Two Partial Orders for Standard Young Tableaux

TL;DR

This paper proves the equivalence of two partial orders on standard Young tableaux using two different algorithmic proofs, linking combinatorics and geometry of nilpotent operators.

Contribution

It provides two proofs of the equivalence between the box order and the dominance order on tableaux, connecting combinatorial orders with geometric representation theory.

Findings

Proves the equivalence of box and dominance orders on tableaux.

Provides algorithmic proofs linking combinatorics and group theory.

Enhances understanding of the geometry of nilpotent operator spaces.

Abstract

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.