aa r X i v : . [ m a t h . M G ] F e b ISO EDGE DOMAINS
MATHIEU DUTOUR SIKIRI´C AND MARIO KUMMER
Abstract.
Iso-edge domains are a variant of the iso-Delaunay decomposition introducedby Voronoi. They were introduced by Baranovskii & Ryshkov in order to solve thecovering problem in dimension 5.In this work we revisit this decomposition and prove the following new results:(1) We review the existing theory and give a general mass-formula for the iso-edgedomains.(2) We prove that the associated toroidal compactification of the moduli space of prin-cipally polarized abelian varieties is projective.(3) We prove the Conway–Sloane conjecture in dimension 5.(4) We prove that the quadratic forms for which the conorms are non-negative areexactly the matroidal ones in dimension 5. Introduction
Voronoi introduced several decompositions of the cone of positive definite quadratic forms.The decomposition [20] uses perfect forms while the decomposition [21] uses Delaunay poly-topes. See [18] for an overview of such decompositions.Let A ∈ S d> be a positive definite quadratic form on R d . A lattice polytope P ⊂ R d iscalled a Delaunay polytope if there is some c ∈ R d and r > A [ x − c ] ≥ r forall x ∈ Z d with equality if and only if x is a vertex of P . The set Del( A ) of all Delaunaypolytopes is the Delaunay subdivision associated to A . The secondary cone of a Delaunaysubdivision D is ∆( D ) = { Q ∈ S d> : Del( Q ) = D} , also known as L -type domain . The secondary cone ∆( A ) of A ∈ S d> is defined to be∆(Del( A )). Voronoi’s second reduction theory [21] states that the set of all secondary cones isa polyhedral subdivision of the cone S d> of positive definite symmetric matrices on which thegroup G L d ( Z ) acts by conjugation. For any fixed d this group action has only finitely manyorbits. The iso-edge domain , or C -type domain , introduced in [17] is a coarser subdivisionof S d> : The iso-edge domain of a Delaunay subdivision D is defined as C ( D ) = { Q ∈ S d> : E ∈ Del( Q ) for all E ∈ D with E centrally symmetric } . In the generic case, the only centrally symmetric Delaunay polytopes are the edges justifyingthe name. The iso-edge domain C ( A ) of A ∈ S d> is defined to be C (Del( A )). It was shownin [17] that the set of all iso-edge domains is a tiling of S d> . Each iso-edge domain is theunion of finitely many secondary cones. Moreover, in each fixed dimension d there are finitelymany iso-edge domains up to G L d ( Z )-equivalence. They classified the iso-edge domains in The terminology “ C -type” comes from “ L -type” introduced by Voronoi. The Cyrillic letter C is pro-nounced like the Latin letter S and refers to “skeleton”, that is the graph determined by the edges of theDelaunay subdivision. d = 5 and found 76 full-dimensional types. This was rediscovered in [11]. Recently, in [9]the classification in dimension 6 yielded 55083358 full-dimensional types.In Subsection 2.1 we give general definitions on the domains. In Subsection 2.2 an algo-rithm for enumerating the iso-edge domains in a fixed dimension is given.In Algebraic Geometry, compactifications of the space of principally polarized Abelianvarieties were considered and they are described by equivariant polyhedral decomposition ofthe cone of positive definite quadratic forms. The most thoroughly studied compactifictionsare the perfect form decomposition, the central cone compactification and the secondarycone compactification (see [15]). In Section 3 we prove that the iso-edge domains definea projective compactification. In Subsection 4.1 a mass formula involving all the iso-edgedomains of a given dimension is given. In Subsection 4.2 the enumeration of all iso-edgedomains in dimension 5 is presented.The notion of iso-edge domain is useful if one is interested in properties that only dependon the edges of the Delaunay polytope or, equivalently, on the facets of the Voronoi polytope,see e.g. [8]. We address here two questions of this kind. For every e = [ u, v ] edge of aDelaunay polytope, the middle ( u + v ) / c = L/L where L is the lattice.We then set f ( c ) = k u − v k . Conway & Sloane conjectured that this function characterizes L up to conjugacy and prove it in dimension at most 3 [5, p. 58]. The conjecture is provedin dimension 4 in [19, Ch 4.4]. We prove this conjecture in dimension 5 in Section 5. Wefurther prove in dimension 4 and 5 that the conorms of a matrix A are non-negative if andonly if A belongs to the matroidal locus. This implies a positive answer to [4, Question 4.11]for d ≤ Definition and Enumeration techniques
Voronoi polytopes and parity vectors.
