# On the Generalised Hermite Constants

Save this PDF as:

Size: px
Start display at page:

## Transcription

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

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

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

4 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 / 35

5 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 / 35

6 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 / 35

7 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 / 35

8 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 / 35

9 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 / 35

10 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 / 35

11 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 / 35

12 Known values Introduction Known values of γ n n 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 / 35

13 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 / 35

14 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 / 35

15 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 / 35 v λ

16 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 / 35

17 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 / 35

18 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 / 35

19 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 / 35

20 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 / 35

21 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 / 35

22 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 / 35

23 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 / 35

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

25 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 / 35

26 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 / 35

27 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 / 35

28 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 / 35

29 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 / 35

30 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 / 35

31 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) B. Meyer (IMB) Hermite constants Jul 10th / 35

32 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 / 35

33 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 / 35

34 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 / 35

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 / 35

36 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 / 35

37 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) = B. Meyer (IMB) Hermite constants Jul 10th / 35

38 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 / 35

39 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 / 35

40 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 / 35

41 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 / 35

42 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 / 35

### 5 Quiver Representations

5 Quiver Representations 5. Problems Problem 5.. Field embeddings. Recall that k(y,..., y m ) denotes the field of rational functions of y,..., y m over a field k. Let f : k[x,..., x n ] k(y,..., y m )

### CSE 206A: Lattice Algorithms and Applications Winter The dual lattice. Instructor: Daniele Micciancio

CSE 206A: Lattice Algorithms and Applications Winter 2016 The dual lattice Instructor: Daniele Micciancio UCSD CSE 1 Dual Lattice and Dual Basis Definition 1 The dual of a lattice Λ is the set ˆΛ of all

### CLASSIFICATION OF QUINTIC EUTACTIC FORMS

MATHEMATICS OF COMPUTATION Volume 70, Number 233, Pages 395 417 S 0025-5718(00)01295-3 Article electronically published on July 21, 2000 CLASSIFICATION OF QUINTIC EUTACTIC FORMS CHRISTIAN BATUT Abstract.

### REPRESENTATION THEORY WEEK 7

REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable

### THE UNIT GROUP OF A REAL QUADRATIC FIELD

THE UNIT GROUP OF A REAL QUADRATIC FIELD While the unit group of an imaginary quadratic field is very simple the unit group of a real quadratic field has nontrivial structure Its study involves some geometry

### REPRESENTATIONS OF S n AND GL(n, C)

REPRESENTATIONS OF S n AND GL(n, C) SEAN MCAFEE 1 outline For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G Although

### 1: Introduction to Lattices

CSE 206A: Lattice Algorithms and Applications Winter 2012 Instructor: Daniele Micciancio 1: Introduction to Lattices UCSD CSE Lattices are regular arrangements of points in Euclidean space. The simplest

### Math 210C. The representation ring

Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let

### HERMITE CONSTANT AND EXTREME FORMS FOR ALGEBRAIC NUMBER FIELDS

HERMITE CONSTANT AND EXTREME FORMS FOR ALGEBRAIC NUMBER FIELDS M I ICAZA 1 Introduction In this paper we consider a generalization to algebraic number fields of the classical Hermite constant γ n For this

### ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

### Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

### REPRESENTATION THEORY WEEK 5. B : V V k

REPRESENTATION THEORY WEEK 5 1. Invariant forms Recall that a bilinear form on a vector space V is a map satisfying B : V V k B (cv, dw) = cdb (v, w), B (v 1 + v, w) = B (v 1, w)+b (v, w), B (v, w 1 +

### MATH Linear Algebra

MATH 304 - Linear Algebra In the previous note we learned an important algorithm to produce orthogonal sequences of vectors called the Gramm-Schmidt orthogonalization process. Gramm-Schmidt orthogonalization

### Voronoi-Dickson Hypothesis on Perfect Forms and L-types

Voronoi-Dickson Hypothesis on Perfect Forms and L-types Robert Erdahl and Konstantin Rybnikov November 3, 2001 Abstract George Voronoi (1908, 1909) introduced two important reduction methods for positive

