On the Generalised Hermite Constants

On the Generalised Hermite Constants NTU SPMS-MAS Seminar Bertrand MEYER IMB Bordeaux Singapore, July 10th, 2009 B. Meyer (IMB) Hermite constants Jul 10th 2009 1 / 35

Outline 1 Introduction 2 The generalised constant 3 Some inequalities 4 Voronoï theory 5 Vexillar designs B. Meyer (IMB) Hermite constants Jul 10th 2009 2 / 35

Outline Introduction 1 Introduction 2 The generalised constant 3 Some inequalities 4 Voronoï theory 5 Vexillar designs B. Meyer (IMB) Hermite constants Jul 10th 2009 3 / 35

Introduction Classical Hermite constant y x Take a lattice L, consider its minimum and determinant. The Hermite invariant is γ n (L) = min L (det L) 1/n. S (L) = { } x L, x 2 = min L Call γ n = sup L R n γ n (L) the Hermite constant. B. Meyer (IMB) Hermite constants Jul 10th 2009 4 / 35

Introduction Classical Hermite constant S (L) = y S (L) { } x L, x 2 = min L x Take a lattice L, consider its minimum and determinant. The Hermite invariant is Call γ n (L) = min L (det L) 1/n. γ n = sup L R n γ n (L) the Hermite constant. B. Meyer (IMB) Hermite constants Jul 10th 2009 4 / 35

Introduction Classical Hermite constant S (L) = y S (L) det L { } x L, x 2 = min L x Take a lattice L, consider its minimum and determinant. The Hermite invariant is Call γ n (L) = min L (det L) 1/n. γ n = sup L R n γ n (L) the Hermite constant. B. Meyer (IMB) Hermite constants Jul 10th 2009 4 / 35

Introduction Classical Hermite constant y x Take a lattice L, consider its minimum and determinant. The Hermite invariant is γ n (L) = min L (det L) 1/n. S (L) = { } x L, x 2 = min L Call γ n = sup L R n γ n (L) the Hermite constant. B. Meyer (IMB) Hermite constants Jul 10th 2009 4 / 35

Introduction Classical Hermite constant S (L) = y { } x L, x 2 = min L x Equivalently, the Hermite constant is γ n = max min A q. f. x Z n 0 A[x] (det A) 1/n, where A is positive definite n-dim. quadratic form, or γ n = max g GL n(r) det g=1 min gγ e 1. γ GL n(z) B. Meyer (IMB) Hermite constants Jul 10th 2009 4 / 35

Introduction What is the Hermite constant good for? Some examples : 1 Sphere packing problem 2 Non zero integer point of bounded size (whatever the norm is) 3 Example of an easily computable systole. B. Meyer (IMB) Hermite constants Jul 10th 2009 5 / 35

Introduction What is the Hermite constant good for? Some examples : 1 Sphere packing problem 2 Non zero integer point of bounded size (whatever the norm is) 3 Example of an easily computable systole. y {3x 2 + 3xy + y 2 < 1} x Z 2 B. Meyer (IMB) Hermite constants Jul 10th 2009 5 / 35

Introduction What is the Hermite constant good for? Some examples : 1 Sphere packing problem 2 Non zero integer point of bounded size (whatever the norm is) 3 Example of an easily computable systole. Lattice L Shortest vector vs minimal non-contractile loop Torus R n /L B. Meyer (IMB) Hermite constants Jul 10th 2009 5 / 35

Known values Introduction Known values of γ n n 1 2 3 4 5 6 7 8 24 Lattice Z A 2 A 3 D 4 D 5 E 6 E 7 E 8 Λ 24 B. Meyer (IMB) Hermite constants Jul 10th 2009 6 / 35

Outline The generalised constant 1 Introduction 2 The generalised constant 3 Some inequalities 4 Voronoï theory 5 Vexillar designs B. Meyer (IMB) Hermite constants Jul 10th 2009 7 / 35

The generalised constant The generalised Hermite constant Some number theoretic notations k : number field of degree d = r 1 + 2r 2 (we assume that the class number is 1) o : ring of integers V : set of Archimedean places ( V = r = r 1 + r 2 ) k v : completion of k ; v : module on k v A = (A v ) v V : Hermite Humbert form. Depending on v, A v is a Euclidean or Hermitian positive definite quadratic form. We define also : det A = v V det A v v. B. Meyer (IMB) Hermite constants Jul 10th 2009 8 / 35

