def test__replace_conj_pairs(self):
q_lst = ring('[ x0-I*x1, x0+I*x1, x0+x2, x3*I, x3+I*x4]')
new_lst = MARing.replace_conj_pairs(q_lst)
chk_lst = sorted(ring('[ x0, x1, x0+x2, x3, x3+I*x4]'))
print(q_lst)
print(new_lst)
print(chk_lst)
assert chk_lst == new_lst
def usecase__invariant_quadratic_forms_experiment():
'''
We output for given subgroup G of Aut(P^1xP^1),
the vectors space of real G-invariant quadratic forms in
the ideal of a (projection of) the double Segre surface
obtained by the method:
"DSegre.get_ideal_lst( exc_idx_lst )"
This method is for experimentation with different
groups and real structures. The parameters are set
in the code.
'''
#
# obtain 1-parameter subgroups whose tangent vectors at the
# identity generates the Lie algebra sl(2)
#
t, q, s, r, e, T = DSegre.get_gens_sl2()
#
# set input parameters
#
exc_idx_lst = []
c_lst_lst = [s + e, e + t]
involution = 'leftright'
MATools.p('exc_idx_lst =', exc_idx_lst)
MATools.p('c_lst_lst =', DSegre.to_str(c_lst_lst))
MATools.p('involution =', involution)
MATools.p('---')
#
# compute vector space of real invariant quadratic forms.
#
iq_lst = DSegre.get_invariant_qf(c_lst_lst, exc_idx_lst)
MATools.p('invariant quadratic forms =\n\t', iq_lst)
iq_lst = DSegre.change_basis(iq_lst, involution)
MATools.p('change basis =\n\t', iq_lst)
iq_lst = MARing.replace_conj_pairs(iq_lst)
MATools.p('replaced conjugate pairs =\n\t', iq_lst)
def usecase__invariant_quadratic_forms_veronese(case):
'''
Let G be a subgroup of Aut(P^2).
Compute the vectors space of real G-invariant quadratic forms in
the ideal of the Veronese surface in projective 5-space P^5
obtained by the method "Veronese.get_ideal_lst()".
The real structure is specified in terms of an involution. We have
two equivalent choices for the involution. Either we can use
"Veronese.get_change_basis()" or we consider the usual complex
conjugation.
See the code below for a description of the cases.
Parameters
----------
case : str
A string representing the name of a group G.
The string should be either:
['1a','1b','sl3','so3']
'''
#
# We initialize 1-parameter subgroups of SL3, whose tangent vectors at the
# identity generate Lie subalgebras of sl3(RR) and sl3(CC).
#
sl3_lst = h1, h2, a1, a2, a3, b1, b2, b3 = Veronese.get_c_lst_lst_dct(
)['SL3(C)']
so3_lst = r1, r2, r3 = Veronese.get_c_lst_lst_dct()['SO3(R)']
if case == '1a':
descr = '''
This example shows that there exists a quadric of signature
(1,5) in the ideal of the Veronese surface, with the involution
that induces the involution (a,b)|->(b,a) on the lattice polygon
of the Veronese surface.
'''
infoG = 'trivial group'
c_lst_lst = []
involution = 'diagonal'
elif case == '1b':
descr = '''
This example shows that there exists a quadric of signature
(1,5) in the ideal of the Veronese surface, with the involution
that induces the identity (a,b)|->(a,b) on the lattice polygon
of the Veronese surface.
'''
infoG = 'trivial group'
c_lst_lst = []
involution = 'identity'
elif case == 'sl3':
descr = '''
This example shows that there exists no quadric that is invariant
under the full automorphism group of the Veronese surface.
'''
infoG = 'SL3(C): [h1, h2, a1, a2, a3, b1, b2, b3]'
c_lst_lst = sl3_lst
involution = 'identity'
elif case == 'so3':
descr = '''
This example shows that there exists a quadric of signature
(1,5) in the ideal of the Veronese surface, such that this
quadric is invariant under the group SO(3)
'''
infoG = 'SO3(R): [r1, r2, r3]'
c_lst_lst = so3_lst
involution = 'identity'
else:
raise ValueError('Unknown case:', case)
#
# compute vector space of invariant quadratic forms
#
iq_lst = Veronese.get_invariant_qf(c_lst_lst)
if involution == 'diagonal':
iq_lst = Veronese.change_basis(iq_lst)
iq_lst = MARing.replace_conj_pairs(iq_lst)
elif involution == 'identity':
pass
else:
raise ValueError('Incorrect involution: ', involution)