For any natural number n we defineParity n = (cid:26) , (cid:27) n − { } the set of parity vectors. Given a positive definite quadratic form A and a vector v we define A [ v ] = v t Av and cvp( A, v ) = min x ∈ Z n A [ x − v ]CVP( A, v ) = { x ∈ Z n s.t. A [ x − v ] = cvp( A, v ) } The mapping v conv(CVP( A, v )) establishes a bijection between Parity n and the setof centrally symmetric Delaunay polytopes. In particular the edges of Delaunay polytopesdetermine centrally symmetric Delaunay polytopes.For a positive definite quadratic form A the Voronoi polytope is defined as V or ( A ) = { x ∈ R n s.t. A [ x ] ≤ A [ x − v ] for v ∈ Z n − { }} . The facets of the polytope correspond to the edges of the Delaunay polytope and there areat most 2(2 n −
1) of them.The iso-edge domain of a Delaunay subdivision is C ( D ) = (cid:8) Q ∈ S d> : D ∈ Del( Q ) for all D ∈ D with D centrally symmetric (cid:9) . When the number of facets of the Voronoi polytope is 2(2 n −
1) the iso-edge domain isfull dimensional and we say that the iso-edge domain is primitive . This is equivalent to
SO EDGE DOMAINS 3 saying that for each v ∈ Parity n we have | CVP(
A, v ) | = 2 or to saying that all the centrallysymmetric Delaunay polytopes are 1-dimensional.2.2. Enumeration of full dimensional iso-edge domains.
A typical strategy used forenumerating polyhedral subdivizions such as perfect domains [20] or secondary cones [21] isa graph traversal algorithm: One starts from one cone and for each cone one computes theadjacent cones and insert them in the complete list of cones if they are not isomorphic toother cones. The enumeration ends when all cones have been treated.A well known full-dimensional Delaunay subdivision is the one of the principal domainof Voronoi. This defines a primitive iso-edge domain which we use as starting cone of theenumeration.For a primitive iso-edge domain D formed by vectors ( v i ) ≤ i ≤ N with N = 2(2 n − { i, j, k } such that v i + v j + v k = 0. Any such tripledefines three inequalities: A [ v i ] ≤ A [ v j ] + A [ v k ] and its permutations. The finite set ofall such inequalities is denoted by F and defines D . From F we can determine by linearprogramming which inequalities determine a facet of D . Thus we have a set of facet defininginequalities as ( φ i ) i ≤ i ≤ m and we need to find the adjacent full-dimensional iso-edge domain.Given a facet defining inequality φ ( A ) ≥
0, it is proportional to a number of inequalities ofthe form A [ v i u ] ≤ A [ v j u ] + A [ v k u ] for 1 ≤ u ≤ r . In the adjacent domain the vectors v i u arereplaced by v j u − v k u and this defines the adjacent domain.A full-dimensional iso-edge domain is encoded by a finite set of vectors ( v i ) ≤ i ≤ N with N =2(2 n −
1) and we consider equivalence under G L n ( Z ). We can thus apply the methodologyof [3, Section 3.1] in order to test isomorphy of full-dimensional iso-edge domains.3. Compactification
We define S nrat, ≥ to be the rational closure of S n> , that is the positive semidefinitematrices whose kernel is defined by rational equations. A polyhedral decomposition of S n ≥ with cones σ α is called G L n ( Z )-admissible (see [1, II.4, Definition 4.10]) if(A1) Each face of a σ α is a σ β .(A2) σ α ∩ σ β is a common face of σ α and σ β .(A3) γσ α is a σ β for all γ ∈ G L n ( Z ).(A4) modulo G L n ( Z ) there are only a finite number of σ β (A5) S nrat, ≥ = ∪ α ( σ α ∩ S nrat, ≥ )An admissible decomposition defines a compactification of the cone of principally polarizedabelian varieties (see [1, III.5, Theorem 5.2]). According to [1, IV.2. Definition 2.1] anadmissible decomposition defines a projective variety if there exists a function φ : S nrat, ≥ such that:(B1) φ is convex, piecewise-linear(B2) φ ( A ) > A = 0(B3) φ is linear on the σ α and those are the maximal area in which they are linear.(B4) φ is integral on Σ Z which in this case are the integral valued matrices. Lemma 3.1.
