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| {{DISPLAYTITLE:E<sub>8</sub> (mathematics)}}
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| {{Group theory sidebar |Topological}}
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| {{Lie groups |Simple}}
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| In [[mathematics]], '''E<sub>8</sub>''' is any of several closely related [[exceptional simple Lie group]]s, linear [[algebraic group]]s or Lie algebras of [[dimension]] 248; the same notation is used for the corresponding [[root lattice]], which has [[Rank of a Lie group|rank]] 8. The designation E<sub>8</sub> comes from the [[Cartan–Killing classification]] of the complex [[simple Lie algebra]]s, which fall into four infinite series labeled A<sub>''n''</sub>, B<sub>''n''</sub>, C<sub>''n''</sub>, D<sub>''n''</sub>, and [[Exceptional simple Lie group|five exceptional cases]] labeled [[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], E<sub>8</sub>, [[F4 (mathematics)|F<sub>4</sub>]], and [[G2 (mathematics)|G<sub>2</sub>]]. The E<sub>8</sub> algebra is the largest and most complicated of these exceptional cases.
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| {{harvs|txt |authorlink=Wilhelm Killing |first=Wilhelm |last=Killing |year1=1888a |year2=1888b |year3=1889 |year4=1890}} discovered the complex Lie algebra E<sub>8</sub> during his classification of simple compact Lie algebras, though he did not prove its existence, which was first shown by [[Élie Cartan]]. Cartan determined that a complex simple Lie algebra of type E<sub>8</sub> admits three real forms. Each of them gives rise to a simple [[Lie group]] of dimension 248, exactly one of which is [[compact Lie group|compact]]. {{harvtxt|Chevalley|1955}} introduced [[algebraic group]]s and Lie algebras of type E<sub>8</sub> over other [[field (mathematics)|fields]]: for example, in the case of [[finite field]]s they lead to an infinite family of [[finite simple group]]s of Lie type.
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| == Basic description ==
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| The Lie group E<sub>8</sub> has dimension 248. Its [[Cartan subgroup|rank]], which is the dimension of its maximal torus, is 8. Therefore the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The [[Weyl group]] of E<sub>8</sub>, which is the [[symmetry group|group of symmetries]] of the maximal torus which are induced by [[conjugacy class|conjugations]] in the whole group, has order 2<sup>14</sup> 3 <sup>5</sup> 5 <sup>2</sup> 7 = 696729600.
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| The compact group E<sub>8</sub> is unique among simple compact Lie groups in that its non-[[trivial (mathematics)|trivial]] representation of smallest dimension is the [[Adjoint representation of a Lie algebra|adjoint representation]] (of dimension 248) acting on the Lie algebra E<sub>8</sub> 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).
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| There is a Lie algebra [[En (Lie algebra)|E<sub>''n''</sub>]] for every integer ''n'' ≥ 3, which is infinite dimensional if ''n'' is greater than 8.
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| ==Real and complex forms==
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| There is a unique complex Lie algebra of type E<sub>8</sub>, corresponding to a complex group of complex dimension 248. The complex Lie group E<sub>8</sub> of [[complex dimension]] 248 can be considered as a simple real Lie group of real dimension 496. This is simply connected, has maximal [[Compact space|compact]] subgroup the compact form (see below) of E<sub>8</sub>, and has an outer automorphism group of order 2 generated by complex conjugation.
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| As well as the complex Lie group of type E<sub>8</sub>, there are three real forms of the Lie algebra, three real forms of the group with trivial center (two of which have non-algebraic double covers, giving two further real forms), all of real dimension 248, as follows:
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| *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.
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| *The split form, EVIII (or E<sub>8(8)</sub>), which has maximal compact subgroup Spin(16)/('''Z'''/2'''Z'''), fundamental group of order 2 (implying that it has a [[Double covering group|double cover]], which is a simply connected Lie real group but is not algebraic, see [[#E8 as an algebraic group|below]]) and has trivial outer automorphism group.
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| *EIX (or E<sub>8(-24)</sub>), which has maximal compact subgroup E<sub>7</sub>×SU(2)/(−1,−1), fundamental group of order 2 (again implying a double cover, which is not algebraic) and has trivial outer automorphism group.
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| For a complete list of real forms of simple Lie algebras, see the [[list of simple Lie groups]].