### CSE 206A: Lattice Algorithms and Applications Spring Minkowski s theorem. Instructor: Daniele Micciancio

CSE 206A: Lattice Algorithms and Applications Spring 2014 Minkowski s theorem Instructor: Daniele Micciancio UCSD CSE There are many important quantities associated to a lattice. Some of them, like the

### ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS

ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS LISA CARBONE Abstract. We outline the classification of K rank 1 groups over non archimedean local fields K up to strict isogeny,

### Dense lattices as Hermitian tensor products

Contemporary Mathematics Volume 58, 013 http://dx.doi.org/10.1090/conm/58/1165 Dense lattices as Hermitian tensor products Renaud Coulangeon and Gabriele Nebe This paper is dedicated to Boris Venkov. Abstract.

### Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

### Logarithmic functional and reciprocity laws

Contemporary Mathematics Volume 00, 1997 Logarithmic functional and reciprocity laws Askold Khovanskii Abstract. In this paper, we give a short survey of results related to the reciprocity laws over the

### Since G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg =

Problem 1 Show that if π is an irreducible representation of a compact lie group G then π is also irreducible. Give an example of a G and π such that π = π, and another for which π π. Is this true for

### A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM CONTENTS

A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM TIMOTHY E. GOLDBERG ABSTRACT. This is a handout for a talk given at Bard College on Tuesday, 1 May 2007 by the author. It gives careful versions

### Multiplicity free actions of simple algebraic groups

Multiplicity free actions of simple algebraic groups D. Testerman (with M. Liebeck and G. Seitz) EPF Lausanne Edinburgh, April 2016 D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity

### Linear Algebra Massoud Malek

CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product

### Linear Algebra. Workbook

Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx

### NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 19, Time Allowed: 150 Minutes Maximum Marks: 30

NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 19, 2015 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions on the following page 1 INSTRUCTIONS

### von Neumann algebras, II 1 factors, and their subfactors V.S. Sunder (IMSc, Chennai)

von Neumann algebras, II 1 factors, and their subfactors V.S. Sunder (IMSc, Chennai) Lecture 3 at IIT Mumbai, April 24th, 2007 Finite-dimensional C -algebras: Recall: Definition: A linear functional tr

### THE EULER CHARACTERISTIC OF A LIE GROUP

THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth

### Twists and residual modular Galois representations

Twists and residual modular Galois representations Samuele Anni University of Warwick Building Bridges, Bristol 10 th July 2014 Modular curves and Modular Forms 1 Modular curves and Modular Forms 2 Residual

### 15 Dirichlet s unit theorem

18.785 Number theory I Fall 2017 Lecture #15 10/30/2017 15 Dirichlet s unit theorem Let K be a number field. The two main theorems of classical algebraic number theory are: The class group cl O K is finite.

### Math 396. An application of Gram-Schmidt to prove connectedness

Math 396. An application of Gram-Schmidt to prove connectedness 1. Motivation and background Let V be an n-dimensional vector space over R, and define GL(V ) to be the set of invertible linear maps V V

### 6 Lecture 6: More constructions with Huber rings

6 Lecture 6: More constructions with Huber rings 6.1 Introduction Recall from Definition 5.2.4 that a Huber ring is a commutative topological ring A equipped with an open subring A 0, such that the subspace

### Irreducible Representations of symmetric group S n

Irreducible Representations of symmetric group S n Yin Su 045 Good references: ulton Young tableaux with applications to representation theory and geometry ulton Harris Representation thoery a first course

### Residual modular Galois representations: images and applications

Residual modular Galois representations: images and applications Samuele Anni University of Warwick London Number Theory Seminar King s College London, 20 th May 2015 Mod l modular forms 1 Mod l modular

### Bruhat Tits buildings and representations of reductive p-adic groups

Bruhat Tits buildings and representations of reductive p-adic groups Maarten Solleveld Radboud Universiteit Nijmegen joint work with Ralf Meyer 26 November 2013 Starting point Let G be a reductive p-adic

