Is there some characterizing property for an element $g$ of minimal length such that $gag^{-1}=b$?
We know that if $g_1$ and $g_2$ are both such that $g_1ag_1^{-1}=b$ and $g_2ag_2^{-1}=b$ then they belong to the same coset of the stabilizer of $a$ under conjugation.
Are there any known results about the structure of the coset decomposition in this case? I am particularly interested in the case of the symmetric group.