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| ==E<sub>8</sub> as an algebraic group==
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| By means of a [[Chevalley basis]] for the Lie algebra, one can define E<sub>8</sub> as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as “untwisted”) form of E<sub>8</sub>. Over an algebraically closed field, this is the only form; however, over other fields, there are often many other forms, or “twists” of E<sub>8</sub>, which are classified in the general framework of [[Galois cohomology]] (over a [[perfect field]] ''k'') by the set H<sup>1</sup>(''k'',Aut(E<sub>8</sub>)) which, because the Dynkin diagram of E<sub>8</sub> (see [[#Dynkin diagram|below]]) has no automorphisms, coincides with H<sup>1</sup>(''k'',E<sub>8</sub>).<ref>{{Citation | last1=Платонов | first1=Владимир П. | last2=Рапинчук | first2=Андрей С. | title=Алгебраические группы и теория чисел | year=1991 | publisher=Наука | isbn=5-02-014191-7 }} (English translation: {{Citation | last1=Platonov | first1=Vladimir P. | last2=Rapinchuk | first2=Andrei S. | title=Algebraic groups and number theory | year=1994 | publisher=Academic Press | isbn=0-12-558180-7 }}), §2.2.4</ref>
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| Over '''R''', the real connected component of the identity of these algebraically twisted forms of E<sub>8</sub> coincide with the three real Lie groups mentioned [[#Real and complex forms|above]], but with a subtlety concerning the fundamental group: all forms of E<sub>8</sub> are simply connected in the sense of algebraic geometry, meaning that they admit no non-trivial algebraic coverings; the non-compact and simply connected real Lie group forms of E<sub>8</sub> are therefore not algebraic and admit no faithful finite-dimensional representations.
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| Over finite fields, the [[Lang–Steinberg theorem]] implies that H<sup>1</sup>(''k'',E<sub>8</sub>)=0, meaning that E<sub>8</sub> has no twisted forms: see [[#Chevalley groups of type E8|below]].
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| ==Representation theory==
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| The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the [[Weyl character formula]]. The dimensions of the smallest irreducible representations are {{OEIS|id=A121732}}:
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| :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…
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| The 248-dimensional representation is the [[Adjoint representation of a Lie group|adjoint representation]]. There are two non-isomorphic irreducible representations of dimension 8634368000 (it is not unique; however, the next integer with this property is 175898504162692612600853299200000 {{OEIS|id=A181746}}). The [[fundamental representation]]s are those with dimensions 3875, 6696000, 6899079264, 146325270, 2450240, 30380, 248 and 147250 (corresponding to the eight nodes in the [[#Dynkin diagram|Dynkin diagram]] in the order chosen for the [[#Cartan matrix|Cartan matrix]] below, i.e., the nodes are read in the seven-node chain first, with the last node being connected to the third).
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| The coefficients of the character formulas for infinite dimensional irreducible [[representation theory|representation]]s of E<sub>8</sub> depend on some large square matrices consisting of polynomials, the [[Lusztig–Vogan polynomial]]s, an analogue of [[Kazhdan–Lusztig polynomial]]s introduced for [[reductive group]]s in general by [[George Lusztig]] and [[David Kazhdan]] (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.
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| These matrices were computed after four years of collaboration by a [[atlas of Lie groups and representations|group of 18 mathematicians and computer scientists]], led by [[Jeffrey Adams (mathematician)|Jeffrey Adams]], with much of the programming done by [[Fokko du Cloux]]. The most difficult case (for exceptional groups) is the split [[real form]] of E<sub>8</sub> (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''<sub>8</sub> 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.
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| The representations of the E<sub>8</sub> groups over finite fields are given by [[Deligne–Lusztig theory]].
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| ==Constructions==
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| One can construct the (compact form of the) E<sub>8</sub> group as the [[automorphism group]] of the corresponding '''e'''<sub>8</sub> Lie algebra. This algebra has a 120-dimensional subalgebra '''so'''(16) generated by ''J''<sub>''ij''</sub> as well as 128 new generators ''Q''<sub>''a''</sub> that transform as a [[Weyl–Majorana spinor]] of '''spin'''(16). These statements determine the commutators
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| :<math>[J_{ij},J_{k\ell}]=\delta_{jk}J_{i\ell}-\delta_{j\ell}J_{ik}-\delta_{ik}J_{j\ell}+\delta_{i\ell}J_{jk}</math>
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| as well as
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| :<math>[J_{ij},Q_a] = \frac 14 (\gamma_i\gamma_j-\gamma_j\gamma_i)_{ab} Q_b,</math>
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| while the remaining commutator (not anticommutator!) is defined as
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| :<math>[Q_a,Q_b]=\gamma^{[i}_{ac}\gamma^{j]}_{cb} J_{ij}.</math>
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| It is then possible to check that the [[Jacobi identity]] is satisfied.