The generalised constant The generalised Hermite constant Some notations Fix a partition λ of depth λ and transpose λ. λ Consider a familly of λ lattice vectors (x i ) 1 i λ o n. Form the flag = ( j ) 1 j λ where j is the sublattice spanned by (x i ) 1 i λj. X = x 3 x 2 x 2 x 2 x 1 x 1 x 1 x 1 Denote the evaluation of X in the Humbert form A A[X ] = det(x i A vx i ) i;i j v V,j λ B. Meyer (IMB) Hermite constants Jul 10th 2009 9 / 35 v λ

The generalised constant The generalised Hermite constant Our concern for the scope of this talk Definition Define the generalised Hermite constant as γ n,λ (k) = max A A(X ) min o n (det A) λ /n Ch. Hermite where A runs over all n-dimensional Humbert forms and over all flags of shape λ. B. Meyer (IMB) Hermite constants Jul 10th 2009 10 / 35

The generalised constant The generalised Hermite constant Our concern for the scope of this talk Definition (Watanabe) Equivalently, the generalised Hermite constant is also γ n,λ (k) = max g GL n(k A ) det g =1 min H(π(gγ) x π) γ GL n(k) for the representation π of GL n (R) of weight λ and where x π denotes a normalised highest weight vector. Remark Here, k A denotes the adel ring of k and H a height on the Schur module associated with π that needs to be carefully chosen. B. Meyer (IMB) Hermite constants Jul 10th 2009 11 / 35

The generalised constant The generalised Hermite constant And Diophantine approximation Let g GL n (k A ), call H g (V ) = H(g V ) the twisted height. Théorème The Hermite constant γ n,λ (k) is the smallest constant C such that for any g GL n (k A ), there exists a rational flag satisfying y {3x 2 + 3xy + y 2 < 1} x Z 2 H g ( ) C 1/2 det g ka. B. Meyer (IMB) Hermite constants Jul 10th 2009 12 / 35

The generalised constant The generalised Hermite constant And its particular cases Example If λ = and k = Q, γ n,λ is the classical Hermite constant If λ =, γ n,λ is called Hermite-Humbert constant If λ = (only one column) and k = Q, γ n,λ is called the Rankin constant. If λ = (only one column), γ n,λ was introduced by Thunder. B. Meyer (IMB) Hermite constants Jul 10th 2009 13 / 35

The generalised constant Purpose of the talk 1 Can one calculate some values of γ n,λ? Answer : Use inequalities 2 How to characterise the extreme forms with respect to λ, i.e. the local maxima of A γ n,λ (A)? Answer :Voronoy theory 3 How to show that expected examples (e.g. Leech lattice and friends) are extreme with respect to λ? Answer : Vexillar designs B. Meyer (IMB) Hermite constants Jul 10th 2009 14 / 35

The generalised constant Purpose of the talk 1 Can one calculate some values of γ n,λ? Answer : Use inequalities 2 How to characterise the extreme forms with respect to λ, i.e. the local maxima of A γ n,λ (A)? Answer :Voronoy theory 3 How to show that expected examples (e.g. Leech lattice and friends) are extreme with respect to λ? Answer : Vexillar designs B. Meyer (IMB) Hermite constants Jul 10th 2009 14 / 35

Outline Some inequalities 1 Introduction 2 The generalised constant 3 Some inequalities 4 Voronoï theory 5 Vexillar designs B. Meyer (IMB) Hermite constants Jul 10th 2009 15 / 35

Some inequalities How to compare the constants for different n? Proposition (Mordell type inequality) For a partition λ of depth λ n n, ( ) λ /n γ n,λ γ n,λ γn,n Proof Start with the best n -dim sublattice L. Chose the best flag of shape λ in L. Write the corresponding inequality. B. Meyer (IMB) Hermite constants Jul 10th 2009 16 / 35