Let A be a quadratic form on R n . If for each coset α ∈ Z n / Z n there is an x α ∈ α with A [ x α ] = 0 , then A = 0 .Proof. Assume for the sake of a contradiction that the claim is not true. This means thatthe vector space N of all quadratic forms A with A [ x α ] = 0 for all α ∈ Z n / Z n has positivedimension. Since x α ∈ Z n , the vector space N must contain a nonzero quadratic form all of M. D. SIKIRI´C AND M. KUMMER whose coefficients are rational numbers. After scaling appropriately, we obtain a quadraticform A ∈ N all of whose coefficients are integers such that at least one coefficient is odd.Reducing modulo 2 gives a quadratic form Q on F n with at least one nonzero coefficientwhich vanishes on all of F n . This cannot happen as Q [ δ i ] = 0 for i = 1 , . . . , n forces thecoefficient of x i to be zero and then Q [ δ i + δ j ] = 0 forces the coefficient of x i x j to be zero. (cid:3) Theorem 3.2.
The iso-edge domains form a G L n ( Z ) -admissible decomposition of S nrat, ≥ and define a projective compactification.Proof. We first define the function φ as φ ( A ) = X x ∈ Parity n A, x ) . The convexity and positivity is clear. Lemma 3.1 implies that A = 0 if φ ( A ) = 0 proving(B2).On a given full dimensional iso-edge domain, φ is linear proving (B1). Let us consider twoadjacent full dimensional iso-edge domains C and C . If an edge vector v in C becomesa vector v ′ in C then there exist a triple ( v, v , v ) in C with v + v + v = 0 such that v ′ = v − v . We then have the relation A [ v ] + A [ v ′ ] = 2 A [ v ] + 2 A [ v ]. Thus we have on theinterior of C the relation A [ v ] < A [ v ′ ] and the reverse on C . This shows that the maximalarea in which φ is linear does not extend beyond C and so is exactly C proving (B3).For an integral matrix and a parity vector x the minimum cvp( A, x ) is realized by a vector v ∈ Z n thus 4 cvp( A, x ) = A [2 x − v ] ∈ Z proving (B4). Therefore we have a characteristicfunction φ .The iso-edge domaina define convex polyhedral cones σ α . Taking a face of such a poly-hedral cone corresponds to extending the set of closest point proving (A1).Let us consider the convex set C defined by φ ( A ) ≥
1. The faces σ α of this cone aredefined by ℓ ( x ) = 1 for ℓ a function such that ℓ ( x ) ≥ x ∈ C . If σ α and σ β are two conesof function ℓ α and ℓ β then σ α ∩ σ β is a cone of the polyhedral tessellation for the function( ℓ α + ℓ β ) / G L n ( Z ) equivalence. Since every secondary cone belongsto an iso-edge domain and each iso-edge domain contains finitely many secondary cones,finiteness (A4) follows. Since every A ∈ S nrat, ≥ belongs to a secondary cone, it belongs to aniso-edge domain. Conversely, since every iso-edge domain is the union of secondary cones, itis included in S nrat, ≥ proving (A5). (cid:3) Starting from n = 4 this compactification is singular. This can be seen from the fact thatin n = 4 there are 3 orbits of full dimensional iso-edge domains, one of them being the unionof two full dimensional simplices. Non-simplicial cones induce fundamental singularities ofthe compactification.4. Lower dimensional cells of iso-edge domain and their mass formula
Mass formula.
We follow here the methodology of [6] where mass formulas are estab-lished in the case of perfect domains over imaginary fields.
SO EDGE DOMAINS 5
Table 1.
Number of G L ( Z )-inequivalent iso-edge cones in S > by theirdimension.dim nr. iso. edge dim nr. iso. edge dim nr. iso. edge1 5 6 2478 11 87962 24 7 5180 12 49053 90 8 8642 13 19274 318 9 11350 14 4785 972 10 11472 15 76 Theorem 4.1.
