def skew(w):
n = w.rank
f = tuple(i for i in range(1, n + 1) if w(i) > i)
r = len(f)
lam = tuple(w(f[-1]) - r - f[i] + i + 1 for i in range(r))
mu = tuple(w(f[-1]) - r - w(f[i]) + i + 1 for i in range(r))
print(Partition.printable(lam))
print()
print(Partition.printable(mu))
return lam, mu
def test_GQ_to_GP_expansion(): # noqa
for mu in Partition.all(25, strict=True):
print('mu =', mu)
print()
print(Partition.printable(mu, shifted=True))
print()
n = len(mu)
q = GQ(n, mu)
expansion = SymmetricPolynomial.GP_expansion(q)
normalized = Vector({
tuple(nu[i] - mu[i] for i in range(len(mu))): c * sgn(mu, nu) *
BETA**(sum(nu) - sum(mu)) / 2**(len(mu) - sum(nu) + sum(mu))
for nu, c in expansion.items()
})
unsigned = all(c > 0 for c in normalized.values())
print(' mu =', mu, 'n =', n)
print(' expansion =', expansion)
print(' normalized expansion =', normalized)
assert all(len(nu) == 0 or max(nu) <= 1 for nu in normalized)
assert all(len(nu) == len(mu) for nu in expansion)
assert all(Partition.contains(nu, mu) for nu in expansion)
assert all(c % 2**(len(mu) - sum(nu) + sum(mu)) == 0
for nu, c in expansion.items())
assert unsigned
expected = {
tuple(mu[i] + a[i] for i in range(len(a)))
for a in zero_one_tuples(len(mu)) if all(
mu[i - 1] + a[i - 1] > mu[i] + a[i] for i in range(1, len(a)))
}
print(' expected =', expected)
assert set(expansion) == expected
print()
print()
def test_refined_gq_to_gp_expansion(k=12): # noqa
for mu in Partition.all(k, strict=True):
print('mu =', mu)
print()
print(Partition.printable(mu, shifted=True))
print()
n = len(mu)
q = SymmetricPolynomial._slow_refined_dual_stable_grothendieck_q(n, mu)
expected = {}
for a in zero_one_tuples(len(mu)):
if not all(mu[i - 1] - a[i - 1] > mu[i] - a[i]
for i in range(1, len(a))):
continue
if not all(mu[i] - a[i] > 0 for i in range(len(a))):
continue
nu = Partition.trim(tuple(mu[i] - a[i] for i in range(len(a))))
coeff = 2**(len(nu) - sum(a)) * sgn(nu, mu) * BETA**sum(a)
assert coeff != 0
expected[nu] = coeff
print(' expected =', expected)
expected = sum([
coeff *
SymmetricPolynomial._slow_refined_dual_stable_grothendieck_p(
n, nu) for (nu, coeff) in expected.items()
])
print(' =', expected)
assert q == expected
print()
print()
def test_gq_to_gp_expansion(): # noqa
for mu in Partition.all(15, strict=True):
print('mu =', mu)
print()
print(Partition.printable(mu, shifted=True))
print()
n = len(mu)
q = gq(n, mu)
expansion = SymmetricPolynomial.gp_expansion(q)
print(' mu =', mu, 'n =', n)
print(' expansion =', expansion)
assert all(len(nu) == len(mu) for nu in expansion)
assert all(Partition.contains(mu, nu) for nu in expansion)
# assert all(c % 2**(len(mu) - sum(nu) + sum(mu)) == 0 for nu, c in expansion.items())
expected = {}
for a in zero_one_tuples(len(mu)):
if not all(mu[i - 1] - a[i - 1] > mu[i] - a[i]
for i in range(1, len(a))):
continue
if not all(mu[i] - a[i] > 0 for i in range(len(a))):
continue
nu = Partition.trim(tuple(mu[i] - a[i] for i in range(len(a))))
coeff = 2**(len(nu) - sum(a)) * sgn(nu, mu) * BETA**sum(a)
assert coeff != 0
expected[nu] = coeff
print(' expected =', expected)
assert expansion == Vector(expected)
print()
print()
def _expansion(n, function, expand, shifted=True, unsigned=True): # noqa
for mu in Partition.all(n, strict=shifted):
n = len(mu)
p = function(n, mu)
ansion = expand(p)
if unsigned:
expansion = {
nu: coeff * (-1)**abs(sum(mu) - sum(nu))
for nu, coeff in ansion.items()
}
else:
expansion = ansion
print('mu =', mu)
print()
print(Partition.printable(mu, shifted=shifted))
print()
print(' mu =', mu, 'n =', n)
print(' expansion =', ansion)
if unsigned:
print(' unsigned expansion =', expansion)
print()
assert all(v > 0 for v in expansion.values())
