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How made the headlines Max Neunhöffer How E8 made the headlines

Introduction Lie groups

What does E8 mean? Real and complex Max Neunhöffer Representations

What they did The Atlas of Lie Groups Classification of Representations Compact Lie groups Non-compact Lie groups 8 The E8 -computation What the press made of it How to explain this? How E8 made the headlines A news story Max Neunhöffer On 20 March 2007 wrote:

Introduction Lie groups The Scientific Promise of Perfect Symmetry What does E8 mean? Real and complex Representations It is one of the most symmetrical mathematical structures What they did in the universe. It may underlie the Theory of Everything The Atlas of Lie Groups Classification of that physicists seek to describe the universe. Representations Compact Lie groups Non-compact Lie groups The E 8 -computation Other quotes: 8 What the press made of it 248-dimension maths puzzle solved How to explain this? The “structure” is described in the form of a vast matrix. Mapping the 248-dimensional structure, called E8, took four years of work and produced more data than the Human Genome Project, researchers said.

Classification The unitary representations of the split real form of the simple complex E8 have been classified. How E8 made the headlines What is a Lie group? Max Neunhöffer Let K be either R or C. Introduction Lie groups A Lie group over is both What does E8 mean? K Real and complex Representations a smooth K-manifold, and What they did The Atlas of Lie Groups a group with smooth multiplication and inversion. Classification of Representations Compact Lie groups (imagine: a closed subgroup of some GL ( )) Non-compact Lie groups n K 8 The E8 -computation Examples: What the press made of it GL ( ) := {A ∈ n×n | det A 6= 0} “general linear” How to explain this? n C C n×n GLn(R) := {A ∈ R | det A 6= 0} “general linear” n×n SLn(K) := {A ∈ K | det A = 1} “special linear” T Un := {A ∈ GLn(C) | A · A = 1} “unitary” SUn := Un ∩ SLn(C) “special unitary”  T On := A ∈ GLn(R) | A · A = 1 “orthogonal” SOn := On ∩ SLn(C) “special orthogonal” How E8 made the headlines The easiest example Max Neunhöffer Let’s look a bit closer at

Introduction    cos(α) − sin(α) Lie groups SO2 = 0 ≤ α < 2π < GL2(R). What does E8 mean? sin(α) cos(α) Real and complex Representations It is S1: Identify points on the sphere with rotations of the What they did The Atlas of Lie Groups plane around the origin. Classification of Representations Compact Lie groups Non-compact Lie groups 8 The E8 -computation What the press made of it How to explain this?

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We see that it is smooth and compact (closed+bounded)! How E8 made the headlines What does E8 mean? Max Neunhöffer

Introduction Lie groups Theorem (Lie correspondence) What does E8 mean? Real and complex Representations A connected Lie group is (up to a discrete, central What they did extension) determined by its , which is the The Atlas of Lie Groups Classification of tangent space at the identity. Representations Compact Lie groups Non-compact Lie groups 8 The E8 -computation Simple complex Lie algebras are classified by the Dynkin What the press made of it type An, Bn, Cn, Dn, , , E8, or (Killing, 1887). How to explain this? Classification This leads to a classification of the connected simple complex Lie groups.

E8 is simply the largest of the 5 exceptional cases. How E8 made the headlines Real and complex Max Neunhöffer Example: The real and complex worlds are related! Introduction Lie groups Look at G := SL ( ) and two automorphisms of order 2: What does E mean? n C 8 Real and complex Representations −T What they did Automorphisms: c : A 7→ A t : A 7→ A The Atlas of Lie Groups Classification of c t Representations Invariants: G = SLn(R) G = SUn Compact Lie groups Non-compact Lie groups Topology: not compact compact 8 The E8 -computation What the press Name: “split” compact made of it How to explain this? Def: SLn(R) and SU(n) are real forms of SLn(C). SLn(C) is the complexification of both.