Let us take n ≥ . Let us take the set S of all iso-edge domains whichcontain a positive definite matrix. Then we have the formula (1) X S ∈S ( − dim S | Stab( S ) | = 0 . Proof.
For each full dimensional iso-edge domain S we compute the list of extreme rays. Foreach extreme ray we choose a generator with integral coefficients whose greatest commondivisor is 1. To S we associate the sum iso ( S ) of those generators which are invariant underthe stabilizer of S .A given iso-edge domain containing a positive definite quadratic form is contained in a fi-nite set of full-dimensional iso-edge domains S , . . . , S L . We encode the set { iso ( S ) , . . . , iso ( S L ) } .This defines a complex which is contractible since the cone of positive definite quadratic formis contractible.The proof then follows the same line as the one of [6, Theorem 4.6] and the conclusioncomes from the fact that the Euler Poincar´e characteristic of G L n ( Z ) is 0 for n ≥ (cid:3) Full decomposition for n = 5 . The classification of possible Delaunay tessellations indimension 5 was obtained in [7]. The full dimensional iso-edge domains were determined in[17]. In order to determine the iso-edge domain of lower dimension, we compute the facets ofthe top-dimensional ones and check for isomorphism. Then we reiterate the procedure untilwe reach the iso-edge domains of dimension 1. We denote by σ nk the set of k -dimensionalcells in dimension n . The classification for n = 5 is given in Table 1. As in [7] a significantcheck for correctness comes from Theorem 4.1.5. The Conway–Sloane conjecture
For any v ∈ Parity n we call Θ v ( A ) = − cvp( A, v ) the tropical theta constant with char-acteristic v because it describes the asymptotic behavior of the classical Riemann thetaconstants. Note that the vonorm of the class 2 v + 2 Z n as defined by Conway and Sloane in[5, §
3] equals − · Θ v ( A ). They state that A is determined up to G L n ( Z )-equivalence by thevonorms for n ≤ n : Conjecture 1 (Conway–Sloane) . The map
Θ : S n> → R n − defined by Θ( A ) = (Θ v ( A )) v ∈ Parity n is injective on any fundamental domain of the G L n ( Z ) -action on S n> . M. D. SIKIRI´C AND M. KUMMER
Another way to phrase Conjecture 1 is that a tropical abelian variety is determined byits theta constants. This gives positive evidence for the conjecture as the correspondingstatement for classical abelian varieties is true [14]. We will prove that it is true for n = 5 aswell as when restricted to the matroidal locus. Lemma 3.1 shows injectivity on each iso-edgedomain. Lemma 5.1.
Let A ∈ S n> and x, y ∈ Z n . The line segment xy is in Del( A ) if and only ifthe closed ball B (with respect to A ) around x + y with radius r = (cid:13)(cid:13) x − y (cid:13)(cid:13) contains no latticepoints other than x and y .Proof. The “if” direction is clear from the definition. In order to show the other direction,let z ∈ B ∩ Z n with z
6∈ { x, y } . Note that z ′ = ( x + y ) − z ∈ B ∩ Z n . Further let w ∈ R n such that A [ w − x ] = A [ w − y ]. We will show that min( A [ w − z ] , A [ w − z ′ ]) ≤ A [ w − x ]which shows the claim. To this end, we first observe that A [ w − x ] = A [ w − y ] implies( x − y ) t · A · ( w − x + y ) = 0. We further have that A [ w − z ] = A (cid:2) w − x + y (cid:3) + A (cid:2) z − x + y (cid:3) − ( w − x + y ) t A ( z − x + y ) ,A [ w − z ′ ] = A (cid:2) w − x + y (cid:3) + A (cid:2) z − x + y (cid:3) + ( w − x + y ) t A ( z − x + y ) . Therefore, we havemin( A [ w − z ] , A [ w − z ′ ]) ≤ A (cid:2) w − x + y (cid:3) + A (cid:2) z − x + y (cid:3) ≤ A (cid:2) w − x + y (cid:3) + A (cid:2) x − y (cid:3) = A [ w − x ] . (cid:3) Lemma 5.2.