Some inequalities Some computed values Rephrasing Korkine and Zolotareff reduction γ 3, = 3 2 γ 4, = 2, 32 15 γ 5, < 32 15 + 6 10 5. By extension using Mordell type inequality n = 4 n = 6 n = 8 3 γ 2 n, 2 2 3 6 γ 2 2 3 2/3 8 n, γ n, 3 9 24 B. Meyer (IMB) Hermite constants Jul 10th 2009 17 / 35

Some inequalities Changing the field Proposition (Field change) γ n,λ (k) D k λ (γ nd,λ (Q)) d d d Proof. Write o as a Z module and use arithmetic-geometric inequality. Example : Gaussian or Eisentein lattices The previous inequality is sharp for Q[i] with D 4, E 8, Leech lattice Q[j] with E 6, E 8, Leech lattice. This enables to compute γ n, (k). B. Meyer (IMB) Hermite constants Jul 10th 2009 18 / 35

Outline Voronoï theory 1 Introduction 2 The generalised constant 3 Some inequalities 4 Voronoï theory 5 Vexillar designs B. Meyer (IMB) Hermite constants Jul 10th 2009 19 / 35

Voronoï theory Perfection and eutaxy Recall that S (A) stands for the set of minimal flags of A with respect to λ. Definition A form A is called perfect with respect to λ if the gradients ( l ) S (A) of the length functions l : A ln A[x ] affinely span the tangent space. A form A is called eutactic with respect to λ if the zero vector lies in the affine interior of the convex hull of the gradients ( l ) S (A). B. Meyer (IMB) Hermite constants Jul 10th 2009 20 / 35

Voronoï theory Voronoy theory A Humbert form is called extreme with respect to λ if it achieves a local maximum of the Hermite invariant γ n,λ (A). Theorem A Humbert form is extreme with respect to λ if and only if it is perfect and eutactic wrt λ. G. Voronoï Was already known for Hermite and Rankin constants (see Voronoï and Coulangeon). B. Meyer (IMB) Hermite constants Jul 10th 2009 21 / 35

Voronoï theory Sketch of proof Proof Check that condition (C) by Bavard is satisfied : study the expansion of the length functions up to the 4th order. Interpretation : det A Ryshkov domain min A 1 eutactic perfect B. Meyer (IMB) Hermite constants Jul 10th 2009 22 / 35

Voronoï theory Consequences Proposition The constant γ n,λ is algebraic. Proposition For fixed integer n and partition λ, there are only finitely many perfect lattices (up to homotheties and rotations). B. Meyer (IMB) Hermite constants Jul 10th 2009 23 / 35

Consequences Voronoï theory Proposition When λ = and r = 1 (i.e. k is rational or imaginary quadratic), there is an algorithm that enumerates all perfect forms and compute γ n, (k). Example (n=3) Q[i] Q[ 2] Q[j] Q[ 7] Q[ 11] Perfect forms D 6 2 E 6, E 6 A 6, A (2) 6 12 γ 3, (k) 2 3 2 4 3 9 4 1890625 15376 B. Meyer (IMB) Hermite constants Jul 10th 2009 24 / 35

Outline Vexillar designs 1 Introduction 2 The generalised constant 3 Some inequalities 4 Voronoï theory 5 Vexillar designs B. Meyer (IMB) Hermite constants Jul 10th 2009 25 / 35

Vexillar designs Vexillar designs Denote by F λ the set of all vectorial flags of shape λ. Definition We say that a finite subset D of the flag variety F λ is a vexillar t-design if for any polynomial f of degree less than t the following equality holds f ( ) d = 1 f ( ). 4-design with the minimal vectors of A 2 F λ D D Already known in the literature : spherical and Grassmanian designs. B. Meyer (IMB) Hermite constants Jul 10th 2009 26 / 35

Vexillar designs Strongly perfect lattices Definition A strongly perfect lattice wrt λ is a lattice the minimal flags of which carry a vexillar 4-design. Theorem (M) A strongly perfect lattice wrt λ is extreme wrt λ. Proof Strongly perfect perfect and eutactic extreme. See Venkov for the classical case and Bachoc, Coulangeon, Nebe for the Grassmaniann case. B. Meyer (IMB) Hermite constants Jul 10th 2009 27 / 35

