def __init__(self, R, n, q=None):
"""
TESTS::
sage: HeckeAlgebraSymmetricGroupT(QQ, 3)
Hecke algebra of the symmetric group of order 3 on the T basis over Univariate Polynomial Ring in q over Rational Field
::
sage: HeckeAlgebraSymmetricGroupT(QQ, 3, q=1)
Hecke algebra of the symmetric group of order 3 with q=1 on the T basis over Rational Field
"""
self.n = n
self._basis_keys = permutation.Permutations(n)
self._name = "Hecke algebra of the symmetric group of order %s"%self.n
self._one = permutation.Permutation(range(1,n+1))
if q is None:
q = PolynomialRing(R, 'q').gen()
R = q.parent()
else:
if q not in R:
raise ValueError, "q must be in R (= %s)"%R
self._name += " with q=%s"%q
self._q = q
CombinatorialAlgebra.__init__(self, R)
# _repr_ customization: output the basis element indexed by [1,2,3] as [1,2,3]
self.print_options(prefix="")
def ubs(f):
r"""
Given a sextic form `f`, return a dictionary of the invariants of Mestre, p 317 [M]_.
`f` may be homogeneous in two variables or inhomogeneous in one.
EXAMPLES::
sage: from sage.schemes.hyperelliptic_curves.invariants import ubs
sage: x = QQ['x'].0
sage: ubs(x^6 + 1)
{'A': 2, 'C': -2/9, 'B': 2/3, 'D': 0, 'f': x^6 + h^6, 'i': 2*x^2*h^2, 'Delta': -2/3*x^2*h^2, 'y1': 0, 'y3': 0, 'y2': 0}
sage: R.<u, v> = QQ[]
sage: ubs(u^6 + v^6)
{'A': 2, 'C': -2/9, 'B': 2/3, 'D': 0, 'f': u^6 + v^6, 'i': 2*u^2*v^2, 'Delta': -2/3*u^2*v^2, 'y1': 0, 'y3': 0, 'y2': 0}
sage: R.<t> = GF(31)[]
sage: ubs(t^6 + 2*t^5 + t^2 + 3*t + 1)
{'A': 0, 'C': -15, 'B': -12, 'D': -15, 'f': t^6 + 2*t^5*h + t^2*h^4 + 3*t*h^5 + h^6, 'i': -4*t^4 + 10*t^3*h + 2*t^2*h^2 - 9*t*h^3 - 7*h^4, 'Delta': -10*t^4 + 12*t^3*h + 7*t^2*h^2 - 5*t*h^3 + 2*h^4, 'y1': 4*t^2 - 10*t*h - 13*h^2, 'y3': 4*t^2 - 4*t*h - 9*h^2, 'y2': 6*t^2 - 4*t*h + 2*h^2}
"""
ub = Ueberschiebung
if f.parent().ngens() == 1:
f = PolynomialRing(f.parent().base_ring(), 1,
f.parent().variable_name())(f)
x1, x2 = f.homogenize().parent().gens()
f = sum([f[i] * x1**i * x2**(6 - i) for i in range(7)])
U = {}
U['f'] = f
U['i'] = ub(f, f, 4)
U['Delta'] = ub(U['i'], U['i'], 2)
U['y1'] = ub(f, U['i'], 4)
U['y2'] = ub(U['i'], U['y1'], 2)
U['y3'] = ub(U['i'], U['y2'], 2)
U['A'] = ub(f, f, 6)
U['B'] = ub(U['i'], U['i'], 4)
U['C'] = ub(U['i'], U['Delta'], 4)
U['D'] = ub(U['y3'], U['y1'], 2)
return U
def ubs(f):
r"""
Given a sextic form `f`, return a dictionary of the invariants of Mestre, p 317 [M]_.
`f` may be homogeneous in two variables or inhomogeneous in one.
EXAMPLES::
sage: from sage.schemes.hyperelliptic_curves.invariants import ubs
sage: x = QQ['x'].0
sage: ubs(x^6 + 1)
{'A': 2, 'C': -2/9, 'B': 2/3, 'D': 0, 'f': x^6 + h^6, 'i': 2*x^2*h^2, 'Delta': -2/3*x^2*h^2, 'y1': 0, 'y3': 0, 'y2': 0}
sage: R.<u, v> = QQ[]
sage: ubs(u^6 + v^6)
{'A': 2, 'C': -2/9, 'B': 2/3, 'D': 0, 'f': u^6 + v^6, 'i': 2*u^2*v^2, 'Delta': -2/3*u^2*v^2, 'y1': 0, 'y3': 0, 'y2': 0}
sage: R.<t> = GF(31)[]
sage: ubs(t^6 + 2*t^5 + t^2 + 3*t + 1)
{'A': 0, 'C': -15, 'B': -12, 'D': -15, 'f': t^6 + 2*t^5*h + t^2*h^4 + 3*t*h^5 + h^6, 'i': -4*t^4 + 10*t^3*h + 2*t^2*h^2 - 9*t*h^3 - 7*h^4, 'Delta': -10*t^4 + 12*t^3*h + 7*t^2*h^2 - 5*t*h^3 + 2*h^4, 'y1': 4*t^2 - 10*t*h - 13*h^2, 'y3': 4*t^2 - 4*t*h - 9*h^2, 'y2': 6*t^2 - 4*t*h + 2*h^2}
"""
ub = Ueberschiebung
if f.parent().ngens() == 1:
f = PolynomialRing(f.parent().base_ring(), 1, f.parent().variable_name())(f)
x1, x2 = f.homogenize().parent().gens()
f = sum([ f[i]*x1**i*x2**(6-i) for i in range(7) ])
U = {}
U['f'] = f
U['i'] = ub(f, f, 4)
U['Delta'] = ub(U['i'], U['i'], 2)
U['y1'] = ub(f, U['i'], 4)
U['y2'] = ub(U['i'], U['y1'], 2)
U['y3'] = ub(U['i'], U['y2'], 2)
U['A'] = ub(f, f, 6)
U['B'] = ub(U['i'], U['i'], 4)
U['C'] = ub(U['i'], U['Delta'], 4)
U['D'] = ub(U['y3'], U['y1'], 2)
return U