The map
Θ : S n> → R n − is linear and injective on every iso-edge domain.Proof. We show that Θ is linear and injective on the closure of every full-dimensional iso-edge domain. This implies the claim, see also Section 3. Let A be a positive definite n × n matrix and u ∈ Parity n . We first assume that Del( A ) is a triangulation. Let v u ∈ Z n bea closest (with respect to A ) lattice point to u and define r = A [ u − v u ]. We clearly have r = A [(2 u − v u ) − u ]. Thus 2 u − v u is also a closest lattice point to u . This means that v u and 2 u − v u are vertices of a Delaunay polytope of A . Since Del( A ) is a triangulation, thisshows that the line segment from v u to 2 u − v u is in Del( A ). Thus it follows from Lemma5.1 that v u is a closest lattice point to u with respect to every Q in the iso-edge domain of A . This means that cvp( Q, u ) = min x ∈ Z n Q [ x − u ] = Q [ v u − u ]for all Q ∈ C ( A ) which shows that Θ is linear on C ( A ). Since both sides of the above equationare continuous in Q , this equality holds on the closure of the iso-edge domain as well. Finally,assume that Θ( A ) = Θ( Q ) for some Q ∈ C ( A ). Then we have ( Q − A )[ v u − u ] = 0 for every u ∈ Parity n . Thus by Lemma 3.1 we have Q − A = 0. (cid:3) Thus in order to prove the Conway–Sloane conjecture it remains to show that the im-age under Θ of any two different representatives of G L n ( Z )-equivalence classes of iso-edgedomains are disjoint. We carry this out for n = 5. Theorem 5.3.
The Conway–Sloane conjecture is true in dimension .Proof. On a given iso-edge domain the map Θ is a linear map. The image of the polyhedralcone of the iso-edge domain is a polyhedral cone.Since we know the iso-edge domains in dimension 5, we can look at the obtained 76different images. We found out the following for each pair C , C .(1) The pairwise intersection Θ( C ) ∩ Θ( C ) does not happens in their relative interiors. SO EDGE DOMAINS 7 (2) The pairwise intersection Θ( C ) ∩ Θ( C ) define a face of qT heta ( C ) and Θ( C ).(3) The corresponding faces of C and C are arithmetically equivalent.Thus if two matrices have the same image then they are arithmetically equivalent. (cid:3) The matroidal locus
Let M be a unimodular n × r matrix with columns v , . . . , v r ∈ Z n . The set of positivedefinite matrices in the conic hull of the rank one matrices v v t , . . . , v r v tr is denoted by σ ( M ).The union of all σ ( M ) for some r and some unimodular n × r matrix M ∈ Z n × r is calledthe matroidal locus and is studied in [13]. In order to study the matroidal locus, we considerthe Fourier transforms of the tropical theta constants. Definition 6.1.
Let A ∈ S n> and v ∈ Parity n . Then we define ϑ v ( A ) = 12 n − X w ∈{ , / } n ( − · v T w · Θ w ( A ) . Following [5] we call ϑ v ( A ) the conorm with characteristic v of A .Conorms naturally appear when tropicalizing the Schottky–Igusa modular form whichcuts out the Schottky locus in genus 4 [4, § Proposition 6.2.
Assume that A ∈ S n> lies in the matroidal locus. Then ϑ u ( A ) ≥ for all u ∈ Parity n . More precisely, let M be a unimodular n × r matrix with columns v , . . . , v r ∈ Z n , no two of which are linearly dependent, such that we can write A = r X i =1 c i · v i v ti for some c i ≥ . Then ϑ u ( A ) = c i if u ≡ v i mod 2 and ϑ u ( A ) = 0 if u is not congruentto any v i .Proof. This was shown in the course of the proof of [4, Thm. 4.2]. (cid:3)
This implies the Conway–Sloane conjecture for matrices in the matroidal locus.
Corollary 6.3.
Let
A, B ∈ S n> in the matroidal locus. If Θ( A ) = Θ( B ) , then there is S ∈ G L n ( Z ) such that A = S t BS .Proof. Let M be a unimodular n × r matrix with columns v , . . . , v r ∈ Z n , no two of whichare linearly dependent, such that we can write A = r X i =1 c i · v i v ti for some c i ≥
0. We first observe that Proposition 6.2 implies that the regular matroidrepresented by M can be recovered as the binary matroid represented by the 2 u with u ∈ Parity n and ϑ u ( A ) = 0. Thus it is uniquely determined by Θ( A ). By [16, Cor. 10.1.4] everyregular matroid is uniquely representable which implies that the rank one matrices v i v ti areuniquely determined by Θ( A ) up to simultaneous G L n ( Z )-action. Finally, Proposition 6.2shows that the coefficients c i are also determined by Θ( A ) which completes the proof. (cid:3) We conjecture that the matroidal locus can be characterized by the conorms.
