def _test_shifted_q(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_q_doublebar(
v, mu, lam1).truncate(n).polynomial('x')
b = SymmetricPolynomial.dual_stable_grothendieck_p(
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_q_doublebar(
v, lam2, kappa).truncate(n)
b = SymmetricPolynomial.dual_stable_grothendieck_p(
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_simple():
f = SymmetricPolynomial._slow_transposed_dual_stable_grothendieck_q(4, (4,))
g = SymmetricPolynomial.dual_stable_grothendieck_q(4, (4,))
assert f.omega_schur_expansion(f) == g.schur_expansion(g)
f = SymmetricPolynomial._slow_transposed_dual_stable_grothendieck_p(4, (4,))
g = SymmetricPolynomial.dual_stable_grothendieck_p(4, (4,))
assert f.omega_schur_expansion(f) == g.schur_expansion(g)
def test_p(n=4):
lam = tuple([n - i for i in range(n)])
for mu in Partition.subpartitions(lam, strict=True):
for nu in Partition.subpartitions(mu, strict=True):
f = SymmetricPolynomial._slow_transposed_dual_stable_grothendieck_p(n, mu, nu)
g = SymmetricPolynomial.dual_stable_grothendieck_p(n, mu, nu)
ef, eg = f.omega_schur_expansion(f), g.schur_expansion(g)
print('n =', n, 'mu =', mu)
print(ef)
print(eg)
print()
assert ef == eg
def test_decompose():
schur = SymmetricPolynomial.schur
schur_P = SymmetricPolynomial.schur_p # noqa
dual_grothendieck = SymmetricPolynomial.dual_stable_grothendieck
grothendieck = SymmetricPolynomial.stable_grothendieck
GP = SymmetricPolynomial.stable_grothendieck_p # noqa
GQ = SymmetricPolynomial.stable_grothendieck_q # noqa
n = 6
mu = (3, 2, 1)
nu = (4, 2, 1)
f = schur(n, mu) * schur(n, mu)
exp = SymmetricPolynomial.schur_expansion(f)
assert f == sum([schur(n, a) * coeff for a, coeff in exp.items()])
f = schur_P(n, nu)
exp = SymmetricPolynomial.schur_expansion(f)
assert f == sum([schur(n, a) * coeff for a, coeff in exp.items()])
f = GP(n, nu)
exp = SymmetricPolynomial.grothendieck_expansion(f)
assert f == sum([grothendieck(n, a) * coeff for a, coeff in exp.items()])
f = GQ(n, nu)
exp = SymmetricPolynomial.grothendieck_expansion(f)
assert f == sum([grothendieck(n, a) * coeff for a, coeff in exp.items()])
f = SymmetricPolynomial.dual_stable_grothendieck_p(n, nu)
exp = SymmetricPolynomial.dual_grothendieck_expansion(f)
assert f == sum(
[dual_grothendieck(n, a) * coeff for a, coeff in exp.items()])
f = SymmetricPolynomial.dual_stable_grothendieck_q(n, nu)
exp = SymmetricPolynomial.dual_grothendieck_expansion(f)
assert f == sum(
[dual_grothendieck(n, a) * coeff for a, coeff in exp.items()])
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 dual_grothendieck_P(num_variables, mu, nu=()): # noqa
return SymmetricPolynomial.dual_stable_grothendieck_p(
num_variables, mu, nu)