### Representation theory and quantum mechanics tutorial Spin and the hydrogen atom

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition

### BRUHAT-TITS BUILDING OF A p-adic REDUCTIVE GROUP

Trends in Mathematics Information Center for Mathematical Sciences Volume 4, Number 1, June 2001, Pages 71 75 BRUHAT-TITS BUILDING OF A p-adic REDUCTIVE GROUP HI-JOON CHAE Abstract. A Bruhat-Tits building

### LECTURE 8: THE SECTIONAL AND RICCI CURVATURES

LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ( V ) the set of 4-tensors that is anti-symmetric with respect to

### Gelfand Pairs and Invariant Distributions

Gelfand Pairs and Invariant Distributions A. Aizenbud Massachusetts Institute of Technology http://math.mit.edu/~aizenr Examples Example (Fourier Series) Examples Example (Fourier Series) L 2 (S 1 ) =

### Convex Optimization & Parsimony of L p-balls representation

Convex Optimization & Parsimony of L p -balls representation LAAS-CNRS and Institute of Mathematics, Toulouse, France IMA, January 2016 Motivation Unit balls associated with nonnegative homogeneous polynomials

### 1 Invariant subspaces

MATH 2040 Linear Algebra II Lecture Notes by Martin Li Lecture 8 Eigenvalues, eigenvectors and invariant subspaces 1 In previous lectures we have studied linear maps T : V W from a vector space V to another

### Tangent spaces, normals and extrema

Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent

### What can be expressed via Conic Quadratic and Semidefinite Programming?

What can be expressed via Conic Quadratic and Semidefinite Programming? A. Nemirovski Faculty of Industrial Engineering and Management Technion Israel Institute of Technology Abstract Tremendous recent

### Elliptic curves and Hilbert s Tenth Problem

Elliptic curves and Hilbert s Tenth Problem Karl Rubin, UC Irvine MAA @ UC Irvine October 16, 2010 Karl Rubin Elliptic curves and Hilbert s Tenth Problem MAA, October 2010 1 / 40 Elliptic curves An elliptic

### These notes are incomplete they will be updated regularly.

These notes are incomplete they will be updated regularly. LIE GROUPS, LIE ALGEBRAS, AND REPRESENTATIONS SPRING SEMESTER 2008 RICHARD A. WENTWORTH Contents 1. Lie groups and Lie algebras 2 1.1. Definition

### Chap. 3. Controlled Systems, Controllability

Chap. 3. Controlled Systems, Controllability 1. Controllability of Linear Systems 1.1. Kalman s Criterion Consider the linear system ẋ = Ax + Bu where x R n : state vector and u R m : input vector. A :

### CSE 206A: Lattice Algorithms and Applications Spring Basis Reduction. Instructor: Daniele Micciancio

CSE 206A: Lattice Algorithms and Applications Spring 2014 Basis Reduction Instructor: Daniele Micciancio UCSD CSE No efficient algorithm is known to find the shortest vector in a lattice (in arbitrary

### Math 164-1: Optimization Instructor: Alpár R. Mészáros

Math 164-1: Optimization Instructor: Alpár R. Mészáros First Midterm, April 20, 2016 Name (use a pen): Student ID (use a pen): Signature (use a pen): Rules: Duration of the exam: 50 minutes. By writing

### Invariant Distributions and Gelfand Pairs

Invariant Distributions and Gelfand Pairs A. Aizenbud and D. Gourevitch http : //www.wisdom.weizmann.ac.il/ aizenr/ Gelfand Pairs and distributional criterion Definition A pair of groups (G H) is called

### YOUNG TABLEAUX AND THE REPRESENTATIONS OF THE SYMMETRIC GROUP

YOUNG TABLEAUX AND THE REPRESENTATIONS OF THE SYMMETRIC GROUP YUFEI ZHAO ABSTRACT We explore an intimate connection between Young tableaux and representations of the symmetric group We describe the construction

### Integration of Rational Functions by Partial Fractions

Integration of Rational Functions by Partial Fractions Part 2: Integrating Rational Functions Rational Functions Recall that a rational function is the quotient of two polynomials. x + 3 x + 2 x + 2 x

### Some notes on Coxeter groups

Some notes on Coxeter groups Brooks Roberts November 28, 2017 CONTENTS 1 Contents 1 Sources 2 2 Reflections 3 3 The orthogonal group 7 4 Finite subgroups in two dimensions 9 5 Finite subgroups in three

### (K + L)(c x) = K(c x) + L(c x) (def of K + L) = K( x) + K( y) + L( x) + L( y) (K, L are linear) = (K L)( x) + (K L)( y).

