Voronoi s algorithm to compute perfect lattices

Transcription

1 Voronoi s algorithm to compute perfect lattices F R n n sym,>0 min(f ) := min{xf x tr 0 x Z n } minimum Min(F ) := {x Z n xf x tr = min(f )}. Vor(F ) := conv (x tr x x Min(F )) Voronoi domain F perfect, if and only if dim(vor(f )) = n(n + )/2. P n := {F R n n sym,>0 min(f ) =, F perfect }. Theorem (Voronoi) T n := {Vor(F ) F P n } is a locally finite, face to face tessellation of R n n sym,>0 on which GL n(z) acts with finitely many orbits. Min(gF g tr ) = {xg x Min(F )} so Vor(gF g tr ) = g tr Vor(F )g

2 Max Koecher: Pair of dual cones Jürgen Opgenorth: Dual cones and the Voronoi Algorithm Experimental Mathematics 200 V, V 2 real vector spaces of same dimension n σ : V V 2 R bilinear and non-degenerate. Definition V >0 V and V >0 2 V 2 are dual cones if (DC) V >0 i is open in V i and non-empty for i=,2. (DC2) For all x V >0 and y V >0 2 one has σ(x, y) > 0. (DC3) For every x V V >0 there is 0 y V >0 2 with σ(x, y) 0 for every y V 2 V >0 2 there is 0 x V >0 with σ(x, y) 0.

3 V >0 and V >0 2 pair of dual cones Let D V 0 2 {0} be discrete in V 2 and x V >0. µ D (x) := min{σ(x, d) d D} the D-minimum of x. M D (x) := {d D µ D (x) = σ(x, d)} the set of D-minimal vectors of x. M D (x) is finite and M D (x) = M D (λx) for all λ > 0. V D (x) := { d a dd d M D (x), a d R >0 } the D-Voronoi domain of x. A vector x V >0 is called D-perfect, if codim(v D (x)) = 0. P D := {x V >0 µ D (x) =, x is D-perfect } Definition D is called admissible if for every sequence (x i ) i N that converges to a point x δv >0 the sequence (µ D (x i )) i N converges to 0.

4 Voronoi tessellation Theorem If D V 0 2 {0} is discrete in V 2 and admissible then the D-Voronoi domains of the D-perfect vectors form an exact tessellation of V >0 Definition The graph Γ D of D-perfect vectors has vertices P D and edges E = {(x, y) P D P D x and y are neighbours }. Here x, y P D are neighbours if codim(v D (x) V D (y)) =. Corollary If D V 0 2 {0} is discrete and admissible then Γ D is a connected, locally finite graph. 2.

5 Discontinuous Groups Aut(V >0 i ) := {g GL(V i ) V >0 i g = V >0 i }. Ω Aut(V >0 ) properly discontinously on V >0. Ω ad := {ω ad ω Ω} Aut(V >0 2 D V 0 2 {0} discrete, admissible and invariant under Ω ad For x V >0 and ω Ω we have µ D (xw) = µ D (x), M D (xw) = M D (x)(ω ad ), V D (xw) = V D (x)(ω ad ). In particular Ω acts on Γ D.

6 Discontinuous Groups (continued) Theorem Assume additionally that the residue graph Γ D /Ω is finite. x,..., x t P D orbit representatives spanning a connected subtree T of Γ D δt := {y P D T y neighbour of some x i T }. ω y Ω with yω y T. Ω = ω y, Stab Ω (x) x T, y δt In particular the group Ω is finitely generated.

7 Applications Jürgen Opgenorth, 200 G GL n (Z) finite. Compute Ω := N GLn(Z)(G). Michael Mertens, 204 L (R n+, n i= x2 i x2 n+) =: H n+ a Z-lattice in hyperbolic space (signature (n, )). Compute Ω := Aut(L) := {g O(H n+ ) Lg = L}. Braun, Coulangeon, N., Schönnenbeck, 205 A finite dimensional semisimple Q-algebra, Λ A order, i.e. a finitely generated full Z-lattice that is a subring of A. Compute Ω := Λ := {g Λ h Λ, gh = hg = }.

