Coloring the Voronoi tessellation of lattices

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1 Coloring the Voronoi tessellation of lattices Frank Vallentin University of Cologne, Germany based on joint work with: M. Dutour-Sikirić, D. Madore A talk in honour of Thomas C. Hales on the occasion of his 60th birthday June 18, 2018

2 The problem Generalize the checkerboard...!??!

3 Definitions R n lattice V (,v)={x 2 R n : kx vk applekx wk 8w 2 },v 2 Voronoi tessellation

4 More definitions Voronoi vectors v 2 Vor( ) () V (, 0) \ V (,v) is (n 1)-dim. facet 0 Cayley(, Vor( )) graph with vertices V and edges E vertices: V =, edges: {v, w} 2 E () v w 2 Vor( ) Determine chromatic number of this infinite graph:? ( )= (Cayley(, Vor( ))

5 Questions 1. What is the chromatic number of some interesting lattices? 2. Is there an algorithm to determine ( ) for a given lattice?? (E 8 ) = 16, ( 24 ) = How fast can ( ) grow depending on the dimension n? ( ) apple 2 n 4. What is ( ) of a random n-dimensional lattice? E[ ( )] n

6 Voronoi s characterization v 2 Vor( ) () V (, 0) \ V (,v) is (n 1)-dim. facet Theorem. (Voronoi 1908) Vor( ) ={u 2 \{0} : ±u only shortest vectors in u +2 }. u 1 2 (u + v) 2 v

7 Upper bounds Graph theory 101: Greedy coloring: G finite graph =) (G) apple + 1 = max. vertex degree Apply to 2 -periodic colorings G =( /2, {{v, w} : 9u 2 : v w + 2u 2 Vor( )} Then ( ) apple (G) apple + 1 = Vor( ) apple 2(2n 1) = 2 n

8 Lower bounds 1) Sphere packing bound 2) Spectral bound

9 Sphere packing bound Theorem. Let R n be lattice determining packing of unit spheres Let µ>0 be so that for all v 2 \{0} if kvk <µ, then v 2 Vor( ). Then, ( ) (µ/2) n ( ) R n. ( )=center density of Proof. Suppose ( )=k. Decompose = C 1 [ C 2 [...[ C k. We may assume k (C 1 ) ( ). For all v, w 2 C 1, v 6= w: kv wk µ. So 2/µC 1 defines packing of unit spheres. So: k R n k (2/µC 1 ) (µ/2) n ( ). Rn = largest sphere packing density in Rn É

10 Consequences of the sphere packing bound Apply to E 8 and 24 : (E 8 )= R 8 (Viazovska, 2017) ( 24 )= R 24 (Cohn, Kumar, Miller, Radchenko, Viazovska, 2017) =) (E 8 )=16, ( 24 )=4096 Apply to generic lattices: Lemma. lattice defining packing of unit p spheres, and v 2 \{0} not a Voronoi vector, then kvk 8. Let be a generic n-dimensional lattice =) ( ) ( p 8/2) n ( ) R n n use Minkowski-Hlawka lower bound for ( ) and Kabatiansky-Levenshtein upper bound for R n.

11 Spectral bound Graph theory 201: Hoffman bound: Let G =(V, E) r-regular graph then, (G) 1 1 m(a). A 2 R V V normalized adjacency operator Af (v)= 1 X f (w) r w2v {v,w}2e m(a)= min (Af, f ) smallest eigenvalue of A k f k=1 v Bachoc, DeCorte, Oliveira Vallentin (2014): A : `2( )! `2( ) 1 Af (v)= Vor( ) X f (v u) u2vor( ) also works for infinite graphs X `2( )= f :! C : f (v) 2 < 1 v2

12 ( ) 1 1 m(a) Fourier transform m(a)= inf (Af, f ) k f k=1 adjacency operator is convolution operator 1 Af (v)= Vor( ) 1 Vor( ) f (v) Fourier transform diagonalizes A (Af, f )=( c Af, b f )= 1 Vor( ) ( b1 Vor( ) b f, b f ) b1 Vor( ) : R n /! R = { y 2 R n : x y 2 Z8x 2 } X b1 Vor( ) (x)= e 2 u x u2vor( ) Theorem. ( ) 1 inf x2r n / 1 Vor( ) X u2vor( ) e 2 iu x! 1.

13 Spectral bound for root lattices R n root lattice: 8v, w 2 : v w 2 Z 8v 2 : v v 2 2Z R( )={v 2 : v v = 2} generates If is a root lattice, then R( )=Vor( ). Classification of Witt (1941): Every root lattice is orthogonal direct sum of irreducible root lattices. The irreducible root lattices are A n, D n, E 6, E 7, E 8. Coxeter-Dynkin diagrams of irreducible root lattices: b i, b j connected iff b i b j = 1 b i, b j not connected iff b i b j = 0

14 First observations Suppose L = L 1? L 2, then (L)=max{ (L 1 ), (L 2 )}. c 1 : L 1! {0,..., k 1 } c 2 : L 2! {0,..., k 2 } k 1 k 2 =) c : L 1? L 2! {0,..., k 1 } v 1 + v 2 7! c 1 (v 1 )+c 2 (v 2 ) mod k 1 colorings of L 1 and L 2 coloring of L

15 A n nx A n = x 2 Z n+1 : has n(n + 1) roots k=0 x k = 0 = x 2 R n+1 : nx k=0 x k = 0 R(A n )={e r Theorem. e s :0apple r, s apple n, r 6= s}, inf x2 /A n X u2r(a n ) e 2 iu x = (n + 1). In particular, (A n ) n + 1. Can also show (A n ) apple n + 1.

