In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
,
E_{8} is any of several closely related exceptional simple Lie groups, linear
algebraic groupIn algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s or Lie algebras of
dimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E
_{8} comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A
_{n}, B
_{n}, C
_{n}, D
_{n}, and five exceptional cases labeled
E_{6},
E_{7}, E
_{8},
F_{4}, and
G_{2}. The E
_{8} algebra is the largest and most complicated of these exceptional cases.
discovered the complex Lie algebra E
_{8} during his classification of simple compact Lie algebras, though he did not prove its existence, which was first shown by
Élie CartanÉlie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
. Cartan determined that a complex simple Lie algebra of type E
_{8} admits three real forms. Each of them gives rise to a simple
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
of dimension 248, exactly one of which is compact. introduced
algebraic groupIn algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s and Lie algebras of type E
_{8} over other
fieldsIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
: for example, in the case of
finite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s they lead to an infinite family of finite simple groups of Lie type.
Basic description
The Lie group E
_{8} has dimension 248. Its
rankIn mathematics, a Cartan subgroup of a Lie group or algebraic group G is one of the subgroups whose Lie algebrais a Cartan subalgebra. The dimension of a Cartan subgroup, and therefore of a Cartan subalgebra, is the rank of G.Conventions:...
, which is the dimension of its maximal torus, is 8. Therefore the vectors of the root system are in eightdimensional Euclidean space: they are described explicitly later in this article. The
Weyl groupIn mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
of E
_{8}, which is the
group of symmetriesThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
of the maximal torus which are induced by
conjugationsIn mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of nonabelian groups reveals many important features of their structure...
in the whole group, has order 696729600.
The compact group E
_{8} is unique among simple compact Lie groups in that its non
trivialIn mathematics, the adjective trivial is frequently used for objects that have a very simple structure...
representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra E
_{8} itself; it is also the unique one which has the following four properties: trivial center, compact, simply connected, and simply laced (all roots have the
same length).
There is a Lie algebra
E_{n}In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1,2, and k, with k=n4....
for every integer
n ≥ 3, which is infinite dimensional if
n is greater than 8.
Real and complex forms
There is a unique complex Lie algebra of type E
_{8}, corresponding to a complex group of complex dimension 248.
The complex Lie group E
_{8} of
complex dimensionIn mathematics, complex dimension usually refers to the dimension of a complex manifold M, or complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d...
248 can be considered as a simple real Lie group of real dimension 496. This is simply connected, has maximal
compactIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
subgroup the compact form (see below) of E
_{8}, and has an outer automorphism group of order 2 generated by complex conjugation.
As well as the complex Lie group of type E
_{8}, there are three real forms of the Lie algebra, three real forms of the group with trivial center (two of which have nonalgebraic double covers, giving two further real forms), all of real dimension 248, as follows:
 The compact form (which is usually the one meant if no other information is given), which is simply connected and has trivial outer automorphism group.
 The split form, sometimes known as EVIII or E_{8(8)}, which has maximal compact subgroup Spin(16)/(Z/2Z), fundamental group of order 2 (implying that it has a double cover, which is a simply connected Lie real group but is not algebraic, see below) and has trivial outer automorphism group.
 A third form, sometimes known as EIX or E_{8(24)}, which has maximal compact subgroup E_{7}×SU(2)/(−1×−1), fundamental group of order 2 (again implying a double cover, which is not algebraic) and has trivial outer automorphism group.
For a complete list of real forms of simple Lie algebras, see the
list of simple Lie groups.
E_{8} as an algebraic group
By means of a
Chevalley basisIn mathematics, a Chevalley basis for a simple complex Lie algebra isa basis constructed by Claude Chevalley with the property that all structure constants are integers...
for the Lie algebra, one can define E
_{8} as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the socalled split (sometimes also known as “untwisted”) form of E
_{8}. Over an algebraically closed field, this is the only form; however, over other fields, there are often many other forms, or “twists” of E
_{8}, which are classified in the general framework of
Galois cohomologyIn mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...
