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_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 stable_grothendieck_doublebar(cls,
num_variables,
mu,
nu=(),
degree_bound=None): # noqa
ans = SymmetricPolynomial()
if Partition.contains(mu, nu):
for x in Partition.remove_inner_corners(nu):
ans += BETA**(sum(nu) - sum(x)) * cls._stable_grothendieck(
num_variables, mu, x, degree_bound)
return ans
def test_contains():
mu = (1, )
nu = ()
assert Partition.contains(mu, mu)
assert Partition.contains(mu, nu)
assert not Partition.contains(nu, mu)
mu = (3, 2, 1)
nu = (3, 2)
assert Partition.contains(mu, mu)
assert Partition.contains(mu, nu)
assert not Partition.contains(nu, mu)
mu = (3, 2, 1)
nu = (1, 1, 1)
assert Partition.contains(mu, nu)
assert not Partition.contains(nu, mu)
mu = (1, 1, 1, 1)
nu = (1, 1, 1)
assert Partition.contains(mu, nu)
assert not Partition.contains(nu, mu)
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())
def test_strict_symmetric_functions():
nn = 5
for mu in Partition.all(nn, strict=True):
for nu in Partition.all(nn, strict=True):
for n in range(nn):
print(n, mu, nu)
print()
# Schur-P and GP
f = SymmetricPolynomial.schur_p(n, mu, nu)
g = SymmetricPolynomial.stable_grothendieck_p(n, mu, nu)
h = SymmetricPolynomial.dual_stable_grothendieck_p(n, mu, nu)
fs = SymmetricPolynomial._slow_schur_p(n, mu, nu)
gs = SymmetricPolynomial._slow_stable_grothendieck_p(n, mu, nu)
hs = SymmetricPolynomial._slow_dual_stable_grothendieck_p(
n, mu, nu)
if f != fs:
print(f)
print(fs)
print()
if g != gs:
print(g)
print(gs)
print()
if h != hs:
print(h)
print(hs)
print()
print()
print()
assert f == fs
assert g == gs
assert h == hs
if mu == nu:
assert f == 1
assert g == 1
assert h == 1
assert fs == 1
assert gs == 1
assert hs == 1
if not Partition.contains(mu, nu):
assert f == 0
assert g == 0
assert h == 0
assert fs == 0
assert gs == 0
assert hs == 0
# Schur-Q and GQ
f = SymmetricPolynomial.schur_q(n, mu, nu)
g = SymmetricPolynomial.stable_grothendieck_q(n, mu, nu)
h = SymmetricPolynomial.dual_stable_grothendieck_q(n, mu, nu)
fs = SymmetricPolynomial._slow_schur_q(n, mu, nu)
gs = SymmetricPolynomial._slow_stable_grothendieck_q(n, mu, nu)
hs = SymmetricPolynomial._slow_dual_stable_grothendieck_q(
n, mu, nu)
if f != fs:
print(f)
print(fs)
print()
if g != gs:
print(g)
print(gs)
print()
if h != hs:
print(h)
print(hs)
print()
print()
print()
assert f == fs
assert g == gs
assert h == hs
if mu == nu:
assert f == 1
assert g == 1
assert h == 1
assert fs == 1
assert gs == 1
assert hs == 1
if not Partition.contains(mu, nu):
assert f == 0
assert g == 0
assert h == 0
assert fs == 0
assert gs == 0
assert hs == 0
def test_symmetric_functions():
nn = 6
for mu in Partition.all(nn):
for nu in Partition.all(nn, strict=True):
for n in range(nn):
print(n, mu, nu)
print()
f = SymmetricPolynomial.schur(n, mu, nu)
g = SymmetricPolynomial.stable_grothendieck(n, mu, nu)
h = SymmetricPolynomial.dual_stable_grothendieck(n, mu, nu)
fs = SymmetricPolynomial._slow_schur(n, mu, nu)
gs = SymmetricPolynomial._slow_stable_grothendieck(n, mu, nu)
hs = SymmetricPolynomial._slow_dual_stable_grothendieck(
n, mu, nu)
if f != fs:
print(f)
print(fs)
print()
if g != gs:
print(g)
print(gs)
print()
if h != hs:
print(h)
print(hs)
print()
print()
assert f == fs
assert g == gs
assert h == hs
hh = SymmetricPolynomial.schur_s(n, mu, nu)
kk = SymmetricPolynomial.stable_grothendieck_s(n, mu, nu)
if mu == nu:
assert f == 1
assert g == 1
assert h == 1
assert fs == 1
assert gs == 1
assert hs == 1
assert hh == 1
assert kk == 1
if not Partition.contains(mu, nu):
assert f == 0
assert g == 0
assert h == 0
assert fs == 0
assert gs == 0
assert hs == 0
assert hh == 0
assert kk == 0
print(f)
print(g)
print()
print(hh)
print(kk)
print()
assert g.lowest_degree_terms() == f
assert kk.lowest_degree_terms() == hh