16 D n D n = x 2 Z n : nx x k is even k=1 has 2n(n 1) roots R(D n )={±(e r + e s ) :1apple r, s apple n, r 6= s} [ {±(e r e s ) :1apple r, s apple n, r 6= s}. Theorem. inf x2r n /D n X u2r(d n ) e 2 iu x = 2n, n even, 2(n 1), n odd. In particular, (D n ) n for even n, and (D n ) n + 1 for odd n. Can also show (D n ) apple n if n = 2 m.

17 E 8 Construction of E 8 using coding theory (extended) Hamming code H Construction A: E 8 = 1 p2 x : x 2 Z 8,xmod 2 2 H 8. Voronoi vectors Vor(E 8 ): 16 = 2 4 vectors: ± p 2e i,i=1,...,8 224 = vectors: 1 p2 P 8 j=1 (±c j)e j,c2 H 8 and wt(c) =4,

18 Serre s Oberwolfach report Algebraische Gruppen 667 (12/2004) Theorem 3. One has inf x G Tr Ad(x) = inf w W Tr T (w). This shows in particular that inf Tr Ad(x) isan integer, a fact which was not a priori obvious. It also shows that inf Tr Ad(x) isequalto rank(g) ifandonly if W contains an element which acts on T by t t 1. When G is connected and simple, Theorem 3 gives: inf Tr Ad(x) = rank(g) if G is of type A 1,B n,c n,d n (n even),g 2,F 4,E 7,E 8, 1 if G is of type A n (n 1) inf Tr Ad(x) = 2 n if G is of type D n (n odd 3) 3 if G is of type E Proofs They are not published yet. Here are some hints for the interested reader : Theorem 1: Use the properties of the principal A 1 subgroup of G. Theorem 2: An exercise on positive-valued trigonometric polynomials. Observe: Theorem 3 : If w W is such that Tr T (w) isminimum,anyrepresentative x of w in N is such that Tr Ad(x) =Tr T (w); this proves the inequality inf Tr Ad(x) inf Tr T (w). The opposite inequality can be checked by a case-by-case explicit computation. The classical groups are easy enough, but F 4,E 6,E 7 and E 8 are not (especially E 6,whichIowetoAlainConnes). Ihopethereisabetterproof. Tr Ad(x)=n + X u2r( ) e 2 iu x

19 Alternative sum of squares proof spectral bound: 0 ( ) 1 inf x2r n / Vor( ) X u2vor( ) e 2 iu x 1 A = 2 4 vectors: ± p 2e i,i=1,...,8 224 = vectors: 1 p2 P 8 j=1 (±c j)e j,c2 H 8 and wt(c) =4, spectral bound = 16 for E 8 () polynomial p(t) = 8X t 2 i +4 X t i t j t k t l i=1 c2h 8,wt(c)=4,supp(c)={i,j,k,l} is nonnegative on the cube t 2 [ 1, +1] 8 Computer verification: 8X p(t) =q(t)+ (1 t 2 i )q i (t) i=1 with q 2 8,8, q 1,...,q 8 2 8,6

20 E 6, E 7, E 8 E 7 = {x 2 E 8 : x v = 0} E 6 = {x 2 E 8 : x v = 0, x w = 0} v 2 E 8 any root, w 2 E 8 root with v w = 1 Theorem. inf x2r n /E n X u2r(e n ) e 2 iu x = 8 < : 9, for n = 6, 14, for n = 7, 16, for n = 8. In particular, (E 6 ) 9, (E 7 ) 10, and (E 8 ) 16.

21 Summary irred. root lattice spectral lower bound exact value A n n + 1 n + 1 D n n, when n even ( 1 2 H n) n + 1, when n odd ( 1 2 H n) E E E vertex-edge graph of half-cube polytope 1 nx 2 H n = conv x 2 {0, 1} n : x k is even k=1

22 References C. Bachoc, P.E.B. DeCorte, F.M. de Oliveira Filho, and F. Vallentin, Spectral bounds for the independence ratio and the chromatic number of an operator, Israel J. Math. 202 (2014), M. Dutour-Sikirić, D. Madore, and F. Vallentin, Coloring the Voronoi tessellation of lattices, work in progress, 2018 J.-P. Serre, On the values of the characters of compact Lie group, Oberwolfach Reports vol.1 (2004),

Sphere packing, lattice packing, and related problems

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