def _omega_pm_spectrum_odd_c(n: int, field: 'Field', sign: int) -> Iterable[int]:
"""Spectra of Omega^e_{2n}(q) for odd q.
Based on [1, Corollary 8] and [3, Lemma 2.3]
Point 3 of [1, Corollary 8] contains an error
that was corrected in [3, Lemma 2.3]
"""
n //= 2
q = field.order
p = field.char
def nk(k):
return (p ** (k - 1) + 3) // 2
# (1)
a1 = [(q ** n - sign) // 2]
# (2)
a2 = SemisimpleElements(q, n, min_length=2, parity=sign)
# (3)
a3 = []
k = 1
while True:
n_k = nk(k)
if n_k >= n:
break
for delta in [1, -1]:
dk = gcd(4, q ** n_k - sign * delta) // 2
a3.append(
p ** k *
lcm(
dk,
(q**(n - n_k) - delta) // dk
)
)
k += 1
# (4)
a4 = MixedElements(q, n, nk, lambda k: p ** k, min_length=2)
# (5)
a5 = []
for elem in SemisimpleElements(q, n - 2, min_length=2, parity=sign):
a5.append(elem.lcm(SpectraElement(p, q, [1], [-1])))
a5.append(elem.lcm(SpectraElement(p, q, [1], [1])))
# (6)
t = (q ** (n - 2) - sign) // 2
a6 = [p * lcm(q - 1, t), p * lcm(q + 1, t)]
# (7)
k = numeric.get_exponent(2 * n - 3, p)
a7 = [] if k is None else [p * (2 * n - 3) * gcd(4, q ** n - sign) // 2]
# (8)
a8 = [p * (q * q - 1), p * (q * q + 1)] if n == 4 and sign == 1 else []
# (9)
a9 = [9 * (q - 1), 9 * (q + 1)] if n == 4 and p == 3 and sign == 1 else []
return itertools.chain(a1, a2, a3, a4, a5, a6, a7, a8, a9)
def spectrum(n, field):
n //= 2
q = field.order
p = field.char
# if gcd(4, q^n-e) != 4, then POmega = Omega
b = (q % 4 == 3 and n % 2 == 1) if e == -1 else (
q % 4 == 1) # true iff gcd(4, q^n-e)=4
if not b:
return _omega_pm_spectrum(e)(n * 2, field)
nk = lambda k: (p**(k - 1) + 3) // 2
# (1)
a1 = [(q**n - sign) // 4]
# (2)
a2 = []
for n1 in xrange(1, n):
for e1 in [-1, 1]:
a = q**n1 - e1
b = q**(n - n1) - e * e1
d = 2 if _equal_two_part(a, b) else 1
a2.append(lcm(a, b) // d)
# (3)
a3 = SemisimpleElements(q, n, min_length=3, parity=sign)
# (4)
a4 = []
k = 1
while True:
n_k = nk(k)
if n_k >= n:
break
a4.append(p**k * (q**(n - n_k) + 1) // 2)
a4.append(p**k * (q**(n - n_k) - 1) // 2)
k += 1
# (5)
a5 = MixedElements(q, n, nk, lambda k: p**k, min_length=2)
# (6)
a6 = []
for elem in SemisimpleElements(q, n - 2, min_length=2, parity=sign):
a6.append(elem.lcm(SpectraElement(p, q, [1], [-1])))
a6.append(elem.lcm(SpectraElement(p, q, [1], [1])))
# (7)
t = (q**(n - 2) - sign) // 2
a7 = [p * lcm(q - 1, t), p * lcm(q + 1, t)]
# (8)
k = numeric.get_exponent(2 * n - 3, p)
a8 = [] if k is None else [p * (2 * n - 3)]
return itertools.chain(a1, a2, a3, a4, a5, a6, a7, a8)
def _omega_pm_spectrum_odd_c(n, field, sign):
"""Spectra of Omega^e_{2n}(q) for odd q.
[1, Corollary 8]
"""
n //= 2
q = field.order
p = field.char
nk = lambda k: (p**(k - 1) + 3) // 2
# (1)
a1 = [(q**n - sign) // 2]
# (2)
a2 = SemisimpleElements(q, n, min_length=2, parity=sign)
# (3)
a3 = []
k = 1
while True:
n_k = nk(k)
if n_k >= n:
break
dk = gcd(4, q**n_k - sign) // 2
a3.append(p**k * lcm(dk, (q**(n - n_k) + 1) // dk))
a3.append(p**k * lcm(dk, (q**(n - n_k) - 1) // dk))
k += 1
# (4)
a4 = MixedElements(q, n, nk, lambda k: p**k, min_length=2)
# (5)
a5 = []
for elem in SemisimpleElements(q, n - 2, min_length=2, parity=sign):
a5.append(elem.lcm(SpectraElement(p, q, [1], [-1])))
a5.append(elem.lcm(SpectraElement(p, q, [1], [1])))
# (6)
t = (q**(n - 2) - sign) // 2
a6 = [p * lcm(q - 1, t), p * lcm(q + 1, t)]
# (7)
k = numeric.get_exponent(2 * n - 3, p)
a7 = [] if k is None else [p * (2 * n - 3) * gcd(4, q**n - sign) // 2]
# (8)
a8 = [p * (q * q - 1), p * (q * q + 1)] if n == 4 and sign == 1 else []
# (9)
a9 = [9 * (q - 1), 9 * (q + 1)] if n == 4 and p == 3 and sign == 1 else []
return itertools.chain(a1, a2, a3, a4, a5, a6, a7, a8, a9)
def _omega_pm_spectrum_even_c(n: int, field: 'Field', sign: int) -> Iterable[int]:
"""Spectra for groups \Omega^{\pm}(2^k).
