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 grothendieck_doublebar(num_variables,
mu,
nu=(),
degree_bound=None): # noqa
return SymmetricPolynomial.stable_grothendieck_doublebar(
num_variables, mu, nu, degree_bound=degree_bound)