right[1].reduced_word(),
inverse=True))
ret *= x**dl[k]
ret = ret.dict()
w = left[1]*right[1]
return self._from_dict({ (left[0], w,
self._h({I[i]: e for i,e in enumerate(k)
if e != 0}) * right[2]
): ret[k]
for k in ret })
# Otherwise dr is non-trivial and we have La Ls Ra Rs Rac,
# so we must commute Ls Ra = Ra' Ls
w = left[1]*right[1]
return self._from_dict({ (left[0] * hd, w, right[2]): c
for hd, c in commute_w_hd(left[1], dr) })