M. D. SIKIRI´C AND M. KUMMER
Conjecture 2.
A matrix A ∈ S n> lies in the matroidal locus if and only if ϑ u ( A ) ≥ forall u ∈ Parity n . Note that a proof of Conjecture 2 would also give a positive answer to [4, Question 4.11].Since for n ≤ n ≤ n = 4 , Theorem 6.4.
Conjecture 2 is true for n = 4 .Proof. One direction is Proposition 6.2. For the other direction note that it follows fromLemma 5.2 that ϑ u is linear on the secondary cone of every Delaunay subdivision of R n . For n = 4 we use the extreme rays R , . . . , R of the three secondary cones D , D , D givenin [19, § D is in the matroidal locus. The same is true for everymatrix in a face of D and D that does not have R as an extreme ray. Every other matrixin the secondary cone of D or D is the sum of a positive multiple of R and a coniccombination of matrices from { R , . . . , R } \ { R } . Letting u = ( , , ,
0) one calculatesthat ϑ u ( R ) = − ϑ u ( R k ) = 0 for all k
6∈ { , } . Therefore, by linearity we have ϑ u ( A ) < A . (cid:3) Theorem 6.5.
Conjecture 2 is true for n = 5 .Proof. On a given iso-edge domain D and for a fixed vector v ∈ Parity n the expression ϑ v ( A )is linear in A . We then consider the following polyhedral cone embedded in D : { A ∈ D and ϑ v ( A ) ≥ v ∈ Parity n } We can compute its description by using linear programming. We considered all the 76domains in dimension 5 and found out that for each of them this set is spanned by rank onematrices that determine an unimodular system of vectors and so belong to the matroidallocus. Furthermore, the 4 maximal unimodular systems in dimension 5 are obtained in thisway which concludes. (cid:3) Generalizations
The construction of iso-edge domains can be generalized to several different contexts. Welist here some that could be of interest for future researchers.7.1.
Generalization to subspaces.
In [2, 10] the perfect form and secondary cones weregeneralized to the case of matrices A belonging to a vector space V ⊂ S n such that V ∩ S n> = ∅ . This kind of setting is used in an enormous number of cases for a multiplicity of purposes.It would be interesting to extend this to the case of iso-edge domains.7.2. Generalization to k -cells. The iso-edge domains concern the case of dimension 1 cellsof the Delaunay polytopes while the secondary cones consider the full dimensional cells ofthe Delaunay tessellation. If we fix the k -dimensional faces of Delaunay polytope then wealso get a polyhedral decomposition of S n> . The first interesting case would be dimension 5where the case k = 1, 2 and 5 would be all different.7.3. Generalization to the case of symmetric faces.
The iso-edge domains correspondto the centrally symmetric faces of the Delaunay tessellation. If we take a finite subgroup G of G L n ( Z ) then we can consider the forms invariant under this group. The group G acts on R n / Z n . If the set of fixed points S is finite, then we can consider the set of closest points tothem. Those define invariant cells of the Delaunay tessellation. If G = {± Id n } , then we getthe case of iso-edge domains. SO EDGE DOMAINS 9 Acknowledgments
The first author thanks Viatcheslav Grishukhin for introducing him to the works ofRyshkov and Baranovskii and Klaus Hulek for introducing him to the theory of compactifi-cations. Both authors thank Frank Vallentin for the invitation to the conference “Discretegeometry with a view on symplectic and tropical geometry”.
