Let $G$ be a simple classical algebraic group with corresponding root system $\Phi$ and natural module $W$ of dimension $n$.

For a (closed) subsystem $\Psi$ of $\Phi$, let $G_\Psi = \{ U_\alpha \mid \alpha \in \Psi\}$, where $U_\alpha$ is the root subgroup corresponding to $\alpha \in \Psi$. Take a regular unipotent element $x_{\Psi} \in G_\Psi$.

It is well known that there is an injective function from the set of classes of unipotent elements in $G$ to the set of partitions $\lambda \vdash n$.

How can I find the partition of $\mu \vdash n$ corresponding to $x_{\Psi}$?

Let $G$ be a simple classical algebraic group with corresponding root system $\Phi$ and natural module $W$ of dimension $n$.

For a (closed) subsystem $\Psi$ of $\Phi$, let $G_\Psi = \{ U_\alpha \mid \alpha \in \Psi\}$, where $U_\alpha$ is the root subgroup corresponding to $\alpha \in \Psi$. Take a regular unipotent element $x_{\Psi} \in G_\Psi$.

It is well known that there is a function from the set of classes of unipotent elements in $G$ to the set of partitions $\lambda \vdash n$, which is injective most of the time (except some of the time in $D_n$).

How can I find the partition of $\mu \vdash n$ corresponding to $x_{\Psi}$?

Let $G$ be a simple classical algebraic group with corresponding root system $\Phi$ and natural module $W$ of dimension $n$.

For a (closed) subsystem $\Psi$ of $\Phi$, let $G_\Psi = \{ U_\alpha \mid \alpha \in \Psi\}$, where $U_\alpha$ is the root subgroup corresponding to $\alpha \in \Psi$. Take a regular unipotent element $x_{\Psi} \in G_\Psi$.

It is well known that there is a one-to-one correspondence between classes of unipotent elements in classical simple algebraic groups and partitions $\lambda \vdash n$.

How can I find the partition of $\mu \vdash n$ corresponding to $x_{\Psi}$?

Let $G$ be a simple classical algebraic group with corresponding root system $\Phi$ and natural module $W$ of dimension $n$.

For a (closed) subsystem $\Psi$ of $\Phi$, let $G_\Psi = \{ U_\alpha \mid \alpha \in \Psi\}$, where $U_\alpha$ is the root subgroup corresponding to $\alpha \in \Psi$. Take a regular unipotent element $x_{\Psi} \in G_\Psi$.

It is well known that there is an injective function from the set of classes of unipotent elements in $G$ to the set of partitions $\lambda \vdash n$.

How can I find the partition of $\mu \vdash n$ corresponding to $x_{\Psi}$?