#
# computes signatures of random quadrics in
# the vector space of invariant quadratic forms.
#
sig_lst = MARing.get_rand_sigs(iq_lst, 5)
#
# output results
#
while descr != descr.replace(' ', ' '):
descr = descr.replace(' ', ' ')
MATools.p('\n' + 80 * '-')
MATools.p('case :', case)
MATools.p('description :\n', descr)
MATools.p('G :', infoG)
MATools.p('involution :', involution)
MATools.p('G-invariant quadratic forms:')
for iq in iq_lst:
MATools.p('\t', iq)
MATools.p('random signatures:')
MATools.p('\t', sig_lst)
for sig in sig_lst:
if 1 in sig:
MATools.p('\t', sig)
MATools.p('\n' + 80 * '-')
def usecase__classification(cache=True):
'''
Prints a classification of quadratic forms
that contain some fixed double Segre surface S
in projective 8-space, such that the quadratic
form is invariant under subgroup of Aut(S).
We consider subgroups equivalent, if they are
real conjugate in Aut(P^1xP^1).
Parameters
----------
cache : bool
If True, then the cached result is used,
otherwise the output is recomputed. The
computation may take about 9 hours.
'''
key = 'usecase__classification'
if cache and key in MATools.get_tool_dct():
MATools.p(MATools.get_tool_dct()[key])
return
out = ''
for involution in ['identity', 'leftright', 'rotate']:
for c_lst_lst in DSegre.get_c_lst_lst_lst():
#
# compute invariant quadratic forms
#
iq_lst = DSegre.get_invariant_qf(c_lst_lst)
iq_lst = DSegre.change_basis(iq_lst, involution)
iq_lst = MARing.replace_conj_pairs(iq_lst)
#
# computes signatures of random quadrics in
# the vector space of invariant quadratic forms.
#
sig_lst = []
if 'I' not in str(iq_lst):
sig_lst = MARing.get_rand_sigs(iq_lst, 10)
tmp = ''
tmp += '\n\t' + 10 * '-'
tmp += '\n\t involution =' + involution
tmp += '\n\t group =' + DSegre.to_str(c_lst_lst)
tmp += '\n\t invariant ideal = <'
for iq in iq_lst:
sig = ''
if 'I' not in str(iq):
sig = MARing.get_sig(iq)
tmp += '\n\t\t' + str(iq) + '\t\t' + str(sig)
tmp += '\n\t\t >'
tmp += '\n\t random signatures ='
tmp += '\n\t\t ' + str(sig_lst)
for sig in sig_lst:
if 1 in sig:
tmp += '\n\t\t' + str(sig)
tmp += '\n\t' + 10 * '-'
# output obtained info to screen
MATools.p(tmp)
# add to output string
out += tmp
# cache output string
MATools.get_tool_dct()[key] = out
MATools.save_tool_dct()
def usecase__invariant_quadratic_forms(case):
'''
Let G be a subgroup of Aut(P^1xP^1) determined by the case-parameter.
We output the vectors space of real G-invariant quadratic forms in
the ideal of a (projection of) the double Segre surface
obtained by the method:
"DSegre.get_ideal_lst( exc_idx_lst )"
The real structure is specified in terms of an involution. See
"DSegre.change_basis()"
for the specification of involutions.
We also output signatures of random G-invariant quadratic forms.
Parameters
----------
case : str
Three numerical characters, which characterize
the projection of a double Segre surface in S^n.
For example
'078',
means
#families=0, #n=7, #degree=8.
If several examples are considered for the same
triple then we extend with a character in [a-z].
These are currently implemented cases:
['087','287','365','265s','265t','443','243ss','243st']
Notes
-----
See the source code below for a description of each of the cases of
projections of double Segre surfaces that we consider.
'''
#
# obtain 1-parameter subgroups whose tangent vectors at the
# identity generates the Lie algebra sl(2)
#
t, q, s, r, e, T = DSegre.get_gens_sl2()
#
# "involution" is in { 'identity', 'leftright', 'rotate', 'diagonal' }
#
# "c_lst_lst" represents a subgroup G of Aut(P^1xP^1) as a list of
# 1-parameter subgroups that generate G.
#
if case == '087':
descr = '''
This example shows that there exists a
quadric of signature (4,5) that is invariant
under the full automorphism group of S.
'''
infoG = 'SL(2) x SL(2): [t + e, q + e, s + e, e + t, e + q, e + s ]'
exc_idx_lst = []
c_lst_lst = [t + e, q + e, s + e, e + t, e + q, e + s]
involution = 'identity'
elif case == '287':
descr = '''
This example shows quadrics of signatures (1,8),
(1,6) and (1,4) in the ideal of the double Segre
surface, that are invariant under a 2-dimensional
group of toric Moebius automorphisms.