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| ==Geometry==
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| The compact real form of E<sub>8</sub> is the [[isometry group]] of the 128-dimensional exceptional compact [[Riemannian symmetric space]] EVIII (in Cartan's [[Riemannian symmetric space#Classification of Riemannian symmetric spaces|classification]]). It is known informally as the "[[octooctonionic projective plane]]" because it can be built using an algebra that is the tensor product of the [[octonion]]s with themselves, and is also known as a [[Rosenfeld projective plane]], though it does not obey the usual axioms of a projective plane. This can be seen systematically using a construction known as the [[Freudenthal magic square|''magic square'']], due to [[Hans Freudenthal]] and [[Jacques Tits]] {{harv|Landsberg|Manivel|2001}}.
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| ==E<sub>8</sub> root system==
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| [[Image:e8 roots zome.jpg|right|thumb|200px|[[Zome]] model of the E<sub>8</sub> root system, projected into 3-space, and represented by the vertices of the [[4 21 polytope|4<sub>21</sub> polytope]], {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}]]
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| A [[root system]] of rank ''r'' is a particular finite configuration of vectors, called ''roots'', which span an ''r''-dimensional [[Euclidean space]] and satisfy certain geometrical properties. In particular, the root system must be invariant under [[reflection (mathematics)|reflection]] through the hyperplane perpendicular to any root.
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| The '''E<sub>8</sub> root system''' is a rank 8 root system containing 240 root vectors spanning '''R'''<sup>8</sup>. It is [[irreducible]] in the sense that it cannot be built from root systems of smaller rank. All the root vectors in E<sub>8</sub> have the same length. It is convenient for a number of purposes to normalize them to have length √2. These 240 vectors are the vertices of a [[semi-regular polytope]] discovered by [[Thorold Gosset]] in 1900, sometimes known as the [[4_21_polytope|4<sub>21</sub> polytope]].
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| ===Construction===
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| In the so-called ''even coordinate system'' E<sub>8</sub> is given as the set of all vectors in '''R'''<sup>8</sup> with length squared equal to 2 such that coordinates are either all [[integer]]s or all [[half-integer]]s and the sum of the coordinates is even.
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| Explicitly, there are 112 roots with integer entries obtained from
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| :<math>(\pm 1,\pm 1,0,0,0,0,0,0)\,</math>
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| by taking an arbitrary combination of signs and an arbitrary [[permutation]] of coordinates, and 128 roots with half-integer entries obtained from
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| :<math>\left(\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12\right) \,</math>
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| 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.
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| [[File:E8-with-thread.jpg|thumb|200px|E8 with thread made by hand]]
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| The 112 roots with integer entries form a D<sub>8</sub> root system. The E<sub>8</sub> root system also contains a copy of A<sub>8</sub> (which has 72 roots) as well as [[E6 (mathematics)|E<sub>6</sub>]] and [[E7 (mathematics)|E<sub>7</sub>]] (in fact, the latter two are usually ''defined'' as subsets of E<sub>8</sub>).
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| In the ''odd coordinate system'' E<sub>8</sub> 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 half-integer entries have an odd number of minus signs rather than an even number.
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| ===Dynkin diagram===
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| The [[Dynkin diagram]] for E<sub>8</sub> is given by [[ File:Dynkin diagram E8.svg|120px]].
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| 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]].
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| ===Cartan matrix===
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| The [[Cartan matrix]] of a rank ''r'' root system is an ''r × r'' [[matrix (mathematics)|matrix]] whose entries are derived from the simple roots. Specifically, the entries of the Cartan matrix are given by
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| :<math>A_{ij} = 2\frac{(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}</math>
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| where (−,−) is the Euclidean [[inner product]] and ''α<sub>i</sub>'' are the simple roots. The entries are independent of the choice of simple roots (up to ordering).
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| The Cartan matrix for E<sub>8</sub> is given by
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| :<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
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| -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
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| 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & -1 & 2 & -1 & 0 & -1 \\
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| 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\
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| 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2
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| \end{smallmatrix}\right ].</math>
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| The [[determinant]] of this matrix is equal to 1.