Exercise 71 We have L( x) = x 1 L( v 1 ) + x 2 L( v 2 ) + + x n L( v n ) n = x i (a 1i w 1 + a 2i w 2 + + a mi w m ) i=1 ( n ) ( n ) ( n ) = x i a 1i w 1 + x i a 2i w 2 + + x i a mi w m i=1 Therefore y

### is an isomorphism, and V = U W. Proof. Let u 1,..., u m be a basis of U, and add linearly independent

Lecture 4. G-Modules PCMI Summer 2015 Undergraduate Lectures on Flag Varieties Lecture 4. The categories of G-modules, mostly for finite groups, and a recipe for finding every irreducible G-module of a

### Algorithmic Treatment of Algebraic Modular Forms Diamant Symposium Fall 2015

Algorithmic Treatment of Algebraic Modular Forms Diamant Symposium Fall 2015 Sebastian Schönnenbeck November 27, 2015 Experimental and constructive algebra Sebastian Schönnenbeck November 27, 2015 Algorithmic

### Normal Fans of Polyhedral Convex Sets

Set-Valued Analysis manuscript No. (will be inserted by the editor) Normal Fans of Polyhedral Convex Sets Structures and Connections Shu Lu Stephen M. Robinson Received: date / Accepted: date Dedicated

### A strongly polynomial algorithm for linear systems having a binary solution

A strongly polynomial algorithm for linear systems having a binary solution Sergei Chubanov Institute of Information Systems at the University of Siegen, Germany e-mail: sergei.chubanov@uni-siegen.de 7th

### A Highly Symmetric Four-Dimensional Quasicrystal * Veit Elser and N. J. A. Sloane AT&T Bell Laboratories Murray Hill, New Jersey

A Highly Symmetric Four-Dimensional Quasicrystal * Veit Elser and N. J. A. Sloane AT&T Bell Laboratories Murray Hill, New Jersey 7974 Abstract A quasiperiodic pattern (or quasicrystal) is constructed in

### JOSEPH ALFANO* Department of Mathematics, Assumption s y i P (x; y) = 0 for all r; s 0 (with r + s > 0). Computer explorations by

A BASIS FOR THE Y SUBSPACE OF DIAGONAL HARMONIC POLYNOMIALS JOSEPH ALFANO* Department of Mathematics, Assumption College 500 Salisbury Street, Worcester, Massachusetts 065-0005 ABSTRACT. The space DH n

### Representations and Linear Actions

Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

### Fundamentals of Coding For Network Coding

Fundamentals of Coding For Network Coding Marcus Greferath and Angeles Vazquez-Castro Aalto University School of Sciences, Finland Autonomous University of Barcelona, Spain marcus.greferath@aalto.fi and

### Elliptic Nets and Points on Elliptic Curves

Department of Mathematics Brown University http://www.math.brown.edu/~stange/ Algorithmic Number Theory, Turku, Finland, 2007 Outline Geometry and Recurrence Sequences 1 Geometry and Recurrence Sequences

### Binomial Exercises A = 1 1 and 1

Lecture I. Toric ideals. Exhibit a point configuration A whose affine semigroup NA does not consist of the intersection of the lattice ZA spanned by the columns of A with the real cone generated by A.

