Overview of Root System of Type E7

By | 2017-11-19

Lattice

\{\sum_{i=1}^{8} n_i \vec{e}_i | \sum_{i=1}^{8} n_i = 0 \} \cup \{\frac{1}{2}\sum_{i=1}^{8} n_i \vec{e}_i | \frac{1}{2}\sum_{i=1}^{8} n_i = 0 \text{ and } n_i \equiv 1 \pmod{2}  \}

Roots

\{ \vec{e}_i - \vec{e}_j | 1 \leq i \ne j \leq 8 \} \cup \{ \frac{1}{2}( \epsilon_1 \vec{e}_1 + \ldots + \epsilon_8 \vec{e}_8) | \epsilon_1 + \ldots + \epsilon_8 = 0 \}
Root count: 8*7 + \binom{8}{4} = 56 + 70 = 126

Short Hand Notation and Rules

e_{i \ominus j} =e_i - e_j where 1 \leq i \neq j \leq 8
e_I = \sum_{i \in I} e_i - \sum_{j \notin I} e_j where I \subset \{ 1, \cdots, 8 \}, card(I)=4

e_{i \ominus j} + e_{j \ominus k} = e_{i \ominus k}
e_I - e_{i \ominus j} = e_{I\setminus\{i\}\cup\{j\}} \text{ when } i \in I, j \notin I

Basis

\alpha_1=\vec{e}_2 - \vec{e}_1 \alpha_3=\vec{e}_3 - \vec{e}_2 \alpha_4=\vec{e}_4 - \vec{e}_3 \alpha_5=\vec{e}_5 - \vec{e}_4 \alpha_6=\vec{e}_6 - \vec{e}_5 \alpha_7=\vec{e}_7 - \vec{e}_6
\alpha_2=\frac{1}{2}(\vec{e}_1 + \vec{e}_2 + \vec{e}_3 - \vec{e}_4 - \vec{e}_5 - \vec{e}_6 - \vec{e}_7 + \vec{e}_8)

High Root

\vec{e}_8 - \vec{e}_1 = 2 3 \underset{2}{4} 3 2 1

Poset of Abelian Radical

Geometrically, Abelian Radical is the face of the Root polyhedron which contains \alpha_7 (at long end of Dynkin graph) and is parallel to the hyperplane spanned by the remaining roots. Algebraically, it consists of roots which \alpha_7-coefficient is 1. Explicitly
\{e_{i\ominus j}|i \in \{7,8\}, j \in \{1,2,3,4,5,6\}\}\cup \{e_{\{m,n\}\cup\{7,8\}}|1\leq m < n \leq 6\}
Size of the radical: 2*6 + \binom{6}{2} = 12 + 15 = 27
The Levi factor of the parabolic subalgebra defined by \alpha_7 is E_6. This gives a 27-dimensional representation of E_6 (its smallest fundamental one).

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