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)