def _test(n_max, v_max):
for n in range(n_max + 1):
for v in range(1, v_max + 1):
for lam1 in Partition.all(n):
for lam2 in Partition.all(n):
print()
print()
print('* v =', v, ', n =', n, ', mu =', lam1, ', nu =',
lam2)
print()
print('Computing LHS . . .')
print()
s = Polynomial()
for mu in Partition.all(n + max(sum(lam1), sum(lam2))):
a = SymmetricPolynomial.stable_grothendieck_doublebar(
v, mu, lam1).truncate(n).polynomial('x')
b = SymmetricPolynomial.dual_stable_grothendieck(
v, mu, lam2).truncate(n).polynomial('y')
s += (a * b).truncate(n)
print(' ', mu, ':', s, '|', a, '|', b)
print()
print('LHS =', s)
print()
print()
print('Computing RHS . . .')
print()
f = Polynomial.one()
x = Polynomial.x
y = Polynomial.y
for i in range(1, v + 1):
for j in range(1, v + 1):
a = x(i) * y(j)
term = Polynomial.one()
for e in range(1, n + 1):
term += a**e
f = (f * term).truncate(n)
print(' ', ' :', f)
print()
t = Polynomial()
for kappa in Partition.subpartitions(lam2):
a = SymmetricPolynomial.stable_grothendieck_doublebar(
v, lam2, kappa).truncate(n)
b = SymmetricPolynomial.dual_stable_grothendieck(
v, lam1, kappa).truncate(n)
t += (f * a.polynomial('x') *
b.polynomial('y')).truncate(n)
print(' ', kappa, ':', t)
print()
print('RHS =', t)
print()
print()
print('diff =', s - t)
print()
assert s == t
def _test_shifted_p(n_max, v_max):
for n in range(n_max + 1):
for v in range(1, v_max + 1):
for lam1 in Partition.all(n, strict=True):
for lam2 in Partition.all(n, strict=True):
print()
print()
print('* v =', v, ', n =', n, ', mu =', lam1, ', nu =',
lam2)
print()
print('Computing LHS . . .')
print()
s = Polynomial()
for mu in Partition.all(n + max(sum(lam1), sum(lam2)),
strict=True):
a = SymmetricPolynomial.stable_grothendieck_p_doublebar(
v, mu, lam1).truncate(n).polynomial('x')
b = SymmetricPolynomial.dual_stable_grothendieck_q(
v, mu, lam2).truncate(n).polynomial('y')
s += (a * b).truncate(n)
print(' ', mu, ':', s, '|', a, '|', b)
print()
print('LHS =', s)
print()
print()
print('Computing RHS . . .')
print()
f = kernel(n, v)
print(' ', ' :', f)
print()
t = Polynomial()
for kappa in Partition.subpartitions(lam2, strict=True):
a = SymmetricPolynomial.stable_grothendieck_p_doublebar(
v, lam2, kappa).truncate(n)
b = SymmetricPolynomial.dual_stable_grothendieck_q(
v, lam1, kappa).truncate(n)
t += (f * a.polynomial('x') *
b.polynomial('y')).truncate(n)
print(' ', kappa, ':', t)
print()
print('RHS =', t)
print()
print()
print('diff =', s - t)
print()
assert s == t
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_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 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 _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())
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
def test_mrt_sp_crystal():
for mu in Partition.all(10, max_part=4, strict=True):
for excess in range(5):
MRT_Symplectic_ShiftedCrystalGenerator(mu, rank=3,
excess=excess).generate()
def test_urt_crystal():
for mu in Partition.all(10, max_part=4, strict=True):
for excess in range(3):
URTShiftedCrystalGenerator(mu, rank=3, excess=excess).generate()