8 Normalizers of finite unimodular groups G GL n (Z) finite. F(G) := {F R n n sym gf g tr = F for all g G} space of invariant forms. B(G) := {g GL n (Z) gf g tr = F for all F F(G)} Bravais group. F(G) always contains a positive definite form g G ggtr. B(G) is finite. N GLn(Z)(G) N GLn(Z)(B(G)) =: Ω acts on F(G). Compute Ω and then the finite index subgroup N GLn(Z)(G). V := F(G) and V 2 := F(G tr ). σ : V V 2 R >0, σ(a, B) := trace(ab). π : R n n sym V 2, F G gtr F g A F(G), B R n n sym σ(a, π(b)) = trace(ab) D := {q x := π(x tr x) x Z n } F F(G) R n n sym,>0 then µ D(F ) = min(f ).

9 Easy example G = diag(, ) F(G) = diag(, ), diag(0, ) B(G) = diag(, ), diag(, ) F = I 2 is G-perfect. V D (F ) = F >0 (G tr ). N GL2 (Z)(G) Ω = N GL2 (Z)(B(G)) = Aut(F ) = D 8.

10 Orders in semi-simple rational algebras. The positive cone K some rational division algebra, A = K n n A R := A Q R semi-simple real algebra A R = direct sum of matrix rings over of H, R or C. A R carries a canonical involution depending on the choice of the isomorphism that we use to define symmetric elements: V = V 2 = V := Sym(A R ) := { F A R F = F } σ(f, F 2 ) := trace(f F 2 ) defines a Euclidean inner product on V. In general the involution will not fix the set A.

11 Orders: Endomorphism rings of lattices. The simple A-module. Let V = K n denote the simple right A-module, V R = V Q R. For x V we have x x V. F V is called positive if F [x] := σ(f, x x) > 0 for all 0 x V R. V >0 := {F V F is positive }. The discrete admissible set O order in K, L some O-lattice in the simple A-module V Λ := End O (L) is an order in A with unit group Λ := GL(L) = {a A al = L}.

12 Minimal vectors. L-minimal vectors Let F V >0. µ(f ) := µ L (F ) = min{f [l] 0 l L} the L-minimum of F M L (F ) := {l L F [l] = µ L (F )} L-minimal vectors Vor L (F ) := { x M L (F ) a xx x a x 0} V 0 Voronoi domain F is called L-perfect dim(vor L (F )) = dim(v). Theorem T := {Vor L (F ) F V >0, L-perfect } forms a locally finite face to face tessellation of V 0. Λ acts on T with finitely many orbits.

13 Generators for Λ Compute R := {F,..., F s } set of representatives of Λ -orbits on the L-perfect forms, such that their Voronoi-graph is connected. For all neighbors F of one of these F i (so Vor(F ) Vor(F i ) has codimension ) compute some g F Λ such that g F F R. Then Λ = Aut(F i ), g F F i R, F neighbor of some F j R. a b 3 2 e f 2 c d so here Λ = Aut(F ), Aut(F 2 ), Aut(F 3 ), a, b, c, d, e, f.

14 Example Q 2,3. Take the rational quaternion algebra ramified at 2 and 3, Q 2,3 = i, j i 2 = 2, j 2 = 3, ij = ji = diag( 2, ( 0 2), 3 0 Maximal order Λ =, i, 2 ( + i + ij), 2 (j + ij) V = A = Q 2,3, A R = R 2 2, L = Λ Embed A into A R using the maximal subfield Q[ 2]. Get three perfect forms: ( 2 2 F = 2 2 ), F 2 = F 3 = diag( , ) ( ) )