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| ===Simple roots===
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| [[File:E8HassePoset.svg|thumb|320px|[[Hasse diagram]] of E8 [[Root_system#The_root_poset|root poset]] with edge labels identifying added simple root position]]
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| A set of [[Simple root (root system)|simple root]]s for a root system Φ is a set of roots that form a [[basis (linear algebra)|basis]] 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.
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| Given the E<sub>8</sub> [[Cartan matrix]] (above) and a [[Dynkin diagram]] node ordering of: [[File:DynkinE8.svg|150px]]
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| :one choice of [[Simple root (root system)|simple root]]s is given by the rows of the following matrix:
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| :<math>\left [\begin{smallmatrix}
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| 1&-1&0&0&0&0&0&0 \\
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| 0&1&-1&0&0&0&0&0 \\
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| 0&0&1&-1&0&0&0&0 \\
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| 0&0&0&1&-1&0&0&0 \\
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| 0&0&0&0&1&-1&0&0 \\
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| 0&0&0&0&0&1&1&0 \\
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| -\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\
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| 0&0&0&0&0&1&-1&0 \\
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| \end{smallmatrix}\right ].</math>
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| ===Weyl group===
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| The [[Weyl group]] of E<sub>8</sub> is of order 696729600, and can be described as O{{su|p=+|b=8}}(2): it is of the form 2.''G''.2 (that is, a [[Schur multiplier|stem extension]] 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 group]] of order 174182400 (which can be described as PSΩ<sub>8</sub><sup>+</sup>(2)).<ref>{{Citation |last1=Conway |first1=John Horton |authorlink1=John Horton Conway |last2=Curtis |first2=Robert Turner |last3=Norton |first3=Simon Phillips |authorlink3=Simon P. Norton |last4=Parker |first4=Richard A |authorlink4=Richard A. Parker |last5=Wilson |first5=Robert Arnott |authorlink5=Robert Arnott Wilson |title=[[ATLAS of Finite Groups|Atlas of Finite Groups]]: Maximal Subgroups and Ordinary Characters for Simple Groups |year=1985 |month= |publisher=Oxford University Press |isbn=0-19-853199-0 |page=85 }}</ref>
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| ===E<sub>8</sub> root lattice===
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| {{main|E8 lattice|l1=E<sub>8</sub> lattice}}
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| The integral span of the E<sub>8</sub> root system forms a [[lattice (group)|lattice]] in '''R'''<sup>8</sup> naturally called the '''[[E8 lattice|E<sub>8</sub> root lattice]]'''. This lattice is rather remarkable in that it is the only (nontrivial) even, [[unimodular lattice]] with rank less than 16.
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| ===Simple subalgebras of E<sub>8</sub>===
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| [[Image:E8subgroups.svg|thumb|200px|An incomplete simple subgroup tree of E<sub>8</sub>]]
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| The Lie algebra E8 contains as subalgebras all the [[exceptional Lie algebra]]s 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.
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| ==Chevalley groups of type E<sub>8</sub>==
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| {{harvtxt|Chevalley|1955}} showed that the points of the (split) algebraic group E<sub>8</sub> (see [[#E8 as an algebraic group|above]]) over a [[finite field]] with ''q'' elements form a finite [[Group of Lie type|Chevalley group]], generally written E<sub>8</sub>(''q''), which is simple for any ''q'',<ref>{{Citation | first=Roger W. | last=Carter | title=Simple Groups of Lie Type | authorlink=Roger Carter (mathematician) | publisher=John Wiley & Sons | series=Wiley Classics Library | isbn=0-471-50683-4 | year=1989 }}</ref><ref>{{Citation | first=Robert A. | last=Wilson | title=The Finite Simple Groups | authorlink=Robert Arnott Wilson | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics]] | volume=251 | isbn=1-84800-987-9 | year=2009 }}</ref> and constitutes one of the infinite families addressed by the [[classification of finite simple groups]]. Its number of elements is given by the formula {{OEIS|id=A008868}}:
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| :<math>q^{120}(q^{30}-1)(q^{24}-1)(q^{20}-1)(q^{18}-1)(q^{14}-1)(q^{12}-1)(q^8-1)(q^2-1)</math>
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| The first term in this sequence, the order of E<sub>8</sub>(2), namely {{gaps|337|804|753|143|634|806|261|388|190|614|085|595|079|991|692|242|467|651|576|160|959|909|068|800|000}} ≈ 3.38×10<sup>74</sup>, is already larger than the size of the [[Monster group]]. This group E<sub>8</sub>(2) is the last one described (but without its character table) in the [[ATLAS of Finite Groups]].<ref>Conway &al, ''op. cit.'', p. 235.</ref>
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| The [[Schur multiplier]] of E<sub>8</sub>(''q'') is trivial, and its outer automorphism group is that of field automorphisms (i.e., cyclic of order ''f'' if ''q''=''p<sup>f</sup>'' where ''p'' is prime).
