Two Partial Orders for Standard Young Tableaux
Justyna Kosakowska, Markus Schmidmeier, Hugh Thomas
TL;DR
The paper addresses when two natural combinatorial orders on standard Young tableaux of fixed shape are the same. It develops two algorithmic proofs—one via the Bruhat order on the symmetric group and a second a direct box-move procedure—to show that the box order and the dominance order are equivalent for tableaux of the same shape (in rook-strip cases). It also analyzes the induced poset structure, connects LR-fillings to nilpotent-operator invariants and boundary relations, and discusses geometric implications for invariant-subspace varieties. These results illuminate the degeneration geometry of nilpotent operator representations and provide concrete, constructive methods for translating between combinatorial and geometric descriptions.
Abstract
In this manuscript we show that two partial orders defined on the set of standard Young tableaux of shape $α$ 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.