15 The tesselation for Q 2,3 Q[ 2] 2 2.

16 Λ / ± = a, b, t a 3, b 2, atbt a 2 2 t 3 2 b t 2 3

17 Λ = a, b, t a 3 = b 2 = atbt =, A = Q 2,3 a = ( ) ( ) b = t = ( ) Note that t = b a + has minimal polynomial x 2 + x and a, b / ± = C 3 C 2 = PSL2 (Z)

18 The tesselation for Q 2,3 Q[ 3] 2 2.

19 A rational division algebra of degree 3 ϑ = ζ 9 + ζ9, σ = Gal(Q(ϑ)/Q), A the Q-algebra generated by ϑ Z := σ(ϑ) and Π := σ 2 (ϑ) A division algebra, Hasse-invariants 3 at 2 and 2 3 at 3. Λ some maximal order in A Γ := Λ has 43 orbits of perfect forms and presentation Γ = a, b b 2 a 2 (b a ) 2, b 2 (a b ) 2 ab 2 a 2 b 3, ab 2 a b 3 a 2 bab 3, a 2 bab 2 ab (a 2 b) 2, a b 2 a b a 5 b 2 a 3, b 2 a 2 b a b a 2 b a b 2 (a b ) 3 a = 3 (( 3Z Z2 ) + (2 + Z 2 )Π + ( Z 2 )Π 2 ), b = 3 (( 3 2Z + Z2 ) + ( 2Z)Π + ( Z 2 )Π 2 )..

20 Quaternion algebras over CM fields K CM-field and A = Q K where Q is a definite quaternion algebra over the rationals. is a positive involution on A. K = Q 7] ( ) A =, Q[ 7] : Q K Q K; a k a k =, i, j, k, Λ maximal order only one orbit of perfect forms Λ = a, b b 3 =, (b a ba) 2 =, (b 2 a 2 ) 3 = a := 4 (( + 7) ( + 7)i + ( + 7)j + (3 7)k), b := 2 ( + i 3j + 7k)

21 Quaternion algebras over imaginary quadratic fields ( ), A =, k = Q( d) k d Number of Runtime Runtime Number of perfect forms Voronoï Presentation generators 7.24s 0.42s s 0.50s s.0s s.78s s 2.57s s 2.52s s 3.02s s 7.54s 6

22 Quaternion algebras over Q( 7) A = ( a, b ) Q( 7) a,b perfect Runtime Runtime Number of forms Voronoï Presentation generators,.24s 0.42s 2, s 4.3s 6, s 5.s 0, s 89.34s 6

23 Easy solution of constructive recognition PX a t P t b

24 Easy solution of constructive recognition PX a Pa P t t b

25 Easy solution of constructive recognition Pa X PX a Pa P t t b

26 Easy solution of constructive recognition Pa X PX a P t t b

27 Isomorphic unit groups Question Given two maximal orders Λ and Γ in A. Does it hold that Λ is isomorphic to Γ if and only if Λ and Γ are conjugate in A? Maximal finite subgroups Λ = Γ they have the same number of conjugacy classes of maximal finite subgroups G of given isomorphism type. These G arise as stabilisers of well rounded faces of the Voronoi tessellation hence may be obtained by the Voronoi algorithm. Integral Homology Many people have used the Λ action on the subcomplex of well rounded faces of the Voronoi tessellation to compute H n (Λ, Z), which is again an invariant of the isomorphism class of Λ.

28 Conclusion Algorithm works quite well for indefinite quaternion algebras over the rationals Obtain presentation and algorithm for constructive recognition of elements For Q 9,37 our algorithm computes the presentation within 5 minutes (288 perfect forms, 88 generators) whereas the MAGMA implementation FuchsianGroup does not return a result after four hours Reasonably fast for quaternion algebras with imaginary quadratic center or matrix rings of degree 2 over imaginary quadratic fields For the rational division algebra of degree 3 ramified at 2 and 3 compute presentation of Λ, 43 perfect forms, 2 generators in about 0 minutes. Quaternion algebra with center Q[ζ 5 ]: > perfect forms. Database available under de/ Oliver.Braun/unitgroups/ Which questions can one answer for unit groups of orders?