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| {{harvtxt|Lusztig|1979}} described the unipotent representations of finite groups of type ''E''<sub>8</sub>.
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| ==Subgroups==
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| The smaller exceptional groups [[E7 (mathematics)|E<sub>7</sub>]] and [[E6 (mathematics)|E<sub>6</sub>]] sit inside E<sub>8</sub>. In the compact group, both E<sub>7</sub>×SU(2)/(−1,−1) and E<sub>6</sub>×SU(3)/('''Z'''/3'''Z''') are [[maximal subgroup]]s of E<sub>8</sub>.
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| The 248-dimensional adjoint representation of E<sub>8</sub> may be considered in terms of its [[restricted representation]] to the first of these subgroups. It transforms under E<sub>7</sub>×SU(2) as a sum of [[tensor product representation]]s, which may be labelled as a pair of dimensions as (3,1) + (1,133) + (2,56) (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:
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| * (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.
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| * (1,133) consists of all roots with (1,1), (−1,−1), (0,0), (−½,−½) or (½,½) in the last two dimensions, together with the Cartan generators corresponding to the first 7 dimensions.
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| * (2,56) consists of all roots with permutations of (1,0), (−1,0) or (½,−½) in the last two dimensions.
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| The 248-dimensional adjoint representation of E<sub>8</sub>, when similarly restricted, transforms under E<sub>6</sub>×SU(3) as: (8,1) + (1,78) + (3,27) + (<u style="text-decoration:overline">3</u>,<u style="text-decoration:overline">27</u>). We may again see the decomposition by looking at the roots together with the generators in the Cartan subalgebra. In this description:
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| * (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.
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| * (1,78) consists of all roots with (0,0,0), (−½,−½,−½) or (½,½,½) in the last three dimensions, together with the Cartan generators corresponding to the first 6 dimensions.
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| * (3,27) consists of all roots with permutations of (1,0,0), (1,1,0) or (−½,½,½) in the last three dimensions.
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| * (<u style="text-decoration:overline">3</u>,<u style="text-decoration:overline">27</u>) consists of all roots with permutations of (−1,0,0), (−1,−1,0) or (½,−½,−½) in the last three dimensions.
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| The finite quasisimple groups that can embed in (the compact form of) E<sub>8</sub> were found by {{harvtxt|Griess|Ryba|1999}}. | |
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| The [[Dempwolff group]] is a subgroup of (the compact form of) E<sub>8</sub>. It is contained in the [[Thompson sporadic group]], which acts on the underlying vector space of the Lie group E<sub>8</sub> 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<sub>8</sub>('''F'''<sub>3</sub>).
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| ==Applications==
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| The E<sub>8</sub> Lie group has applications in [[theoretical physics]], in particular in [[string theory]] and [[supergravity]]. E<sub>8</sub>×E<sub>8</sub> is the [[gauge group]] of one of the two types of [[heterotic string]] and is one of two [[anomaly (physics)|anomaly-free]] gauge groups that can be coupled to the ''N'' = 1 supergravity in 10 dimensions. E<sub>8</sub> is the [[U-duality]] group of supergravity on an eight-torus (in its split form).
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| One way to incorporate the [[standard model]] of particle physics into heterotic string theory is the [[spontaneous symmetry breaking|symmetry breaking]] of E<sub>8</sub> to its maximal subalgebra SU(3)×E<sub>6</sub>.
