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