Minimal and maximal elements in two-sided cells of image and Robinson–Schensted correspondence

In symmetric groups, a two-sided cell is the set of all permutations which are mapped by the Robinson–Schensted correspondence on a pair of tableaux of the same shape. In this article, we show that the set of permutations in a two-sided cell which have a minimal number of inversions is the set of permutations which have a maximal number of inversions in conjugated Young subgroups. We also give an interpretation of these sets with particular tableaux, called reading tableaux. As a corollary, we give the set of elements in a two-sided cell which have a maximal number of inversions.