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 spectrum(n: int, field: 'Field') -> Iterable[int]:
q = field.order
p = field.char
# (1)
eps = 1 if n % 2 == 0 else e
a1 = [(q ** n - eps) // (q - e)]
# (2)
a2 = SemisimpleElements(q, n, min_length=2, sign=e)
# (3)
a3 = []
k = 1
d = gcd(n, q - e)
while True:
n1 = n - p ** (k - 1) - 1
if n1 < 1:
break
eps = 1 if n1 % 2 == 0 else e
a3.append(p ** k * (q ** n1 - eps) // gcd(d, n1))
k += 1
# (4)
a4 = MixedElements(q, n,
lambda k: p ** (k - 1) + 1,
lambda k: p ** k, min_length=2, sign=e)
# (5)
k = numeric.get_exponent(n - 1, p)
a5 = [] if k is None else [p * (n - 1) * d]
# (6)
a6 = [p * gcd(2, q - 1) * (q + e)] if n == 4 else []
return itertools.chain(a1, a2, a3, a4, a5, a6)
def _omega_spectrum_odd_c(n: int, field: 'Field') -> Iterable[int]:
"""Spectra of groups \Omega_{2n+1}(q) for odd q.
[1, Corollary 6]
"""
n = (n - 1) // 2
q = field.order
p = field.char
# (1)
t = (q ** n - 1) // 2
a1 = [t, t + 1]
# (2)
a2 = SemisimpleElements(q, n, min_length=2)
# (3)
k = 1
a3 = []
while True:
n1 = n - (p ** (k - 1) + 1) // 2
if n1 < 1:
break
t = (q ** n1 - 1) // 2
a3.extend([t * p ** k, (t + 1) * p ** k])
k += 1
# (4)
a4 = MixedElements(q, n,
lambda k: (p ** (k - 1) + 1) // 2,
lambda k: p ** k, min_length=2)
# (5)
k = numeric.get_exponent(2 * n - 1, p)
a5 = [] if k is None else [p * (2 * n - 1)]
return itertools.chain(a1, a2, a3, a4, a5)
def _symplectic_spectrum_even_c(n, field):
"""Spectra of symplectic groups in characteristic 2.
[2, Corollary 3]
"""
n //= 2
q = field.order
# (1)
a1 = SemisimpleElements(q, n)
# (2)
a2 = (2 * elem for elem in SemisimpleElements(q, n - 1))
# (3)
a3 = MixedElements(q, n, lambda k: 2**(k - 1) + 1, lambda k: 2**(k + 1))
# (4)
k = numeric.get_exponent(n - 1, 2)
a4 = [] if k is None else [(n - 1) * 4]
return itertools.chain(a1, a2, a3, a4)
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 spectrum(n, field):
q = field.order
p = field.char
d = gcd(n, q - e)
# (1)
eps = 1 if n % 2 == 0 else e
a1 = [(q**n - eps) // ((q - e) * d)]
# (2)
a2 = []
eps = lambda s: 1 if s % 2 == 0 else e
for n1 in xrange(1, (n + 2) // 2):
pair = (n1, n - n1)
signs = (-eps(n1), -eps(n - n1))
a2.append(
lcm(q**pair[0] + signs[0], q**pair[1] + signs[1]) //
gcd(n // gcd(n1, n - n1), q - e))
# (3)
a3 = SemisimpleElements(q, n, min_length=3, sign=e)
# (4)
a4 = []
k = 1
while True:
n1 = n - p**(k - 1) - 1
if n1 < 1:
break
eps = 1 if n1 % 2 == 0 else e
a4.append(p**k * (q**n1 - eps) / d)
k += 1
# (5)
a5 = MixedElements(q,
n,
lambda k: p**(k - 1) + 1,
lambda k: p**k,
min_length=2,
sign=e)
# (6)
k = numeric.get_exponent(n - 1, p)
a6 = [] if k is None else [p * (n - 1)]
return itertools.chain(a1, a2, a3, a4, a5, a6)
def _symplectic_spectrum_odd_c(n, field):
"""Spectra of symplectic groups in odd characteristic.
