Sends a tableau to its corresponding Gelfand-Tsetlin pattern.
To obtain this Gelfand-Tsetlin pattern, fill in the first row of the pattern with the shape of the tableau.
Then remove the maximal entry from the tableau to obtain a smaller tableau, and repeat the process until the tableau is empty.

The cycle action on a Gelfand-Tsetlin pattern.
Regard a Gelfand-Tsetlin pattern $p$ as an element of the $A_{m-1}$ crystal of shape $\lambda$, where $\lambda$ is the partition given by the first row of $p$ and $m$ is the width of $p$. Let $s_i(p)$ be the reflection of $p$ along its $i$-string, for $i\in\{1,\dots,m-1\}$. Then this map returns $s_1 s_2 \dots s_{m-1}(p)$.
The order of this action is the width of the pattern. As a map on semistandard tableaux, it preserves the shape. Moreover, the weight of $c(p)$ is the same as the weight of the promotion of $p$.