E6 (Mathematics)

In mathematics, E6 is the name of a Lie group and also its Lie algebra \mathfrak{e}_6. It is one of the five exceptional simple Lie groups as well as one of the simply laced groups. E6 has rank 6 and dimension 78. Its center is the cyclic group Z3. Its outer automorphism group is the cyclic group Z2. Its fundamental representation is 27-dimensional (complex) and its dual representation, which is inequivalent to it is also 27-dimensional. In particle physics, E6 plays a role in some grand unified theories.

Algebra

Dynkin diagram

Dynkin diagram of E_6

Roots of E6

Although they span a six-dimensional space, it's much more symmetrical to consider them as vectors in a six-dimensional subspace of a nine-dimensional space.
(1,-1,0;0,0,0;0,0,0), (-1,1,0;0,0,0;0,0,0),
(-1,0,1;0,0,0;0,0,0), (1,0,-1;0,0,0;0,0,0),
(0,1,-1;0,0,0;0,0,0), (0,-1,1;0,0,0;0,0,0),
(0,0,0;1,-1,0;0,0,0), (0,0,0;-1,1,0;0,0,0),
(0,0,0;-1,0,1;0,0,0), (0,0,0;1,0,-1;0,0,0),
(0,0,0;0,1,-1;0,0,0), (0,0,0;0,-1,1;0,0,0),
(0,0,0;0,0,0;1,-1,0), (0,0,0;0,0,0;-1,1,0),
(0,0,0;0,0,0;-1,0,1), (0,0,0;0,0,0;1,0,-1),
(0,0,0;0,0,0;0,1,-1), (0,0,0;0,0,0;0,-1,1),
All 27 combinations of (\bold{3};\bold{3};\bold{3}) where \bold{3} is one of (\frac{2}{3},-\frac{1}{3},-\frac{1}{3}), (-\frac{1}{3},\frac{2}{3},-\frac{1}{3}), (-\frac{1}{3},-\frac{1}{3},\frac{2}{3}) All 27 combinations of (\bold{\bar{3}};\bold{\bar{3}};\bold{\bar{3}}) where \bold{\bar{3}} is one of (-\frac{2}{3},\frac{1}{3},\frac{1}{3}), (\frac{1}{3},-\frac{2}{3},\frac{1}{3}), (\frac{1}{3},\frac{1}{3},-\frac{2}{3}) Simple roots
(0,0,0;0,0,0;0,1,-1)
(0,0,0;0,0,0;1,-1,0)
(0,0,0;0,1,-1;0,0,0)
(0,0,0;1,-1,0;0,0,0)
(0,1,-1;0,0,0;0,0,0)
(\frac{1}{3},-\frac{2}{3},\frac{1}{3};-\frac{2}{3},\frac{1}{3},\frac{1}{3};-\frac{2}{3},\frac{1}{3},\frac{1}{3})

Weyl/Coxeter group

Its Weyl/Coxeter group is symmetry group of the E6 polytope.

Cartan matrix

\begin{pmatrix} 2&-1&0&0&0&0\\ -1&2&-1&0&0&0\\ 0&-1&2&-1&-1&0\\ 0&0&-1&2&0&0\\ 0&0&-1&0&2&-1\\ 0&0&0&0&-1&2 \end{pmatrix}

 

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