Theorem (Cartan, 1890) A connected, semisimple, complex Lie group has only finitely many real forms. There is always a split one which is not compact. How E8 made the 8 headlines The split real form E8 of E8 ... Max Neunhöffer ... is a 248-dimensional real non-compact Lie group. Introduction 8 Lie groups E8 is combinatorially described by its in R :

What does E8 mean? Real and complex 1 8 P Representations Φ := {a ∈ {± } | ai even} 2 ⊆ R8 What they did ∪ {a ∈ {0, ±1}8 | P(a )2 = 2} The Atlas of Lie Groups i Classification of Representations  8 1 8 P hΦi is the E8-lattice: a ∈ ∪ ( + ) | ai even Compact Lie groups Z Z Z 2 Non-compact Lie groups 8 The E8 -computation What the press made of it How to explain this?

8 E8 is a group of symmetries of a 57-dimensional variety. How E8 made the headlines Representations Max Neunhöffer Definition (Representation) Introduction Lie groups A representation of a Lie group G is a continuous group What does E8 mean? Real and complex Representations homomorphism D : G → GL(V ),(V a C-vector space). What they did irreducible :⇐⇒ only {0} and V are D(G)-invariant The Atlas of Lie Groups Classification of closed subspaces Representations Compact Lie groups Non-compact Lie groups unitary :⇐⇒ D(G) ⊆ U(V ) < GL(V ) 8 The E8 -computation (for this V must be a Hilbert space) What the press made of it How to explain this? Fact An irreducible representation D : G → GLn(C) is determined up to equivalence by its character:

χD : G → C, g 7→ Tr(D(g)). So characters are functions on G which are constant on conjugacy classes.

Def: Irr(G) := {characters of irreducible representations} How E8 made the headlines The Atlas project Max Neunhöffer

Introduction Aims Lie groups

What does E8 mean? Real and complex Collect and make available information on semisimple Representations Lie groups and their (unitary) representations. What they did The Atlas of Lie Groups Classification of Representations http://www.liegroups.org/ Compact Lie groups Non-compact Lie groups 8 The E8 -computation What the press People: The core group made of it How to explain this? Jeffrey Adams (University of Maryland), Dan Barbasch (Cornell), John Stembridge (University of Michigan), Peter Trapa (University of Utah), Marc van Leeuwen (Poitiers), David Vogan (MIT), and Fokko du Cloux (Lyons, until his death in 2006). How E8 made the headlines Questions about character tables Max Neunhöffer

Introduction Lie groups Fundamental questions What does E8 mean? Real and complex 1 Representations How can a character be described in a finite way, if What they did G has infinitely many conjugacy classes? The Atlas of Lie Groups Classification of 2 Representations How can a finite computation or data collection Compact Lie groups Non-compact Lie groups describe infinitely many, possibly infinite-dimensional 8 The E8 -computation irreducible unitary representations? What the press made of it How to explain this? Good news about compact Lie groups: Every finite-dimensional, irreducible representation is equivalent to a unitary one. Every unitary representation is a direct sum of finite-dimensional irreducible ones (Peter and Weyl). How E8 made the headlines Maximal Tori Max Neunhöffer Definition () Introduction Lie groups 1 What does E8 mean? Let S := {z ∈ C | |z| = 1}.A torus is a direct product Real and complex Representations := S1 × S1 × · · · × S1 What they did T The Atlas of Lie Groups | {z } Classification of r factors Representations Compact Lie groups This is a compact, abelian Lie group, Non-compact Lie groups 8 The E8 -computation  Qr ai r with set of characters χa(s) = s | a ∈ . What the press i=1 i Z made of it How to explain this? Theorem (Maximal Tori) Let G be a connected, compact Lie group. Then G has a T The normaliser NG(T) is a closed subgroup and NG(T)/T is a finite group, the W. Every of G meets T. x, y ∈ T conjugate in G ⇐⇒ x, y conjugate in NG(T) How E8 made the headlines Character tables for compact Lie groups Max Neunhöffer Theorem () Introduction Lie groups G: connected, compact Lie group, T: a maximal torus. What does E8 mean? Real and complex 1−1 n o Representations Irr(G) ←→ “highest weights” What they did | {z } The Atlas of Lie Groups come from the root system Classification of Representations Compact Lie groups Each character is given explicitly as a quotient of two Non-compact Lie groups 8 The E8 -computation finite integral linear combinations of characters of T. What the press    made of it z −w 2×2 2 2 How to explain this? G = SU = ∈ |z| + |w| = 1 2 w z C  z 0   = S1 = ∈ 2×2 |z|2 = 1 T 0 z C The characters of T are indexed by Z, the highest weights are N ∪ {0}, the character χn on T is zn+1 − z−(n+1) χ (z) = . n z − z−1 How E8 made the headlines Trouble ahead Max Neunhöffer

