# Explicit expressions of inner / outer automorphism of symplectic group

We know that one may identify the automorphism group of the Dynkin diagram of G with a subgroup of Out(G).

In this way, the outer automorphism of symplectic group $Sp(n)$ (some people may denote it as $Sp(2n)$) is related to the Dynkin diagrams $C_{n}$. We use the convention:

$$Sp(1)=SU(2)=Spin(3)=C_1$$

$$Sp(2)=Spin(5)=C_2$$

Question 1 (Examples):

• $Sp(1)=SU(2)$ has only an SO(3) ($\supseteq \mathbb{Z}_2$)-inner automorphism, inner automorphism, but no outer automorphism (because no symmetry for Dynkin diagram $C_1$). Is this true?

• $Sp(2)=Spin(5)=C_{5}$ seems not to have any outer automorphism (because no symmetry for Dynkin diagram $C_5$).

• For a general $Sp(n)$, do we have any inner automorphism, but no outer automorphism?

Question 2: Are there some nice expressions we can write for the mapping

• inner automorphism

How do we find the explicit $k$ such that $$k g_{{Sp(n)}} k^{-1} = g_{{Sp(n)}}^{'}?$$ which is the inner automorphism arisen by the conjugation of $k \in Sp(n)$?

• outer automorphism

Look at the symmetries of Dynkin diagram $C_n$, it looks that there is NO any outer automorphism. Is this true? Alternatively, if it is false, how do we find the explicit map $$g_{{Sp(n)}} \to g_{{Sp(n)}}^{'}?$$ which explicit map is the outer automorphism?