(over a
perfect fieldIn algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
k) by the set
which, because the Dynkin diagram of E
_{8} (see below) has no automorphisms, coincides with
.
Over the field of real numbers, the real connected component of the identity of these algebraically twisted forms of E
_{8} coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all forms of E
_{8} are simply connected in the sense of algebraic geometry, meaning that they admit no nontrivial algebraic coverings; the noncompact and simply connected real Lie group forms of E
_{8} are therefore not algebraic and admit no faithful finitedimensional representations.
Over finite fields, the Lang–Steinberg theorem implies that
, meaning that E
_{8} has no twisted forms: see below.
Representation theory
The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the
Weyl character formulaIn mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
. The dimensions of the smallest irreducible representations are :
 1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250, 4825673125, 6899079264, 8634368000 (twice), 12692520960…
The 248dimensional representation is the
adjoint representationIn mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
. There are two nonisomorphic irreducible representations of dimension 8634368000 (it is not unique; however, the next integer with this property is 175898504162692612600853299200000 ). The
fundamental representationIn representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finitedimensional representation of a semisimple Lie group...
s are those with dimensions 3875, 6696000, 6899079264, 146325270, 2450240, 30380, 248 and 147250 (corresponding to the eight nodes in the Dynkin diagram in the order chosen for the Cartan matrix below, i.e., the nodes are read in the sevennode chain first, with the last node being connected to the third).
The coefficients of the character formulas for infinite dimensional irreducible
representationRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
s of E
_{8} depend on some large square matrices consisting of polynomials, the Lusztig–Vogan polynomials, an analogue of
Kazhdan–Lusztig polynomialIn representation theory, a Kazhdan–Lusztig polynomial Py,w is a member of a family of integral polynomials introduced by . They are indexed by pairs of elements y, w of a Coxeter group W, which can in particular be the Weyl group of a Lie group. Motivation and history:In the spring of 1978...
s introduced for
reductive groupIn mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...
s in general by
George Lusztig and
David KazhdanDavid Kazhdan or Každan, Kazhdan, formerly named Dmitry Aleksandrovich Kazhdan , is a Soviet and Israeli mathematician known for work in representation theory.Life:...
(1983).
The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations (whose characters are easy to describe) with the irreducible representations.
These matrices were computed after four years of collaboration by a
group of 18 mathematicians and computer scientistsThe Atlas of Lie Groups and Representations is a mathematical project to solve the problem of the unitary dual for real reductive Lie groups., the following mathematicians are listed as members:*Jeffrey Adams*Dan Barbasch*Birne Binegar*Bill Casselman...
, led by
Jeffrey AdamsJeffrey Adams is a mathematician at the University of Maryland who works on unitary representations of reductive Lie groups, and who led the project Atlas of Lie groups and representations that calculated the characters of the representations of E8. The project to calculate the representations of...
, with much of the programming done by
Fokko du ClouxFokko du Cloux was a Dutch mathematician and computer scientist. He worked on the Atlas of Lie groups and representations until his death.Career in mathematics:...
. The most difficult case (for exceptional groups) is the split real form of E
_{8} (see above), where the largest matrix is of size 453060×453060. The Lusztig–Vogan polynomials for all other exceptional simple groups have been known for some time; the calculation for the split form of
E_{8} is far longer than any other case. The announcement of the result in March 2007 received extraordinary attention from the media (see the external links), to the surprise of the mathematicians working on it.
The representations of the E
_{8} groups over finite fields are given by
Deligne–Lusztig theoryIn mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓadic cohomology with compact support, introduced by ....
.