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| In 1982, [[Michael Freedman]] used the [[E8 lattice|E<sub>8</sub> lattice]] to construct an example of a [[topological manifold|topological]] [[4-manifold]], the [[E8 manifold|E<sub>8</sub> manifold]], which has no [[Differential structure|smooth structure]]. | |
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| [[Antony Garrett Lisi]]'s incomplete theory "[[An Exceptionally Simple Theory of Everything]]" attempts to describe all known [[fundamental interaction]]s in physics as part of the E<sub>8</sub> Lie algebra.<ref name="SciAm">{{cite journal |doi=10.1038/scientificamerican1210-54 |author1=A. G. Lisi |authorlink1=Antony Garrett Lisi |author2=J. O. Weatherall |authorlink2=James Owen Weatherall |year=2010 |title=A Geometric Theory of Everything |journal=[[Scientific American]] |volume=303 |issue=6 |pages=54–61 |url=http://www.scientificamerican.com/article.cfm?id=a-geometric-theory-of-everything |pmid=21141358}}</ref><ref name="seed">{{cite news |author= Greg Boustead |title=Garrett Lisi's Exceptional Approach to Everything |url=http://seedmagazine.com/news/2008/11/garrett_lisis_exceptional_appr.php |work=SEED Magazine |date=2008-11-17 }}</ref>
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| {{harvs | txt|last1=Coldea | first1=R. | last2=Tennant | first2=D. A. | last3=Wheeler | first3=E. M. | last4=Wawrzynska | first4=E. | last5=Prabhakaran | first5=D. | last6=Telling | first6=M. | last7=Habicht | first7=K. | last8=Smeibidl | first8=P. | last9=Kiefer | first9=K. | title=Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E<sub>8</sub> Symmetry | doi=10.1126/science.1180085 | year= 2010 | journal=[[Science (journal)|Science]] | volume=327 | issue=5962 | pages=177–180}} reported that in an experiment with a [[cobalt]]-[[niobium]] crystal, under certain physical conditions the [[electron spin]]s in it exhibited two of the 8 peaks related to E<sub>8</sub> predicted by {{harvtxt|Zamolodchikov|1989}}.<ref>[http://www.newscientist.com/article/dn18356-most-beautiful-math-structure-appears-in-lab-for-first-time.html "Most beautiful math structure appears in lab for first time], ''[[New Scientist]]'', January 2010. Retrieved January 8, 2010.</ref><ref>[http://www.ams.org/notices/201108/rtx110801055p.pdf Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?], ''[[Notices of the American Mathematical Society]]'', September 2011.</ref>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation | last1=Adams | first1=J. Frank | title=Lectures on exceptional Lie groups | url=http://books.google.com/books?isbn=0226005275 | publisher=[[University of Chicago Press]] | series=Chicago Lectures in Mathematics | isbn=978-0-226-00526-3 | mr=1428422 | year=1996}}
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| *{{Citation | last1=Baez | first1=John C. | title=The octonions | url=http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html | doi=10.1090/S0273-0979-01-00934-X | mr=1886087 | year=2002 | journal=American Mathematical Society. Bulletin. New Series | volume=39 | issue=2 | pages=145–205}}
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| *{{Citation | last1=Chevalley | first1=Claude | title=Sur certains groupes simples | url=http://projecteuclid.org/euclid.tmj/1178245104 | doi=10.2748/tmj/1178245104 | mr=0073602 | year=1955 | journal=The Tohoku Mathematical Journal. Second Series | issn=0040-8735 | volume=7 | pages=14–66}}
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| *{{Citation | last1=Coldea | first1=R. | last2=Tennant | first2=D. A. | last3=Wheeler | first3=E. M. | last4=Wawrzynska | first4=E. | last5=Prabhakaran | first5=D. | last6=Telling | first6=M. | last7=Habicht | first7=K. | last8=Smeibidl | first8=P. | last9=Kiefer | first9=K. | displayauthors=9| title=Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E<sub>8</sub> Symmetry | doi=10.1126/science.1180085 | year= 2010 | journal=[[Science (journal)|Science]] | volume=327 | issue=5962 | pages=177–180}}
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| *{{Citation | last1=Griess | first1=Robert L. | last2=Ryba | first2=A. J. E. | title=Finite simple groups which projectively embed in an exceptional Lie group are classified! | url=http://www.ams.org/bull/1999-36-01/S0273-0979-99-00771-5/home.html | mr=1653177 | year=1999 | journal=American Mathematical Society. Bulletin. New Series | volume=36 | issue=1 | pages=75–93 | doi=10.1090/S0273-0979-99-00771-5}}
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| *{{citation|first=Wilhelm|last=Killing|authorlink=Wilhelm Killing|title=Die Zusammensetzung der stetigen endlichen Transformationsgruppen|journal=Mathematische Annalen|volume=31|issue=2|pages= 252–290|doi=10.1007/BF01211904|year=1888a|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002250810&L=1}}
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| *{{citation|first=Wilhelm|last=Killing|authorlink=Wilhelm Killing|title=Die Zusammensetzung der stetigen endlichen Transformationsgruppen|journal=Mathematische Annalen|volume=33|issue=1|pages= 1–48|doi=10.1007/BF01444109|year=1888b|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002251337&L=1}}
