What is the Lie theoretic interpretation of conjugate of a partition? For a partition $\lambda$ it is very well-known operation to take its conjugate partition $\lambda'$ which is obtained by transposing the Young diagram of $\lambda$.
A partition $\lambda$ can be viewed as a dominant weight of $GL_n$ for some $n \geq l(\lambda)$.
Suppose $m \geq l(\lambda),l(\lambda')$. Is there a 'natural' operation on $\mathbb{Z}^m$, the weight lattice of $GL_m$, which takes $\lambda \mapsto \lambda'$ ?
 A: Computing some simple examples it doesn't appear to be a particularly nice operation. It certainly isn't linear in the weights or anything so simple as that.
Let $\omega_1,\dots,\omega_n$ be the fundamental weights.
Here are some examples of the "transpose the young diagram" operation on weights which I computed using LiE:
$$\begin{array}{|c|c|}
\hline
\omega_1& \omega_1  \\ \hline
 k\omega_1& \omega_k \\ \hline
 \omega_1 + \omega_2 & \omega_1 + \omega_2 \\ \hline
\omega_1 + \cdots + \omega_n & \omega_1 + \cdots + \omega_n \\ \hline
 2\omega_1 + \omega_2&  \omega_1 + \omega_3 \\ \hline
\omega_1 + 2\omega_2 & \omega_2 + \omega_3 \\ \hline
2\omega_1 + 2\omega_2 & \omega_2 + \omega_4 \\ \hline
k\omega_k& k\omega_k\\\hline
(k-i)\omega_k & k\omega_{k-i}\\\hline
k\omega_1 + k\omega_2 & \omega_k + \omega_{2k} \\ \hline
k\omega_1 + k\omega_3 & 2\omega_k + \omega_{2k} \\ \hline
k\omega_2 + k\omega_3 & \omega_k + 2\omega_{2k} \\ \hline
\end{array}$$
Here $i<k\leq n$ (and $2k \leq n$ where it is used).
Caveat: these are patterns I observed and extrapolated so are not proved true for all $k,n,i$.
As you can see there are certainly some patterns but I would be hard-pressed to give a general rule for any weight.