Constructions
One can construct the (compact form of the) E
_{8} group as the automorphism group of the corresponding
e_{8} Lie algebra. This algebra has a 120dimensional subalgebra
so(16) generated by
J_{ij} as well as 128 new generators
Q_{a} that transform as a Weyl–Majorana spinor of
spin(16). These statements determine the commutators
as well as
while the remaining commutator (not anticommutator!) is defined as
It is then possible to check that the
Jacobi identityIn mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...
is satisfied.
Geometry
The compact real form of E
_{8} is the
isometry groupIn mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...
of a 128dimensional
Riemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
known informally as the 'octooctonionic projective plane' because it can be built using an algebra that is the tensor product of the
octonionIn mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
s with themselves. This can be seen systematically using a construction known as the
magic squareIn mathematics, the Freudenthal magic square is a construction relating several Lie groups. It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie group to a pair of division algebras A, B...
, due to
Hans FreudenthalHans Freudenthal was a Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education....
and
Jacques TitsJacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group. Career :Tits received his doctorate in mathematics at the age of 20...
.
E_{8} root system
A
root systemIn mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
of rank
r is a particular finite configuration of vectors, called
roots, which span an
rdimensional
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
and satisfy certain geometrical properties. In particular, the root system must be invariant under
reflectionIn mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
through the hyperplane perpendicular to any root.
The
E_{8} root system is a rank 8 root system containing 240 root vectors spanning
R^{8}. It is irreducible in the sense that it cannot be built from root systems of smaller rank. All the root vectors in E
_{8} have the same length. It is convenient for many purposes to normalize them to have length √2.
Construction
In the socalled
even coordinate system E
_{8} is given as the set of all vectors in
R^{8} with length squared equal to 2 such that coordinates are either all
integerThe integers are formed by the natural numbers together with the negatives of the nonzero natural numbers .They are known as Positive and Negative Integers respectively...
s or all
halfintegerIn mathematics, a halfinteger is a number of the formn + 1/2,where n is an integer. For example,are all halfintegers. Note that a half of an integer is not always a halfinteger: half of an even integer is an integer but not a halfinteger...
s and the sum of the coordinates is even.
Explicitly, there are 112 roots with integer entries obtained from
by taking an arbitrary combination of signs and an arbitrary
permutationIn mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
of coordinates, and 128 roots with halfinteger entries obtained from
by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even). There are 240 roots in all.
The 112 roots with integer entries form a D
_{8} root system. The E
_{8} root system also contains a copy of A
_{8} (which has 72 roots) as well as
E_{6} and
E_{7} (in fact, the latter two are usually
defined as subsets of E
_{8}).
In the
odd coordinate system E
_{8} is given by taking the roots in the even coordinate system and changing the sign of any one coordinate. The roots with integer entries are the same while those with halfinteger entries have an odd number of minus signs rather than an even number.
Simple roots
A set of
simple rootin mathematics the term simple root can refer to one of two unrelated notions:*A simple root of a polynomial is a root of multiplicity one*A simple root in a root system is a member of a subset determined by a choice of positive roots...
s for a root system Φ is a set of roots that form a
basisIn linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
for the Euclidean space spanned by Φ with the special property that each root has components with respect to this basis that are either all nonnegative or all nonpositive.
One choice of simple roots for E
_{8} is given by the rows of the following matrix:
The set of simple roots is by no means unique (the number of possible choices of positive roots is the order of the Weyl group); however, the particular choice displayed above has the unique property that the positive roots are then exactly those whose rightmost nonzero coordinate is positive.
Dynkin diagram
The Dynkin diagram for E
_{8} is given by
This diagram gives a concise visual summary of the root structure. Each node of this diagram represents a simple root. A line joining two simple roots indicates that they are at an angle of 120° to each other. Two simple roots which are not joined by a line are orthogonal.
Weyl group
The
Weyl groupIn mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
of E
_{8} is of order 696729600, and can be described as O
_{8}^{+}(2): it is of the form 2.
G.2 (that is, a
stem extensionIn mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.Examples and properties:...
by the cyclic group of order 2 of an extension of the cyclic group of order 2 by a group
G) where
G is the unique
simple groupIn mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
of order 174182400 (which can be described as PSΩ
_{8}^{+}(2)).