'''
infoG = 'SO(2) x SO(2): [ s + e, e + s ]'
exc_idx_lst = []
c_lst_lst = [s + e, e + s]
involution = 'rotate'
elif case == '365':
descr = '''
This example shows that there exists a smooth sextic Del Pezzo
surface in S^5 that is invariant under a 2-dimensional group
of toric Moebius automorphisms.
'''
infoG = 'SO(2) x SO(2): [ s + e, e + s ]'
exc_idx_lst = [5, 6]
c_lst_lst = [s + e, e + s]
involution = 'rotate'
elif case == '265s':
descr = '''
This example shows that there exists a singular sextic
Del Pezzo surface in S^5 that is invariant under a 1-dimensional
group of toric Moebius automorphisms. Random signatures of type
[1,n+1] with n>=3 found so far are: [1,4], [1,6].
If we extend the group with [e+s] then we obtain a quadric of
signature [1,4]; this is the horn cyclide case with
generators: x0^2 - x3*x4, x0^2 - x1^2 - x2^2,
If we extend the group with [e+t] then we obtain a quadric of
signature [1,4]; this is the spindle cyclide case with
generators: x0^2 - x3*x4, x4^2 - x6^2 - x7^2.
'''
infoG = 'SO(2): [ s + e ]'
exc_idx_lst = [5, 8]
c_lst_lst = [s + e]
involution = 'leftright'
elif case == '265t':
descr = '''
This example indicates that there does not exists a singular
sextic Del Pezzo surface in S^5 that is invariant under a
1-dimensional group of Moebius translations. Random signatures
of type [1,n+1] with n>=3 found so far are: [1,4].
'''
infoG = 'SE(1): [ e + t ]'
exc_idx_lst = [5, 8]
c_lst_lst = [e + t]
involution = 'leftright'
elif case == '443':
descr = '''
This example shows that the ring cyclide in S^3 admits a
2-dimensional family of toric Moebius automorphisms.
'''
infoG = 'SO(2) x SO(2): [ s + e, e + s ]'
exc_idx_lst = [5, 6, 7, 8]
c_lst_lst = [s + e, e + s]
involution = 'rotate'
elif case == '243ss':
descr = '''
This example shows that the spindle cyclide in S^3 admits a
2-dimensional family of Moebius automorphisms.
'''
infoG = 'SO(2) x SX(1): [ s + e, e + s ]'
exc_idx_lst = [5, 6, 7, 8]
c_lst_lst = [s + e, e + s]
involution = 'leftright'
elif case == '243st':
descr = '''
This example shows that the horn cyclide in S^3 admits a
2-dimensional family of Moebius automorphisms.
'''
infoG = 'SO(2) x SE(1): [ s + e, e + t ]'
exc_idx_lst = [1, 2, 5, 8]
c_lst_lst = [s + e, e + t]
involution = 'leftright'
else:
raise ValueError('Unknown case: ', case)
#
# compute vector space of invariant quadratic forms
#
iq_lst = DSegre.get_invariant_qf(c_lst_lst, exc_idx_lst)
iq_lst = DSegre.change_basis(iq_lst, involution)
iq_lst = MARing.replace_conj_pairs(iq_lst)
#
# computes signatures of random quadrics in
# the vector space of invariant quadratic forms.
#
sig_lst = MARing.get_rand_sigs(iq_lst, 10)
#
# output results
#
while descr != descr.replace(' ', ' '):
descr = descr.replace(' ', ' ')
mat_str = str(sage_matrix([[8, 3, 5], [2, 0, 1], [6, 4, 7]]))
new_mat_str = mat_str
for ei in exc_idx_lst:
new_mat_str = new_mat_str.replace(str(ei), ' ')
for ei in range(0, 9):
new_mat_str = new_mat_str.replace(str(ei), '*')
MATools.p('\n' + 80 * '-')
MATools.p('case :', case)
MATools.p('description :\n', descr + '\n' + mat_str + '\n\n' + new_mat_str)
MATools.p('G :', infoG)
MATools.p('exc_idx_lst :', exc_idx_lst)
MATools.p('involution :', involution)
MATools.p('G-invariant quadratic forms:')
for iq in iq_lst:
MATools.p('\t', iq)
MATools.p('random signatures:')
MATools.p('\t', sig_lst)
for sig in sig_lst:
if 1 in sig:
MATools.p('\t', sig)
MATools.p('\n' + 80 * '-')