[1, Corollary 4]
"""
n //= 2
q = field.order
# (1)
a1 = SemisimpleElements(q, n, parity=sign)
# (2)
a2 = MixedElements(q, n,
lambda k: 2 ** (k - 1) + 2,
lambda k: 2 ** (k + 1))
# (3)
a3 = (2 * elem for elem in SemisimpleElements(q, n - 2))
# (4)
a4 = []
for elem in SemisimpleElements(q, n - 2, parity=sign):
a4.append(2 * lcm(q - 1, elem))
a4.append(2 * lcm(q + 1, elem))
# (5)
a5 = []
signMod = 0 if sign == 1 else 1
for ni in FullBoundedSets(n - 3):
if len(ni) % 2 != signMod:
continue
a5.append(4 * SpectraElement(q=q, partition=[1] + ni,
signs=[-1] + [1] * len(ni)))
# (6)
a6 = (elem.lcm(SpectraElement(4, q, [1], [1])) for elem in SemisimpleElements(q, n - 3, parity=-sign))
# (7)
k = numeric.get_exponent(n - 2, 2)
a7 = [] if k is None else [4 * (n - 2)]
return itertools.chain(a1, a2, a3, a4, a5, a6, a7)
def spectrum(n, field):
n //= 2
q = field.order
p = field.char
# (1)
a1 = SemisimpleElements(q, n, parity=e)
# (2)
a2 = MixedElements(q, n, lambda k: (p**(k - 1) + 3) // 2,
lambda k: p**k)
# (3)
a3 = []
for elem in SemisimpleElements(q, n - 2, parity=e):
a3.append(elem.lcm(SpectraElement(p, q, [1], [-1])))
a3.append(elem.lcm(SpectraElement(p, q, [1], [1])))
# (4)
k = numeric.get_exponent(2 * n - 3, p)
a4 = [] if k is None else [2 * p * (2 * n - 3)]
return itertools.chain(a1, a2, a3, a4)
def test_lcm(self):
elem1 = SpectraElement(q=2, partition=[2, 3], signs=[1, -1])
elem2 = SpectraElement(q=2, partition=[3, 4], signs=[1, -1])
expected = "[2^4 - 1, 2^3 + 1, 2^3 - 1, 2^2 + 1]"
self.assertEqual(expected, elem1.lcm(elem2).str_verbose())
def test_str(self):
elem = SpectraElement(3, 2, [2, 3, 4], [1, -1, 1])
expected = "3 * [2^4 + 1, 2^3 - 1, 2^2 + 1]"
self.assertEqual(expected, elem.str_verbose())
elem = SpectraElement(3)
self.assertEqual("3", elem.str_verbose())
elem = SpectraElement(q=2, partition=[2, 3, 4], signs=[1, -1, 1])
expected = "[2^4 + 1, 2^3 - 1, 2^2 + 1]"
self.assertEqual(expected, elem.str_verbose())
elem = SpectraElement(q=2, partition=[1, 1, 4], signs=[1, -1, 1])
expected = "[2^4 + 1, 2 + 1, 2 - 1]"
self.assertEqual(expected, elem.str_verbose())
elem = SpectraElement(q=2, partition=[1], signs=[1])
expected = "2 + 1"
self.assertEqual(expected, elem.str_verbose())
elem = SpectraElement(quotient=2, q=2, partition=[1], signs=[1])
expected = "2 * (2 + 1)"
self.assertEqual(expected, elem.str_verbose())
def test_mult(self):
elem = SpectraElement(q=2, partition=[2, 3], signs=[1, -1])
expected = "2 * [2^3 - 1, 2^2 + 1]"
self.assertEqual(expected, (elem * 2).str_verbose())
def test_lcm(self):
elem1 = SpectraElement(q=2, partition=[2, 3], signs=[1, -1])
elem2 = SpectraElement(q=2, partition=[3, 4], signs=[1, -1])
expected = "[2^2 + 1, 2^3 - 1, 2^3 + 1, 2^4 - 1]"
self.assertEqual(expected, elem1.lcm(elem2).str_verbose())
def test_str(self):
elem = SpectraElement(3, 2, [2, 3, 4], [1, -1, 1])
expected = "3 * [2^2 + 1, 2^3 - 1, 2^4 + 1]"
self.assertEqual(expected, elem.str_verbose())
elem = SpectraElement(3)
self.assertEqual("3", elem.str_verbose())
elem = SpectraElement(q=2, partition=[2, 3, 4], signs=[1, -1, 1])
expected = "[2^2 + 1, 2^3 - 1, 2^4 + 1]"
self.assertEqual(expected, elem.str_verbose())
elem = SpectraElement(q=2, partition=[1, 1, 4], signs=[1, -1, 1])
expected = "[2 - 1, 2 + 1, 2^4 + 1]"
self.assertEqual(expected, elem.str_verbose())
elem = SpectraElement(q=2, partition=[1], signs=[1])
expected = "2 + 1"
self.assertEqual(expected, elem.str_verbose())
elem = SpectraElement(quotient=2, q=2, partition=[1], signs=[1])
expected = "2 * (2 + 1)"
self.assertEqual(expected, elem.str_verbose())