Cartan matrix
The
Cartan matrixIn mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. In fact, Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan. Lie algebras :A generalized...
of a rank
r root system is an
r ×
r matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
whose entries are derived from the simple roots. Specifically, the entries of the Cartan matrix are given by
where (−,−) is the Euclidean inner product and
α_{i} are the simple roots. The entries are independent of the choice of simple roots (up to ordering).
The Cartan matrix for E
_{8} is given by
The
determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of this matrix is equal to 1.
E_{8} root lattice
The integral span of the E
_{8} root system forms a
latticeIn mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
in
R^{8} naturally called the
E_{8} root latticeIn mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positivedefinite, even, unimodular lattice of rank 8...
. This lattice is rather remarkable in that it is the only (nontrivial) even,
unimodular latticeIn mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples. Definitions :...
with rank less than 16.
Simple subalgebras of E_{8}
The Lie algebra E8 contains as subalgebras all the exceptional Lie algebras as well as many other important Lie algebras in mathematics and physics. The height of the Lie algebra on the diagram approximately corresponds to the rank of the algebra. A line from an algebra down to a lower algebra indicates that the lower algebra is a subalgebra of the higher algebra.
Chevalley groups of type E_{8}
showed that the points of the (split) algebraic group E
_{8} (see above) over a
finite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
with
q elements form a finite
Chevalley groupIn mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
, generally written E
_{8}(
q), which is simple for any
q, and constitutes one of the infinite families addressed by the
classification of finite simple groupsIn mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
. Its number of elements is given by the formula :
The first term in this sequence, the order of E
_{8}(2), namely 337804753143634806261388190614085595079991692242467651576160959909068800000 ≈ 3.38×10
^{74}, is already larger than the size of the
Monster groupIn the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
. This group E
_{8}(2) is the last one described (but without its character table) in the
ATLAS of Finite GroupsThe ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway, Robert Turner Curtis, Simon Phillips Norton, Richard Alan Parker and Robert Arnott Wilson , published in December 1985 by Oxford University Press and reprinted with corrections in 2003...
.
The
Schur multiplierIn mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.Examples and properties:...
of E
_{8}(
q) is trivial, and its outer automorphism group is that of field automorphisms (i.e., cyclic of order
f if
q=
p^{f} where
p is prime).
described the unipotent representations of finite groups of type
E_{8}.
Subgroups
The smaller exceptional groups
E_{7} and
E_{6} sit inside E
_{8}. In the compact group, both (E
_{7}×SU(2)) / (
Z/2
Z) and (E
_{6}×SU(3)) / (
Z/3
Z) are
maximal subgroupIn mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words H is a maximal element of the partially...
s of
E
_{8}.
The 248dimensional adjoint representation of E
_{8} may be considered in terms of its
restricted representationIn mathematics, restriction is a fundamental construction in representation theory of groups. Restriction forms a representation of a subgroup from a representation of the whole group. Often the restricted representation is simpler to understand...
to the first of these subgroups. It transforms under SU(2)×E
_{7} as a sum of tensor product representations, which may be labelled as a pair of dimensions as
(Since there is a quotient in the product, these notations may strictly be taken as indicating the infinitesimal (Lie algebra) representations.)
Since the adjoint representation can be described by the roots together with the generators in the
Cartan subalgebra, we may see that decomposition by looking at these. In this description:
 The (3,1) consists of the roots (0,0,0,0,0,0,1,−1), (0,0,0,0,0,0,−1,1) and the Cartan generator corresponding to the last dimension.
 The (1,133) consists of all roots with (1,1), (−1,−1), (0,0), (−1/2,−1/2) or (1/2,1/2) in the last two dimensions, together with the Cartan generators corresponding to the first 7 dimensions.
 The (2,56) consists of all roots with permutations of (1,0), (−1,0) or (1/2,−1/2) in the last two dimensions.