Vexillar designs Strategy and technical details 1 Decompose L 2 (F λ ) into irreducible O n -modules. 2 Define zonal functions for each irreducible space. 3 Express being a design as a condition on zonal functions. 4 Check the implication strongly perfect perfect & eutactic with zonal functions. 5 (Bonus) Give an additionnal criterion on finite groups to obtain a design as the orbit of some flag under the group. B. Meyer (IMB) Hermite constants Jul 10th 2009 28 / 35

Vexillar designs A word on representation theory How to work in practice? z 1,1, z 1,m Set Z =... Then R[Z ] is a GL n (R) GL m (R) module z n,1 z n,m under left and right multiplication that can be handled with bitableaux. [ ] For {T, Θ} a bitableau, define the polynomial of R[Z ] = R (z i,j ) 1 i n 1 j m M {T,Θ} = det(z T (j,i),θ(j,i )) 1 i,i λ j, 1 j λ also called determinantal monomial. Example 4 Soit T = 1 3 2, Θ = 1 1, M {T,Θ} = z 1,1 z 1,2 z 4,1 z 4,2 z 3,1. B. Meyer (IMB) Hermite constants Jul 10th 2009 29 / 35

Vexillar designs Decomposition of R[Z ] As a GL n (R) GL m (R) module, R[Z ] decomposes as R[Z ] = λ; λ m,n S λ (R n ) S λ (R m ) where S λ (R n ) denotes the representation space (Schur module) of GL n (R) of weight λ. Further more, S λ (R n ) x λ has a basis given by the determinantal monomial M {T,U(λ)} where x λ is a highest weight vector, T is a λ standard tableau and U(λ) = 1 1 1 1. B. Meyer (IMB) Hermite constants Jul 10th 2009 30 / 35

Vexillar designs Decomposition of L 2 (F λ ) Represent a flag by a n m matrix (with m = λ ). This gives the inclusion L 2 (F λ ) R[Z ] = S λ (R n ) S λ (R m ). Under the action of O n (R) λ; λ m,n L 2 (F λ ) = µ S [µ] (R n ) S µ (R m ) Om }{{} :=N µ with S [µ] (R n ) O n -irreducible representation of weight µ. Define n µ = dim N µ. In particular, polynomials up to degree 4 are R S [ ] (R n ) n S [ ] (R n ) n S [ ] (R n ) n B. Meyer (IMB) Hermite constants Jul 10th 2009 31 / 35

Zonal functions Vexillar designs We call zonal function a function F λ F λ R such that τ O n (R), Z (τ, τ ) = Z (, ) Example (Computations) The zonal functions of S [ ] (R n ) n are spanned by ( ) Z i,i (, ) = Tr pr j pr λ jλ j j n where pr j is the projection on the jth subspace of the flag. B. Meyer (IMB) Hermite constants Jul 10th 2009 32 / 35

Vexillar designs Equivalences How to know that we have a vexillar design Theorem The following conditions are equivalent: 1 The set D is a t-design. 2 For any polynomial of degree t, for any τ O n, we have f ( ) = f (τ ). D D 3 For any µ with 0 < µ t, for any Ξ N µ, for any function f of S [µ] (R n ) Ξ, the sum D f ( ) is zero. 4 For any µ with 0 < µ t, for any Ξ and Ξ N µ and for any F λ, Z Ξ,Ξ (, ) = 0. D B. Meyer (IMB) Hermite constants Jul 10th 2009 33 / 35

Vexillar designs Group theory to the rescue of designs When does the orbit of a flag carry a design Theorem Let G be a finite subgroup of the orthogonal group O n (R), then the following properties are equivalent : 1 The decomposition of the vector space S [µ] (R n ) nµ as µ t µ m invariant G-modules discloses the trivial representation 1 G only once. 2 For any flag, the orbite G of under the action of G forms a t-design. B. Meyer (IMB) Hermite constants Jul 10th 2009 34 / 35

Vexillar designs Results Examples The root lattices D 4, E 6, E 7, E 8, the Leech lattice, the Barnes-Wall lattices (BW k ) k 3, the Thompson-Smith lattice are extreme for any λ. Proof. Check by character theory the group theoretic criterion for being a vexillar t-design (with t 4) on the automorphism group of the lattices or see classification by Tiep. B. Meyer (IMB) Hermite constants Jul 10th 2009 35 / 35