Frédéric Wang
About Me
Blog Archive
Mathematics
Computer Science
A
Théorie Classique
A
Théorie Classique
A.2
Formule des caractères de Weyl
A.1
Algèbres de Lie simples de dimension finie
Type
Matrice de Symétrisation
d
i
d_{i}
Matrice de Cartan
a
i
j
a_{{ij}}
Nombres de Coxeter
A
n
A_{n}
I
n
=
(
1
⋱
1
)
I_{n}=\begin{pmatrix}1&&\\ &\ddots&\\ &&1\end{pmatrix}
(
2
-
1
-
1
2
⋱
⋱
⋱
-
1
-
1
2
)
\begin{pmatrix}2&-1&&\\ -1&2&\ddots&\\ &\ddots&\ddots&-1\\ &&-1&2\end{pmatrix}
h
=
h
∨
=
n
+
1
h=h^{\vee}=n+1
B
n
B_{n}
(
1
⋱
1
2
)
\begin{pmatrix}1&&&\\ &\ddots&&\\ &&1&\\ &&&2\\ \end{pmatrix}
(
2
-
1
-
1
2
⋱
⋱
⋱
-
1
-
1
2
-
2
-
1
2
)
\begin{pmatrix}2&-1&&&\\ -1&2&\ddots&&\\ &\ddots&\ddots&-1&\\ &&-1&2&-2\\ &&&-1&2\end{pmatrix}
h
=
2
n
,
h
∨
=
2
n
-
1
h=2n,h^{\vee}=2n-1
C
n
C_{n}
(
2
⋱
2
1
)
\begin{pmatrix}2&&&\\ &\ddots&&\\ &&2&\\ &&&1\\ \end{pmatrix}
(
2
-
1
-
1
2
⋱
⋱
⋱
-
1
-
1
2
-
1
-
2
2
)
\begin{pmatrix}2&-1&&&\\ -1&2&\ddots&&\\ &\ddots&\ddots&-1&\\ &&-1&2&-1\\ &&&-2&2\end{pmatrix}
h
=
2
n
,
h
∨
=
n
+
1
h=2n,h^{\vee}=n+1
D
n
D_{n}
I
n
=
(
1
⋱
1
)
I_{n}=\begin{pmatrix}1&&\\ &\ddots&\\ &&1\end{pmatrix}
(
2
-
1
-
1
2
⋱
⋱
⋱
-
1
-
1
-
1
2
0
-
1
0
2
)
\begin{pmatrix}2&-1&&&\\ -1&2&\ddots&&\\ &\ddots&\ddots&-1&-1\\ &&-1&2&0\\ &&-1&0&2\end{pmatrix}
h
=
h
∨
=
2
n
-
2
h=h^{\vee}=2n-2
E
6
E_{6}
I
6
=
(
1
⋱
1
)
I_{6}=\begin{pmatrix}1&&\\ &\ddots&\\ &&1\end{pmatrix}
(
2
0
-
1
0
0
0
0
2
0
-
1
0
0
-
1
0
2
-
1
0
0
0
-
1
-
1
2
-
1
0
0
0
0
-
1
2
-
1
0
0
0
0
-
1
2
)
\begin{pmatrix}2&0&-1&0&0&0\\ 0&2&0&-1&0&0\\ -1&0&2&-1&0&0\\ 0&-1&-1&2&-1&0\\ 0&0&0&-1&2&-1\\ 0&0&0&0&-1&2\end{pmatrix}
h
=
h
∨
=
12
h=h^{\vee}=12
E
7
E_{7}
I
7
=
(
1
⋱
1
)
I_{7}=\begin{pmatrix}1&&\\ &\ddots&\\ &&1\end{pmatrix}
(
2
0
-
1
0
0
0
0
0
2
0
-
1
0
0
0
-
1
0
2
-
1
0
0
0
0
-
1
-
1
2
-
1
0
0
0
0
0
-
1
2
-
1
0
0
0
0
0
-
1
2
-
1
0
0
0
0
0
-
1
2
)
\begin{pmatrix}2&0&-1&0&0&0&0\\ 0&2&0&-1&0&0&0\\ -1&0&2&-1&0&0&0\\ 0&-1&-1&2&-1&0&0\\ 0&0&0&-1&2&-1&0\\ 0&0&0&0&-1&2&-1\\ 0&0&0&0&0&-1&2\end{pmatrix}
h
=
h
∨
=
18
h=h^{\vee}=18
E
8
E_{8}
I
8
=
(
1
⋱
1
)
I_{8}=\begin{pmatrix}1&&\\ &\ddots&\\ &&1\end{pmatrix}
(
2
0
-
1
0
0
0
0
0
0
2
0
-
1
0
0
0
0
-
1
0
2
-
1
0
0
0
0
0
-
1
-
1
2
-
1
0
0
0
0
0
0
-
1
2
-
1
0
0
0
0
0
0
-
1
2
-
1
0
0
0
0
0
0
-
1
2
-
1
0
0
0
0
0
0
-
1
2
)
\begin{pmatrix}2&0&-1&0&0&0&0&0\\ 0&2&0&-1&0&0&0&0\\ -1&0&2&-1&0&0&0&0\\ 0&-1&-1&2&-1&0&0&0\\ 0&0&0&-1&2&-1&0&0\\ 0&0&0&0&-1&2&-1&0\\ 0&0&0&0&0&-1&2&-1\\ 0&0&0&0&0&0&-1&2\end{pmatrix}
h
=
h
∨
=
30
h=h^{\vee}=30
F
4
F_{4}
(
1
0
0
0
0
1
0
0
0
0
2
0
0
0
0
2
)
\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&2&0\\ 0&0&0&2\end{pmatrix}
(
2
-
1
0
0
-
1
2
-
2
0
0
-
1
2
-
1
0
0
-
1
2
)
\begin{pmatrix}2&-1&0&0\\ -1&2&-2&0\\ 0&-1&2&-1\\ 0&0&-1&2\end{pmatrix}
h
=
12
,
h
∨
=
9
h=12,h^{\vee}=9
G
2
G_{2}
(
3
0
0
1
)
\begin{pmatrix}3&0\\ 0&1\end{pmatrix}
(
2
-
1
-
3
2
)
\begin{pmatrix}2&-1\\ -3&2\end{pmatrix}
h
=
6
,
h
∨
=
4
h=6,h^{\vee}=4