The 248dimensional adjoint representation of E
_{8}, when similarly restricted, transforms under SU(3)×E
_{6} as:
We may again see the decomposition by looking at the roots together with the generators in the Cartan subalgebra. In this description:
 The (8,1) consists of the roots with permutations of (1,−1,0) in the last three dimensions, together with the Cartan generator corresponding to the last two dimensions.
 The (1,78) consists of all roots with (0,0,0), (−1/2,−1/2,−1/2) or (1/2,1/2,1/2) in the last three dimensions, together with the Cartan generators corresponding to the first 6 dimensions.
 The (3,27) consists of all roots with permutations of (1,0,0), (1,1,0) or (−1/2,1/2,1/2) in the last three dimensions.
 The (3,27) consists of all roots with permutations of (−1,0,0), (−1,−1,0) or (1/2,−1/2,−1/2) in the last three dimensions.
The finite quasisimple groups that can embed in (the compact form of) E
_{8} were found by .
The
Dempwolff groupIn mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension 25·GL5 of GL5 by its natural module of order 25...
is a subgroup of (the compact form of) E
_{8}. It is contained in the Thompson sporadic group, which acts on the underlying vector space of the Lie group E
_{8} but does not preserve the Lie bracket. The Thompson group fixes a lattice and does preserve the Lie bracket of this lattice mod 3, giving an embedding of the Thompson group into E
_{8}(
F_{3}).
Invariant polynomial
E8 is the automorphism group of an octic polynomial invariant, thought to be the lowest order symmetric polynomial invariant of E8.
Applications
The E
_{8} Lie group has applications in
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, in particular in
string theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
and
supergravityIn theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry...
. The group E
_{8}×E
_{8} (the
Cartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of two copies of E
_{8}) serves as the gauge group of one of the two types of
heterotic stringIn physics, a heterotic string is a peculiar mixture of the bosonic string and the superstring...
and is one of two
anomalyfreeIn quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics an anomaly is the failure of a symmetry to be restored in the limit in which the symmetrybreaking...
gauge groups that can be coupled to the
N = 1 supergravity in 10 dimensions.
E
_{8} is the
UdualityIn physics, Uduality is a symmetry of string theory or Mtheory combining Sduality and Tduality transformations. The term is most often met in the context of the "Uduality group" of Mtheory as defined on a particular background space . This is the union of all the S and Tdualities...
group of supergravity on an eighttorus (in its split form).
One way to incorporate the
standard modelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
of particle physics into heterotic string theory is the
symmetry breakingSpontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
of E
_{8} to its maximal subalgebra SU(3)×E
_{6}.
In 1982,
Michael FreedmanMichael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
used the
E_{8} latticeIn mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positivedefinite, even, unimodular lattice of rank 8...
to construct an example of a
topologicalIn mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
4manifoldIn mathematics, 4manifold is a 4dimensional topological manifold. A smooth 4manifold is a 4manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
, the
E_{8} manifoldIn mathematics, the E8 manifold is the unique compact, simply connected topological 4manifold with intersection form the E8 lattice.The E8 manifold was discovered by Michael Freedman in 1982...
, which has no
smooth structureIn mathematics, an ndimensional differential structure on a set M makes M into an ndimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold...
.
reported that in an experiment with a
cobaltCobalt is a chemical element with symbol Co and atomic number 27. It is found naturally only in chemically combined form. The free element, produced by reductive smelting, is a hard, lustrous, silvergray metal....

niobiumNiobium or columbium , is a chemical element with the symbol Nb and atomic number 41. It's a soft, grey, ductile transition metal, which is often found in the pyrochlore mineral, the main commercial source for niobium, and columbite...
crystal, under certain physical conditions the electron spins in it exhibited two of the 8 peaks related to E
_{8} predicted by .
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Related to the calculation of the Lusztig–Vogan polynomials:
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