Introduction There are irreducible representations which are not Lie groups equivalent to unitary ones. What does E8 mean? Real and complex Representations There are infinite-dimensional, unitary, irreducible What they did representations. The Atlas of Lie Groups Classification of Representations The operators D(g) ∈ GL(V ) do not necessarily have Compact Lie groups Non-compact Lie groups a trace, if dim(V ) = ∞! 8 The E8 -computation Characters are now distributions (Harish-Chandra). What the press made of it How to explain this? Theorem (Langlands, Knapp-Zuckerman) Let G be a reductive, algebraic Lie group. Then the irreducible characters are classified.

Big questions: Which of them are unitary and what are their dimensions and character values? How E8 made the headlines “Character tables” of non-compact Lie groups Max Neunhöffer

Introduction Theorem (Jantzen-Zuckerman “translation principle”) Lie groups . What does E8 mean? [ N Real and complex Irr(G) = i=1 Fi (“translation families”) Representations What they did Each F is parametrised by the set IChar(G) of The Atlas of Lie Groups i Classification of λ Representations infinitesimal characters: Fi = {χi | λ ∈ IChar(G)}. Compact Lie groups Non-compact Lie groups 8 The E8 -computation N λ X λ What the press χi = Pi,j (1) · Θj for 1 ≤ i ≤ N made of it How to explain this? j=1 λ λ for certain functions Θ1,..., ΘN (Harish-Chandra). −1 The polynomials Pi,j ∈ Z[q, q ] are the famous Kazhdan-Lusztig-Vogan-polynomials.

The Pi,j do not depend on λ, and Pi,j = 0 for i < j. → Very deep results about intersection homology by Kazhdan, Lusztig, Beilinson, Bernstein, Vogan, ... How E8 made the 8 headlines The E8 -computation Max Neunhöffer

Introduction Facts: Lie groups 8 What does E8 mean? For E8 there are 453060 translation families. Real and complex 2 Representations =⇒ 453060 /2 polynomials, degree up to 31. What they did The Atlas of Lie Groups They can be computed by the Classification of Representations Kazhdan-Lusztig-Algorithm. Compact Lie groups Non-compact Lie groups No bound for the integer coefficients was known! 8 The E8 -computation What the press made of it How to explain this? Size of the computation: At first sight, 480 GB of RAM seemed to be needed! This was reduced to about 150 GB by various tricks. Finally, computing modulo 251, 253, 255 and 256 separately did the job. Altogether, the computation now needs about 50 hours and produces 60 GB of output. How E8 made the headlines How to explain this to the public? Max Neunhöffer Recipe: Introduction Lie groups The idea was to promote mathematics in public.

What does E8 mean? Real and complex Original plan: reach science and technology media, Representations

What they did like “Science” The Atlas of Lie Groups Classification of Press release by AIM and researchers involved Representations Compact Lie groups Professional public relations help was hired. Non-compact Lie groups 8 The E8 -computation It was a lot of work, with long preparations. What the press made of it How to explain this? Ingredients: Needed: “hooks” to be understood by the public: Symmetry, size, team effort, implications for research Critically important: a good accompanying web page Necessary: a good picture Good: associate news release with an event Extremely valuable: quotes from external experts How E8 made the headlines Lesson learned: Max Neunhöffer

Introduction Lie groups

What does E8 mean? Real and complex Representations

What they did The Atlas of Lie Groups Classification of With enough effort, it can be done! Representations Compact Lie groups Non-compact Lie groups 8 The E8 -computation What the press made of it How to explain this? Thanks!