[Paper Review] Two Partial Orders for Littlewood-Richardson Tableaux

Justyna Kosakowska, Markus Schmidmeier+1 more|arXiv (Cornell University)|Mar 31, 2015

Advanced Combinatorial Mathematics7 references3 citations

TL;DR

This paper proves the equivalence of the box order and dominance order on Littlewood-Richardson fillings of shape $(\alpha,\beta,\gamma)$ when $\beta\setminus\gamma$ is a horizontal and vertical strip, using two algorithmic proofs: one linking to the Bruhat order on the symmetric group, and another via direct box move construction. The result strengthens combinatorial foundations for the geometry of invariant subspaces in nilpotent operator representations.

ABSTRACT

In this manuscript we show that two partial orders defined on the set of Littlewood-Richardson fillings of a~given shape $(\alpha,\beta,\gamma)$ are equivalent if $\beta\setminus\gamma$ is a horizontal and vertical strip. In fact, we give two proofs for the equivalence of the box order and the dominance order for fillings. 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 box moves. 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.

Motivation & Objective

• To establish the equivalence of two partial orders—box order and dominance order—on Littlewood-Richardson fillings of a fixed shape $(\alpha,\beta,\gamma)$.
• To investigate the conditions under which these two combinatorial orders coincide, particularly when $\beta\setminus\gamma$ forms a horizontal and vertical strip.
• To provide two algorithmic proofs of this equivalence, one rooted in symmetric group Bruhat order and the other in constructive box move operations.
• To connect this combinatorial equivalence to the geometry of representation spaces of nilpotent linear operators, as previously known in special cases.

Proposed method

• The first proof establishes a connection between the box order and the Bruhat order on the symmetric group, using permutation-based arguments to show order equivalence.
• The second proof constructs a direct sequence of box moves transforming one filling into another, preserving order relations under the dominance order.
• Both proofs are algorithmic, emphasizing constructive transformations rather than abstract order-theoretic arguments.
• The analysis focuses on fillings where $\beta\setminus\gamma$ is a horizontal and vertical strip, a condition that ensures the combinatorial structure remains well-behaved.
• The proofs rely on properties of Littlewood-Richardson tableaux, including the lattice word condition and the semistandard filling constraints.
• The equivalence is verified by showing that the dominance order refines the box order and vice versa under the given shape constraints.

Experimental results

Research questions

• RQ1 Under what conditions are the box order and dominance order equivalent on Littlewood-Richardson fillings of shape $(\alpha,\beta,\gamma)$?
• RQ2 How can the equivalence between these two partial orders be proven algorithmically, without relying on abstract order theory?
• RQ3 What is the role of the symmetric group's Bruhat order in understanding the combinatorics of LR tableaux?
• RQ4 How does the structure of $\beta\setminus\gamma$ as a horizontal and vertical strip influence the order equivalence?
• RQ5 What geometric implications does this order equivalence have for the representation space of nilpotent linear operators?

Key findings

• The box order and dominance order on Littlewood-Richardson fillings are equivalent when $\beta\setminus\gamma$ is a horizontal and vertical strip.
• Two distinct algorithmic proofs are provided: one linking the orders to the Bruhat order on the symmetric group, and another using explicit box move constructions.
• The equivalence holds specifically under the strip condition, which ensures that no two boxes in $\beta\setminus\gamma$ are in the same row or column.
• The constructive proof demonstrates a step-by-step transformation between fillings using valid box moves, confirming order preservation.
• The result provides a combinatorial foundation for describing the geometry of invariant subspaces in nilpotent operator representations.
• The work generalizes known results by offering a constructive and group-theoretic justification for order equivalence in this specific case.