  In this manuscript we show that two partial orders defined on the set of
standard Young tableaux of shape $\alpha$ are equivalent. In fact, we give two
proofs for the equivalence of the box order and the dominance order for
{tableaux}. Both are algorithmic. The first of these proofs emphasizes links to
the Bruhat order for the symmetric group and the second provides a more
straightforward construction of the cover relations. This work is motivated by
the known result that the equivalence of the two combinatorial orders leads to
a description of the geometry of the representation space of invariant
subspaces of nilpotent linear operators.
