def Shi(self, data, K=QQ, names=None, m=1):
r"""
Return the Shi arrangement.
INPUT:
- ``data`` -- either an integer or a Cartan type (or coercible
into; see "CartanType")
- ``K`` -- field (default:``QQ``)
- ``names`` -- tuple of strings or ``None`` (default); the
variable names for the ambient space
- ``m`` -- integer (default: 1)
OUTPUT:
- If ``data`` is an integer `n`, return the Shi arrangement in
dimension `n`, i.e. the set of `n(n-1)` hyperplanes:
`\{ x_i - x_j = 0,1 : 1 \leq i \leq j \leq n \}`. This corresponds
to the Shi arrangement of Cartan type `A_{n-1}`.
- If ``data`` is a Cartan type, return the Shi arrangement of given
type.
- If `m > 1`, return the `m`-extended Shi arrangement of given type.
The `m`-extended Shi arrangement of a given crystallographic
Cartan type is defined by the inner product
`\langle a,x \rangle = k` for `-m < k \leq m` and
`a \in \Phi^+` is a positive root of the root system `\Phi`.
EXAMPLES::
sage: hyperplane_arrangements.Shi(4)
Arrangement of 12 hyperplanes of dimension 4 and rank 3
sage: hyperplane_arrangements.Shi("A3")
Arrangement of 12 hyperplanes of dimension 4 and rank 3
sage: hyperplane_arrangements.Shi("A3",m=2)
Arrangement of 24 hyperplanes of dimension 4 and rank 3
sage: hyperplane_arrangements.Shi("B4")
Arrangement of 32 hyperplanes of dimension 4 and rank 4
sage: hyperplane_arrangements.Shi("B4",m=3)
Arrangement of 96 hyperplanes of dimension 4 and rank 4
sage: hyperplane_arrangements.Shi("C3")
Arrangement of 18 hyperplanes of dimension 3 and rank 3
sage: hyperplane_arrangements.Shi("D4",m=3)
Arrangement of 72 hyperplanes of dimension 4 and rank 4
sage: hyperplane_arrangements.Shi("E6")
Arrangement of 72 hyperplanes of dimension 8 and rank 6
sage: hyperplane_arrangements.Shi("E6",m=2)
Arrangement of 144 hyperplanes of dimension 8 and rank 6
If the Cartan type is not crystallographic, the Shi arrangement
is not defined::
sage: hyperplane_arrangements.Shi("H4")
Traceback (most recent call last):
...
NotImplementedError: Shi arrangements are not defined for non crystallographic Cartan types
The characteristic polynomial is pre-computed using the results
of [Ath1996]_::
sage: hyperplane_arrangements.Shi("A3").characteristic_polynomial()
x^4 - 12*x^3 + 48*x^2 - 64*x
sage: hyperplane_arrangements.Shi("A3",m=2).characteristic_polynomial()
x^4 - 24*x^3 + 192*x^2 - 512*x
sage: hyperplane_arrangements.Shi("C3").characteristic_polynomial()
x^3 - 18*x^2 + 108*x - 216
sage: hyperplane_arrangements.Shi("E6").characteristic_polynomial()
x^8 - 72*x^7 + 2160*x^6 - 34560*x^5 + 311040*x^4 - 1492992*x^3 + 2985984*x^2
sage: hyperplane_arrangements.Shi("B4",m=3).characteristic_polynomial()
x^4 - 96*x^3 + 3456*x^2 - 55296*x + 331776
TESTS::
sage: h = hyperplane_arrangements.Shi(4)
sage: h.characteristic_polynomial()
x^4 - 12*x^3 + 48*x^2 - 64*x
sage: h.characteristic_polynomial.clear_cache() # long time
sage: h.characteristic_polynomial() # long time
x^4 - 12*x^3 + 48*x^2 - 64*x
sage: h = hyperplane_arrangements.Shi("A3",m=2)
sage: h.characteristic_polynomial()
x^4 - 24*x^3 + 192*x^2 - 512*x
sage: h.characteristic_polynomial.clear_cache()
sage: h.characteristic_polynomial()
x^4 - 24*x^3 + 192*x^2 - 512*x
sage: h = hyperplane_arrangements.Shi("B3",m=3)
sage: h.characteristic_polynomial()
x^3 - 54*x^2 + 972*x - 5832
sage: h.characteristic_polynomial.clear_cache()
sage: h.characteristic_polynomial()
x^3 - 54*x^2 + 972*x - 5832
"""
if data in NN:
cartan_type = CartanType(["A", data - 1])
else:
cartan_type = CartanType(data)
if not cartan_type.is_crystallographic():
raise NotImplementedError(
"Shi arrangements are not defined for non crystallographic Cartan types"
)
n = cartan_type.rank()
h = cartan_type.coxeter_number()
Ra = RootSystem(cartan_type).ambient_space()
PR = Ra.positive_roots()
d = Ra.dimension()
H = make_parent(K, d, names)
x = H.gens()
hyperplanes = []
for a in PR:
for const in range(-m + 1, m + 1):
hyperplanes.append(sum(a[j] * x[j] for j in range(d)) - const)
A = H(*hyperplanes)
x = polygen(QQ, 'x')
charpoly = x**(d - n) * (x - m * h)**n
A.characteristic_polynomial.set_cache(charpoly)
return A
