Trac #27640: Shi arrangement for other types

m-extended Shi arrangement defined for any finite crystallograpic root
system

URL: https://trac.sagemath.org/27640
Reported by: etzanaki
Ticket author(s): Eleni Tzanaki, Viviane Pons
Reviewer(s): Frédéric Chapoton, Travis Scrimshaw

src/doc/en/reference/references/index.rst

@@ -225,6 +225,10 @@ REFERENCES:
              by F. Morain.
              http://citeseer.ist.psu.edu/atkin92probabilistic.html
 
+.. [Ath1996] \C. A. Athanasiadis,
+            *Characteristic polynomials of subspace arrangements and finite fields*.
+            Advances in Mathematics, 122(2):193-233, 1996.
+
 .. [Av2000] \D. Avis, *A revised implementation of the reverse search
             vertex enumeration algorithm.* Polytopes-combinatorics and
             computation. Birkhauser Basel, 2000.

src/sage/geometry/hyperplane_arrangement/library.py

@@ -14,10 +14,12 @@
 
 from sage.graphs.all import graphs
 from sage.matrix.constructor import matrix, random_matrix
-from sage.rings.all import QQ, ZZ
+from sage.rings.all import QQ, ZZ, NN
 from sage.misc.misc_c import prod
 
 from sage.combinat.combinat import stirling_number2
+from sage.combinat.root_system.cartan_type import CartanType
+from sage.combinat.root_system.root_system import RootSystem
 from sage.arith.all import binomial
 from sage.rings.polynomial.polynomial_ring import polygen
 
@@ -27,7 +29,7 @@
 def make_parent(base_ring, dimension, names=None):
     """
     Construct the parent for the hyperplane arrangements.
-    
+
     For internal use only.
 
     INPUT:
@@ -375,8 +377,8 @@ def Ish(self, n, K=QQ, names=None):
 
         .. MATH::
 
-            \{ x_i - x_j = 0 : 1 \leq i \leq j \leq n \} 
-            \cup 
+            \{ x_i - x_j = 0 : 1 \leq i \leq j \leq n \}
+            \cup
             \{ x_1 - x_j = i : 1 \leq i \leq j \leq n \}.
 
         EXAMPLES::
@@ -488,7 +490,7 @@ def semiorder(self, n, K=QQ, names=None):
             sage: h.characteristic_polynomial()
             x^5 - 20*x^4 + 180*x^3 - 790*x^2 + 1380*x
             sage: h.characteristic_polynomial.clear_cache()  # long time
-            sage: h.characteristic_polynomial()              # long time 
+            sage: h.characteristic_polynomial()              # long time
             x^5 - 20*x^4 + 180*x^3 - 790*x^2 + 1380*x
         """
         H = make_parent(K, n, names)
@@ -500,33 +502,86 @@ def semiorder(self, n, K=QQ, names=None):
                     hyperplanes.append(x[i] - x[j] - k)
         A = H(*hyperplanes)
         x = polygen(QQ, 'x')
-        charpoly = x * sum([stirling_number2(n, k) * prod([x - k - i for i in range(1, k)]) 
+        charpoly = x * sum([stirling_number2(n, k) * prod([x - k - i for i in range(1, k)])
                             for k in range(1, n+1)])
         A.characteristic_polynomial.set_cache(charpoly)
         return A
 
-    def Shi(self, n, K=QQ, names=None):
+    def Shi(self, data, K=QQ, names=None, m=1):
         r"""
         Return the Shi arrangement.
 
         INPUT:
 
-        - ``n`` -- integer
+        - ``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:
 
-        The Shi arrangement is the set of `n(n-1)` hyperplanes: `\{ x_i - x_j
-        = 0,1 : 1 \leq i \leq j \leq n \}`.
+        - 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::
 
@@ -536,21 +591,41 @@ def Shi(self, n, K=QQ, names=None):
             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
         """
-        H = make_parent(K, n, names)
+        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 i in range(n):
-            for j in range(i+1, n):
-                for const in [0, 1]:
-                    hyperplanes.append(x[i] - x[j] - const)
+
+        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 * sum([(-1)**k * stirling_number2(n, n-k) *
-                            prod([(x - 1 - j) for j in range(k, n-1)]) for k in range(0, n)])
+        charpoly = x**(d-n) * (x-m*h)**n
         A.characteristic_polynomial.set_cache(charpoly)
         return A
 
-
 hyperplane_arrangements = HyperplaneArrangementLibrary()
-