[1, Corollary 1]
"""
n //= 2
q = field.order
p = field.char
# (1)
a1 = SemisimpleElements(q, n)
# (2)
a2 = MixedElements(q, n, lambda k: (p**(k - 1) + 1) // 2, lambda k: p**k)
# (3)
k = numeric.get_exponent(2 * n - 1, p)
a3 = [] if k is None else [2 * p * (2 * n - 1)]
return itertools.chain(a1, a2, a3)
def _special_orthogonal_odd_c_spectrum(n, field):
"""Spectra of groups SO_{2n+1}(q) for odd q.
[1, Corollary 5]
"""
n = (n - 1) // 2
q = field.order
p = field.char
# (1)
a1 = SemisimpleElements(q, n)
# (2)
a2 = MixedElements(q, n, lambda k: (p**(k - 1) + 1) // 2, lambda k: p**k)
# (3)
k = numeric.get_exponent(2 * n - 1, p)
a3 = [] if k is None else [p * (2 * n - 1)]
return itertools.chain(a1, a2, a3)
def spectrum(n: int, field: 'Field') -> Iterable[int]:
q = field.order
p = field.char
# (1)
eps = 1 if n % 2 == 0 else e
a1 = [(q ** n - eps) // (q - e)]
# (2)
a2 = SemisimpleElements(q, n, min_length=2, sign=e)
# (3)
a3 = MixedElements(q, n,
lambda k: p ** (k - 1) + 1,
lambda k: p ** k, sign=e)
# (4)
k = numeric.get_exponent(n - 1, p)
a4 = [] if k is None else [p * (n - 1)]
return itertools.chain(a1, a2, a3, a4)
def test_minus(self, params):
"""
Test elements [q^{n_1}-1, ..., q^{n_k}-1] for all n_1+...+n_k=n
"""
n, q = params
ss = SemisimpleElements(q, n, sign=1, verbose=False)
divisible = set()
for ni in Partitions(n):
elem = evaluate(q, ni)
found = False
for x in ss:
if x % elem == 0:
if x == elem:
divisible.add(x)
found = True
self.assertTrue(found,
msg="element with base {}, partition {} = {} "\
"doesn't divide any of {}".format(str(q), str(ni),
str(elem), str(ss)))
self.assertSetEqual(divisible, set(ss))
def _projective_symplectic_spectrum_odd_c(n, field):
"""Spectra of projective symplectic groups in characteristic 2.
[1, Corollary 2]
"""
n //= 2
q = field.order
p = field.char
# (1)
t = (q**n - 1) // 2
a1 = [t, t + 1]
# (2)
a2 = SemisimpleElements(q, n, min_length=2)
# (3)
a3 = MixedElements(q, n, lambda k: (p**(k - 1) + 1) // 2, lambda k: p**k)
# (4)
k = numeric.get_exponent(2 * n - 1, p)
a4 = [] if k is None else [p * (2 * n - 1)]
return itertools.chain(a1, a2, a3, a4)
def all_semisimple(self, n, q, min_length=1, parity=0, sign=0):
# very slow! For every partition of size n it calculates 2^n parity tuples.
ss = list(
SemisimpleElements(q,
n,
min_length=min_length,
parity=parity,
sign=sign,
verbose=False))
signsMod = 0 if parity == 1 else 1
divisible = set()
for ni in Partitions(n, min_length=min_length):
if sign:
signs = [[-sign**nk for nk in ni]]
else:
signs = Signs(len(ni))
for ei in signs:
# skip needless signs
if parity and ei.count(1) % 2 != signsMod: continue
elem = evaluate(q, ni, ei)
# every element must divide at least one of items in ss
# also every item in ss must be equal to at least one element
found = False
for x in ss:
if x % elem == 0:
if x == elem:
divisible.add(x)
found = True
#self.assertTrue(any(x % elem == 0 for x in ss),
self.assertTrue(found,
msg="element with base {}, partition {} and signs {} = {} "\
"doesn't divide any of {}".format(str(q), str(ni),
str(ei), str(elem), str(set(ss))))
self.assertSetEqual(divisible, set(ss))