References [1] A. Ash, D. Mumford, M. Rapoport, and Y.-S. Tai,
Smooth compactifications of locally symmetricvarieties , second ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010,With the collaboration of Peter Scholze.[2] A.-M. Berg´e, J. Martinet, and F. Sigrist,
Une g´en´eralisation de l’algorithme de Vorono˘ı pour les formesquadratiques , no. 209, 1992, Journ´ees Arithm´etiques, 1991 (Geneva), pp. 12, 137–158.[3] D. Bremner, M. Dutour Sikiri´c, D. V. Pasechnik, T. Rehn, and A. Sch¨urmann,
Computing symmetrygroups of polyhedra , LMS J. Comput. Math. (2014), no. 1, 565–581.[4] L. Chua, M. Kummer, and B. Sturmfels, Schottky algorithms: classical meets tropical , Math. Comp. (2019), no. 319, 2541–2558.[5] J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. VI. Vorono˘ı reduction of three-dimensionallattices , Proc. Roy. Soc. London Ser. A (1992), no. 1896, 55–68.[6] M. Dutour Sikiri´c, H. Gangl, P. E. Gunnells, J. Hanke, A. Sch¨urmann, and D. Yasaki,
On the cohomologyof linear groups over imaginary quadratic fields , J. Pure Appl. Algebra (2016), no. 7, 2564–2589.[7] M. Dutour Sikiri´c, A. Garber, A. Sch¨urmann, and C. Waldmann,
The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices , Acta Crystallogr. Sect. A (2016),no. 6, 673–683.[8] M. Dutour Sikiri´c, D. Madore, P. Moustrou, and F. Vallentin, Coloring the voronoi tessellation oflattices , preprint at arxiv:arXiv:1907.09751 , July 2019.[9] M. Dutour Sikiri´c, A. Magazinov, and W. van Woerden,
Enumerating the Iso-edge domains in dimensionsix , in preparation.[10] M. Dutour Sikiri´c, A. Sch¨urmann, and F. Vallentin,
A generalization of Voronoi’s reduction theory andits application , Duke Math. J. (2008), no. 1, 127–164.[11] P. Engel,
On fedorov’s parallelohedra - a review and new results , Cryst. Res. Technol. (2015), no. 12,929–943.[12] G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups , Ann. Sci. ´Ecole Norm.Sup. (4) (1971), 409–455.[13] M. Melo and F. Viviani, Comparing perfect and 2nd Voronoi decompositions: the matroidal locus ,Math. Ann. (2012), no. 4, 1521–1554.[14] D. Mumford,
Tata lectures on theta. III , Modern Birkh¨auser Classics, Birkh¨auser Boston, Inc., Boston,MA, 2007, With collaboration of Madhav Nori and Peter Norman, Reprint of the 1991 original.[15] Y. Namikawa,
Toroidal compactification of Siegel spaces , Lecture Notes in Mathematics, vol. 812,Springer, Berlin, 1980.[16] J. G. Oxley,
Matroid theory , Oxford Science Publications, The Clarendon Press, Oxford UniversityPress, New York, 1992.[17] S. S. Ryˇskov and E. P. Baranovski˘ı, C -types of n -dimensional lattices and -dimensional primitiveparallelohedra (with application to the theory of coverings) , Proc. Steklov Inst. Math. (1978), no. 4, 140,Cover to cover translation of Trudy Mat. Inst. Steklov
37 (1976), Translated by R. M. Erdahl.[18] A. Sch¨urmann,
Computational geometry of positive definite quadratic forms , University Lecture Series,vol. 48, American Mathematical Society, Providence, RI, 2009, Polyhedral reduction theories, algo-rithms, and applications.[19] F. Vallentin,
Sphere covering, lattices, and tilings (in low dimensions) , Ph.D. thesis, Technische Uni-versit¨at M¨unchen, Universit¨atsbibliothek, 2003.[20] G. Voronoi,
Nouvelles applications des param`etres continues `a la th´eorie des formes quadratiques 1:Sur quelques propri´et´es des formes quadratiques positives parfaites , J. Reine Angew. Math (1908),no. 1, 97–178.[21] G. Voronoi,
Nouvelles applications des param`etres continus `a la th´eorie des formes quadratiques.Deuxi`eme M´emoire. Recherches sur les parall´ello`edres primitifs. , J. Reine Angew. Math (1908),no. 1, 198–287.
Mathieu Dutour Sikiri´c, Rudjer Boskovi´c Institute, Bijeniˇcka 54, 10000 Zagreb, Croatia
Email address : [email protected] Technische Universit¨at Dresden, Fakult¨at Mathematik, Institut f¨ur Geometrie, ZellescherWeg 12-14, 01062 Dresden, Germany
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