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 kernel(n, v):
t = 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)
base = 1 + BETA * x(i) + x(i) * y(j)
term = Polynomial.one()
for e in range(1, n + 1):
term += a**e
base *= term
term = Polynomial.one()
for e in range(1, n + 1):
term += (-BETA * x(i))**e
base *= term
t = (t * base).truncate(n)
return t
def polynomial(self, variable='x'):
x = Polynomial.x if variable == 'x' else Polynomial.y
ans = Polynomial()
for alpha in set(itertools.permutations(self.mu)):
k = len(alpha)
for index in itertools.combinations([i + 1 for i in range(self.n)],
k):
term = Polynomial.one()
for i in range(k):
term *= x(index[i])**alpha[i]
ans += term
return ans
def test_polynomials():
m = SymmetricMonomial(5, ())
assert m.polynomial() == Polynomial.one()
m = SymmetricMonomial(5, (1, ))
x = Polynomial.x
assert m.polynomial() == x(1) + x(2) + x(3) + x(4) + x(5)
m = SymmetricMonomial(3, (
2,
1,
))
x = Polynomial.y
assert m.polynomial('y') == \
x(1) * x(2)**2 + x(1)**2 * x(2) + \
x(1) * x(3)**2 + x(1)**2 * x(3) + \
x(3) * x(2)**2 + x(3)**2 * x(2)
