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_skew_GP_positivity(): # noqa
k = 10
for mu in Partition.all(k, strict=True):
for nu in Partition.all(k, strict=True):
if not Partition.contains(mu, nu):
continue
n = len(mu)
f = GP(n, mu, nu)
expansion = SymmetricPolynomial.GP_expansion(f)
normalized = Vector({
lam: c * BETA**(sum(lam) - sum(mu) + sum(nu))
for lam, c in expansion.items()
})
print('GP_{%s/%s}(x_%s) =' % (mu, nu, n), normalized)
print()
assert all(c > 0 for c in normalized.values())
def test_staircase_grothendieck_GP_positivity(): # noqa
r = 6
for k in range(r):
delta = tuple(k - i for i in range(k))
for nu in Partition.all(sum(delta)):
if not Partition.contains(delta, nu):
continue
n = len(delta)
f = G(n, delta, nu)
expansion = SymmetricPolynomial.GP_expansion(f)
normalized = Vector({
lam: c * BETA**(sum(lam) - sum(delta) + sum(nu))
for lam, c in expansion.items()
})
print('G_{%s/%s}(x_%s) =' % (delta, nu, n), normalized)
print()
assert all(c > 0 for c in normalized.values())
