def lift_map(target):
r"""
Create a lift map, to be used for lifting the cross ratios of a matroid
representation.
.. SEEALSO::
:meth:`lift_cross_ratios() <sage.matroids.utilities.lift_cross_ratios>`
INPUT:
- ``target`` -- a string describing the target (partial) field.
OUTPUT:
- a dictionary
Depending on the value of ``target``, the following lift maps will be created:
- "reg": a lift map from `\GF3` to the regular partial field `(\ZZ, <-1>)`.
- "sru": a lift map from `\GF7` to the
sixth-root-of-unity partial field `(\QQ(z), <z>)`, where `z` is a sixth root
of unity. The map sends 3 to `z`.
- "dyadic": a lift map from `\GF{11}` to the dyadic partial field `(\QQ, <-1, 2>)`.
- "gm": a lift map from `\GF{19}` to the golden mean partial field
`(\QQ(t), <-1,t>)`, where `t` is a root of `t^2-t-1`. The map sends `5` to `t`.
The example below shows that the latter map satisfies three necessary conditions stated in
:meth:`lift_cross_ratios() <sage.matroids.utilities.lift_cross_ratios>`
EXAMPLES::
sage: from sage.matroids.utilities import lift_map
sage: lm = lift_map('gm')
sage: for x in lm:
....: if (x == 1) is not (lm[x] == 1):
....: print('not a proper lift map')
....: for y in lm:
....: if (x+y == 0) and not (lm[x]+lm[y] == 0):
....: print('not a proper lift map')
....: if (x+y == 1) and not (lm[x]+lm[y] == 1):
....: print('not a proper lift map')
....: for z in lm:
....: if (x*y==z) and not (lm[x]*lm[y]==lm[z]):
....: print('not a proper lift map')
"""
if target == "reg":
R = GF(3)
return {R(1): ZZ(1)}
if target == "sru":
R = GF(7)
z = ZZ['z'].gen()
S = NumberField(z * z - z + 1, 'z')
z = S(z)
return {R.one(): S.one(), R(3): z, R(3)**(-1): z**5}
if target == "dyadic":
R = GF(11)
return {R(1): QQ(1), R(-1): QQ(-1), R(2): QQ(2), R(6): QQ((1, 2))}
if target == "gm":
R = GF(19)
t = QQ['t'].gen()
G = NumberField(t * t - t - 1, 't')
return {
R(1): G(1),
R(5): G(t),
R(1) / R(5): G(1) / G(t),
R(-5): G(-t),
R(-5)**(-1): G(-t)**(-1),
R(5)**2: G(t)**2,
R(5)**(-2): G(t)**(-2)
}
raise NotImplementedError(target)
