The Maximality of Standard Tableaux
Any such comes from , where . We will make our induction on the number of “column inversions” in . That is, the number of pairs of entries that are in the same column of , but which are “out of order”, in the sense that is in a lower row than .
Given any such pair, the dominance lemma tells us that . That is, by “untwisting” the column inversion, we can move up the dominance order while preserving the columns. It should also be clear that has fewer column inversions than does. But if we undo all the column inversions, the tableau we’re left with must be standard. That is, it must be itself.