### Packing, coding, and ground states From information theory to physics. Lecture III. Packing and energy minimization bounds in compact spaces

Packing, coding, and ground states From information theory to physics Lecture III. Packing and energy minimization bounds in compact spaces Henry Cohn Microsoft Research New England Pair correlations For

### Numerical Methods I Eigenvalue Problems

Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 October 2nd, 2014 A. Donev (Courant Institute) Lecture

### Vector Calculus. Lecture Notes

Vector Calculus Lecture Notes Adolfo J. Rumbos c Draft date November 23, 211 2 Contents 1 Motivation for the course 5 2 Euclidean Space 7 2.1 Definition of n Dimensional Euclidean Space........... 7 2.2

### Lecture 23: 6.1 Inner Products

Lecture 23: 6.1 Inner Products Wei-Ta Chu 2008/12/17 Definition An inner product on a real vector space V is a function that associates a real number u, vwith each pair of vectors u and v in V in such

### Lattices for Communication Engineers

Lattices for Communication Engineers Jean-Claude Belfiore Télécom ParisTech CNRS, LTCI UMR 5141 February, 22 2011 Nanyang Technological University - SPMS Part I Introduction Signal Space The transmission

### Weyl Group Representations and Unitarity of Spherical Representations.

Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 β ν 1 = ν 2 S α S β ν S β ν S α ν S α S β S α S β ν

### Gröbner Complexes and Tropical Bases

Gröbner Complexes and Tropical Bases Jan Verschelde University of Illinois at Chicago Department of Mathematics, Statistics, and Computer Science http://www.math.uic.edu/ jan jan@math.uic.edu Graduate

### Problems in Linear Algebra and Representation Theory

Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific

### In English, this means that if we travel on a straight line between any two points in C, then we never leave C.

Convex sets In this section, we will be introduced to some of the mathematical fundamentals of convex sets. In order to motivate some of the definitions, we will look at the closest point problem from

### 47-831: Advanced Integer Programming Lecturer: Amitabh Basu Lecture 2 Date: 03/18/2010

47-831: Advanced Integer Programming Lecturer: Amitabh Basu Lecture Date: 03/18/010 We saw in the previous lecture that a lattice Λ can have many bases. In fact, if Λ is a lattice of a subspace L with

### The Cartan Decomposition of a Complex Semisimple Lie Algebra

The Cartan Decomposition of a Complex Semisimple Lie Algebra Shawn Baland University of Colorado, Boulder November 29, 2007 Definition Let k be a field. A k-algebra is a k-vector space A equipped with

### Review (Probability & Linear Algebra)

Review (Probability & Linear Algebra) CE-725 : Statistical Pattern Recognition Sharif University of Technology Spring 2013 M. Soleymani Outline Axioms of probability theory Conditional probability, Joint

### Lattices and Codes (a brief introduction)

Lattices and Codes (a brief introduction) Ian Coley May 28, 2014 0 Preamble The goal here to is to introduce irreducible root systems, and through them try to classify certain lattices. As we ve seen,

### Singular Value Decomposition (SVD)

School of Computing National University of Singapore CS CS524 Theoretical Foundations of Multimedia More Linear Algebra Singular Value Decomposition (SVD) The highpoint of linear algebra Gilbert Strang

### Voronoi-Dickson Hypothesis on Perfect Forms and L-types

Voronoi-Dickson Hypothesis on Perfect Forms and L-types arxiv:math/0112097v1 [mathnt] 11 Dec 2001 Robert Erdahl and Konstantin Rybnikov November 19, 2001 Short Version Abstract George Voronoi (1908, 1909)

### ALGEBRA 8: Linear algebra: characteristic polynomial

ALGEBRA 8: Linear algebra: characteristic polynomial Characteristic polynomial Definition 8.1. Consider a linear operator A End V over a vector space V. Consider a vector v V such that A(v) = λv. This

### arxiv: v1 [math.sg] 6 Nov 2015

A CHIANG-TYPE LAGRANGIAN IN CP ANA CANNAS DA SILVA Abstract. We analyse a simple Chiang-type lagrangian in CP which is topologically an RP but exhibits a distinguishing behaviour under reduction by one