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| *{{citation|first=Wilhelm|last=Killing|authorlink=Wilhelm Killing|title=Die Zusammensetzung der stetigen endlichen Transformationsgruppen|journal=Mathematische Annalen|volume=34|issue=1|pages= 57–122 |doi=10.1007/BF01446792|year=1889|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0034&DMDID=DMDLOG_0009&L=1}}
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| *{{citation|first=Wilhelm|last=Killing|authorlink=Wilhelm Killing|title=Die Zusammensetzung der stetigen endlichen Transformationsgruppen|journal=Mathematische Annalen|volume=36|issue=2|pages= 161–189|doi=10.1007/BF01207837|year=1890|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002252392&L=1}}
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| *J.M. Landsberg and L. Manivel (2001), ''The projective geometry of Freudenthal's magic square'', Journal of Algebra, Volume 239, Issue 2, pages 477–512, {{doi|10.1006/jabr.2000.8697}}, [http://www.arxiv.org/abs/math/9908039 arXiv:math/9908039v1].
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| *{{Citation | last1=Lusztig | first1=George | title=Unipotent representations of a finite Chevalley group of type E8 | doi=10.1093/qmath/30.3.301 | mr=R545068 | year=1979 | journal=The Quarterly Journal of Mathematics. Oxford. Second Series | issn=0033-5606 | volume=30 | issue=3 | pages=315–338}}
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| *{{Citation
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| | last1=Lusztig
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| | first1=George
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| | author1-link=George Lusztig
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| | last2=Vogan
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| | first2=David
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| | author2-link=David Vogan
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| | title=Singularities of closures of K-orbits on flag manifolds.
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| | year=1983
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| | month=
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| | volume=71
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| | issue=2
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| | journal=[[Inventiones Mathematicae]]
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| | publisher=[[Springer-Verlag]]
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| | pages=365–379
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| | doi=10.1007/BF01389103
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| }}
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| *{{Citation | last1=Zamolodchikov | first1=A. B. | authorlink=Alexander Zamolodchikov|title=Integrals of motion and S-matrix of the (scaled) T=T<sub>c</sub> Ising model with magnetic field | doi=10.1142/S0217751X8900176X | mr=1017357 | year=1989 | journal=International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. Nuclear Physics | volume=4 | issue=16 | pages=4235–4248}}
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| ==External links==
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| ;Related to the calculation of the Lusztig–Vogan polynomials:
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| *[http://www.liegroups.org/ Atlas of Lie groups]
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| *[http://www.liegroups.org/kle8.html Kazhdan–Lusztig–Vogan Polynomials for E<sub>8</sub>]
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| *D. Vogan, [http://atlas.math.umd.edu/kle8.narrative.html Narrative of the Project to compute Kazhdan–Lusztig Polynomials for E<sub>8</sub>]
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| *D. Vogan, [http://math.mit.edu/~dav/E8TALK.pdf ''The Character Table for E<sub>8</sub>, or How We Wrote Down a 453,060 × 453,060 Matrix and Found Happiness''] Slides for a popular talk on E<sub>8</sub>.
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| * {{Citation | author=[[American Institute of Mathematics]] | title=Mathematicians Map E<sub>8</sub> |date=March 2007 | url=http://aimath.org/E8/ }}
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| * [http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html The ''n''-Category Café] — [[University of Texas]] blog posting by [[John Baez]] on E<sub>8</sub>
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| ;Other:
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| * [http://www-math.mit.edu/~dav/e8plane.html Graphic representation of E<sub>8</sub> root system] from [[MIT]]
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| * The list of dimensions of [[irreducible representation]]s of the complex form of E<sub>8</sub> is sequence [[OEIS:A121732|A121732]] in the [[OEIS]].
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| {{Exceptional Lie groups}}
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| {{DEFAULTSORT:E8 (Mathematics)}}
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| [[Category:Algebraic groups| ]]
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| [[Category:Lie groups]]
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