def test__change_basis(self):
l_lst = l0, l1, l2, l3, l4, l5 = Veronese.change_basis(
Veronese.get_ideal_lst())
print('l_lst =', l_lst)
#
# l_lst = [
# x1^2 - x4^2 - x5^2,
# x0*x1 - x2^2 - x3^2,
# x2^2 + (2*I)*x2*x3 - x3^2 - x0*x4 + ((-I))*x0*x5,
# x2^2 + ((-2*I))*x2*x3 - x3^2 - x0*x4 + (I)*x0*x5,
# x1*x2 + (I)*x1*x3 - x2*x4 + (I)*x3*x4 + ((-I))*x2*x5 - x3*x5,
# x1*x2 + ((-I))*x1*x3 - x2*x4 + ((-I))*x3*x4 + (I)*x2*x5 - x3*x5
# ]
#
m0 = l0
m1 = l1
m2 = (l2 + l3) * ring('1/2')
m3 = (l2 - l3) * ring('1/2*I')
m4 = (l4 + l5) * ring('1/2')
m5 = (l4 - l5) * ring('1/2*I')
m_lst = [m0, m1, m2, m3, m4, m5]
print('m_lst =', m_lst)
x0, x1 = ring('x0,x1')
dct = {x1: -x0 - x1}
n_lst = n0, n1, n2, n3, n4, n5 = [m.subs(dct) for m in m_lst]
print('n_lst =', n_lst)
print(n0 + 2 * n1)
chk = ring('x1^2-x0^2-2*x2^2-2*x3^2-x4^2-x5^2')
assert n0 + 2 * n1 == chk
def test__get_pmz_lst(self):
id_lst = Veronese.get_ideal_lst()
p = Veronese.get_pmz_lst()
x = ring('x0,x1,x2,x3,x4,x5')
dct = {x[i]: p[i] for i in range(5 + 1)}
for id in id_lst:
ids = id.subs(dct)
print(id, '--->', ids)
assert ids == 0
def test__get_invariant_q_lst__SO2(self):
h1, h2, a1, a2, a3, b1, b2, b3 = Veronese.get_c_lst_lst_dct()['SL3(C)']
q_lst = Veronese.get_invariant_q_lst(h1)
out = sorted(list(set(q_lst)))
print('q_lst =', q_lst)
print('out =', out)
q0, q1, q2, q3, q4, q5 = MARing.q()[:6]
assert out == ring(
'[-1/2*q5, 1/2*q5, -1/2*q4, 1/2*q4, -2*q3, q3, -q2, 2*q2]')
def test__get_c_lst_lst_dct(self):
k = ring('k')
for key in Veronese.get_c_lst_lst_dct():
for c_lst in Veronese.get_c_lst_lst_dct()[key]:
print('key =', key)
print('c_lst =', c_lst)
print('det =', sage_matrix(3, 3, c_lst).det())
n_lst = []
for c in c_lst:
if c in sage_QQ:
n_lst += [c]
else:
n_lst += [c.subs({k: 0})]
print('n_lst =', n_lst)
print
if key != 'SO3(R)':
assert sage_matrix(3, 3, c_lst).det() == 1
assert n_lst == [1, 0, 0, 0, 1, 0, 0, 0, 1]
def test__get_aut_P5__abcdefghk(self):
chk_mat = ''
chk_mat += '['
chk_mat += '( k^2 , 2*g*h , 2*g*k , 2*h*k , g^2 , h^2 ),'
chk_mat += '( c*f , b*d + a*e , c*d + a*f , c*e + b*f , a*d , b*e ),'
chk_mat += '( c*k , b*g + a*h , c*g + a*k , c*h + b*k , a*g , b*h ),'
chk_mat += '( f*k , e*g + d*h , f*g + d*k , f*h + e*k , d*g , e*h ),'
chk_mat += '( c^2 , 2*a*b , 2*a*c , 2*b*c , a^2 , b^2 ),'
chk_mat += '( f^2 , 2*d*e , 2*d*f , 2*e*f , d^2 , e^2 )'
chk_mat += ']'
chk_mat = 'matrix(' + chk_mat + ')'
a, b, c, d, e, f, g, h, k = ring('a, b, c, d, e, f, g, h, k')
out = Veronese.get_aut_P5([a, b, c, d, e, f, g, h, k])
assert out == ring(chk_mat)
def test__get_invariant_qf__SO3(self):
r1, r2, r3 = Veronese.get_c_lst_lst_dct()['SO3(R)']
iqf_lst = Veronese.get_invariant_qf([r1, r2, r3])
print(iqf_lst)
assert iqf_lst != []
def test__get_invariant_qf__SL3(self):
sl3_lst = Veronese.get_c_lst_lst_dct()['SL3(C)']
iqf_lst = Veronese.get_invariant_qf(sl3_lst)
print(iqf_lst)
assert iqf_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 * '-')