Let $X \in \mathbb{C}[S_n]$ be an element of the group algebra of $S_n$ expressible as the sum of some Jucys-Murphy elements. Then let $\lambda$ be any irreducible representation of $S_n$, with the usual basis given by the standard Young tableaux of shape $\lambda$. Each of these tableaux is of course an eigenvector for the action of $X$, with the eigenvalue easily computable from the contents of the relevant boxes.
My question is this: Given a particular $X$ and $\lambda$, is there an easy combinatorial way to construct the tableau $T$ that will have the maximal eigenvalue for the action of $X$? (Again, for the moment, I only care about $X$ that are sums of Jucys-Murphy elements - i.e. each has coefficient 0 or 1. Allowing arbitrary linear combinations of them would seem much harder....)
Ideally, it would be nice if there were a way to "build up" this tableau by adding the boxes $1, \ldots n$ in order, or by placing the numbers in the tableau from $n$ on down to 1, but I'm not convinced this is possible.