### COMPLEX MULTIPLICATION: LECTURE 15

COMPLEX MULTIPLICATION: LECTURE 15 Proposition 01 Let φ : E 1 E 2 be a non-constant isogeny, then #φ 1 (0) = deg s φ where deg s is the separable degree of φ Proof Silverman III 410 Exercise: i) Consider

### CHAPTER 6. Representations of compact groups

CHAPTER 6 Representations of compact groups Throughout this chapter, denotes a compact group. 6.1. Examples of compact groups A standard theorem in elementary analysis says that a subset of C m (m a positive

### Kac-Moody Algebras. Ana Ros Camacho June 28, 2010

Kac-Moody Algebras Ana Ros Camacho June 28, 2010 Abstract Talk for the seminar on Cohomology of Lie algebras, under the supervision of J-Prof. Christoph Wockel Contents 1 Motivation 1 2 Prerequisites 1

### NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 25, Time Allowed: 150 Minutes Maximum Marks: 30

NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 25, 2010 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions on the following page 1 INSTRUCTIONS

### MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.

MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.

### Alexander Ostrowski

Ostrowski p. 1/3 Alexander Ostrowski 1893 1986 Walter Gautschi wxg@cs.purdue.edu Purdue University Ostrowski p. 2/3 Collected Mathematical Papers Volume 1 Determinants Linear Algebra Algebraic Equations

### EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2

EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 2 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 5, 2012 Andre Tkacenko

### LINEAR EQUATIONS WITH UNKNOWNS FROM A MULTIPLICATIVE GROUP IN A FUNCTION FIELD. To Professor Wolfgang Schmidt on his 75th birthday

LINEAR EQUATIONS WITH UNKNOWNS FROM A MULTIPLICATIVE GROUP IN A FUNCTION FIELD JAN-HENDRIK EVERTSE AND UMBERTO ZANNIER To Professor Wolfgang Schmidt on his 75th birthday 1. Introduction Let K be a field

### A tropical approach to secant dimensions

A tropical approach to secant dimensions Jan Draisma Madrid, 30 March 2007 1/16 Typical example: polynomial interpolation in two variables Set up: d N p 1,..., p k general points in C 2 codim{f C[x, y]

### CHARACTERISTIC CLASSES

1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact

### Reduction of Smith Normal Form Transformation Matrices

Reduction of Smith Normal Form Transformation Matrices G. Jäger, Kiel Abstract Smith normal form computations are important in group theory, module theory and number theory. We consider the transformation

### AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY

1 AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY Andrew Ranicki and Daniele Sepe (Edinburgh) http://www.maths.ed.ac.uk/ aar Maslov index seminar, 9 November 2009 The 1-dimensional Lagrangians

### arxiv: v1 [math.ag] 17 Apr 2015

FREE RESOLUTIONS OF SOME SCHUBERT SINGULARITIES. MANOJ KUMMINI, V. LAKSHMIBAI, PRAMATHANATH SASTRY, AND C. S. SESHADRI arxiv:1504.04415v1 [math.ag] 17 Apr 2015 Abstract. In this paper we construct free

### Objective Mathematics

Multiple choice questions with ONE correct answer : ( Questions No. 1-5 ) 1. If the equation x n = (x + ) is having exactly three distinct real solutions, then exhaustive set of values of 'n' is given

### Test Codes : MIA (Objective Type) and MIB (Short Answer Type) 2007

Test Codes : MIA (Objective Type) and MIB (Short Answer Type) 007 Questions will be set on the following and related topics. Algebra: Sets, operations on sets. Prime numbers, factorisation of integers

### Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Dennis S. Bernstein

Matrix Mathematics Theory, Facts, and Formulas with Application to Linear Systems Theory Dennis S. Bernstein PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Contents Special Symbols xv Conventions, Notation,

### Math Linear Algebra II. 1. Inner Products and Norms

Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,

### Convex Functions. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST)

Convex Functions Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline of Lecture Definition convex function Examples

### Grassmann Coordinates

Grassmann Coordinates and tableaux Matthew Junge Autumn 2012 Goals 1 Describe the classical embedding G(k, n) P N. 2 Characterize the image of the embedding quadratic relations. vanishing polynomials.