[Paper Review] The involutive nature of the Littlewood-Richardson commutativity bijection

Olga Azenhas, Ronald C. King+1 more|arXiv (Cornell University)|Mar 16, 2016

Algebraic structures and combinatorial models18 references3 citations

TL;DR

This paper introduces two combinatorial maps—ρ(n) on Littlewood-Richardson tableaux and σ(n) on LR hives—that provide explicit, invertible, and involutive bijections between tableaux of shape λ/μ and weight ν and those of shape λ/ν and weight μ, thereby manifestly proving the symmetry cλμν = cλνμ. The maps are constructed via iterative deletion of entries and path removals, respectively, preserving semistandard and hive conditions while generating a shared Gelfand-Tsetlin pattern that determines the symmetric partner object.

ABSTRACT

Littlewood-Richardson (LR) coefficients $c_{μν}^λ$ may be evaluated by means of several combinatorial models, including the original LR tableaux of skew shape $λ/μ$ and weight $ν$ and the LR hives with boundary edge labels $λ$, $μ$ and $ν$. Unfortunately, neither of these reveal in any obvious way the well-known symmetry property $c_{μν}^λ=c_{νμ}^λ$. Here we introduce two maps, $ρ^{(n)}$ on LR tableaux and $σ^{(n)}$ on LR hives, that each interchange contributions to $c_{μν}^λ$ and $c_{νμ}^λ$ for any partitions $λ$, $μ$, $ν$ of lengths no greater than $n$, and then prove not only that each of them is a bijection, thereby making manifest the required symmetry property, but also that both maps are involutions. The map $ρ^{(n)}$ involves the iterative action of deletion operators on a given LR tableau of skew shape $λ/μ$ and weight $ν$, that produce a sequence of successively smaller tableaux whose consecutive inner shapes define a certain Gelfand-Tsetlin pattern and determine a partner LR tableau of skew shape $λ/ν$ and weight $μ$. Similarly, the map $σ^{(n)}$ involves repeated path removals from a given LR hive with boundary edge labels $(λ,μ,ν)$ that give rise to a sequence of hives whose left-hand boundary edge labels define the same Gelfand-Tsetlin pattern as before, which is sufficient to determine a partner LR hive with boundary edge labels $(λ,ν,μ)$. The deletions in tableaux are organised so as to preserve the semistandard and lattice permutation properties of LR tableaux, while the path removals in hives are designed to preserve both the triangle condition on edge labels and the hive rhombus gradient positivity conditions. At all stages illustrative examples are provided.

Motivation & Objective

• To provide a direct combinatorial proof of the symmetry cλμν = cλνμ in Littlewood-Richardson coefficients.
• To construct explicit, invertible maps ρ(n) and σ(n) that realize this symmetry in the LR tableau and hive models.
• To show that both maps are involutions, meaning applying them twice recovers the original object.
• To establish a self-contained, bijective, and involutive mechanism that makes the symmetry manifest without relying on abstract representation theory.

Proposed method

• The map ρ(n) acts on LR tableaux by iteratively deleting entries from right to left in each row, starting from the bottom row, using a sequence of deletion operators that preserve semistandard and lattice permutation properties.
• The resulting sequence of inner shapes forms a Gelfand-Tsetlin pattern of type μ, which defines the image tableau S ∈ LR(λ/ν, μ).
• The map σ(n) acts on LR hives by successively removing lattice paths from the base, with each path's endpoint recorded to form a new hive, preserving the triangle and rhombus gradient positivity conditions.
• The path removals generate a sequence of left-hand boundary edge labels that form the same Gelfand-Tsetlin pattern as in the tableau case, defining the symmetric hive K ∈ H(n)(λ, ν, μ).
• Both maps are shown to be inverses of themselves (involutive), with inverse constructions building up from the GT pattern of type ν to recover the original object.
• The two models are connected via the known bijection f(n) between LR tableaux and LR hives, though the paper develops each map independently to highlight their respective structural insights.

Experimental results

Research questions

• RQ1 Can the symmetry cλμν = cλνμ be proven via an explicit, invertible, and involutive combinatorial map on LR tableaux?
• RQ2 How can a similar symmetry-preserving bijection be constructed in the LR hive model, and does it yield the same result as the tableau model?
• RQ3 What structural properties (e.g., Gelfand-Tsetlin patterns) are preserved and used to define the symmetric partner object in both models?
• RQ4 Is the map on tableaux or hives truly involutive, i.e., does applying it twice return the original object?
• RQ5 How do the deletion and path removal operations maintain the required combinatorial conditions (semistandardness, hive inequalities) throughout the transformation?

Key findings

• The map ρ(n) is a well-defined bijection from LR(λ/μ, ν) to LR(λ/ν, μ), explicitly constructing the symmetric partner tableau via iterative right-to-left, bottom-to-top deletion.
• The map σ(n) is a bijection from H(n)(λ, μ, ν) to H(n)(λ, ν, μ), constructed by path removals from the base of the hive, preserving all hive conditions.
• Both maps ρ(n) and σ(n) are involutions: applying the map twice returns the original object, i.e., ρ(n)² = id and σ(n)² = id.
• The sequence of inner shapes in the tableau deletion process and the sequence of left-hand boundary edges in the path removal process both generate the same Gelfand-Tsetlin pattern of type μ, which uniquely determines the symmetric partner object.
• The construction of the symmetric partner in both models is fully compatible with the known bijection f(n) between LR tableaux and LR hives, as confirmed by explicit examples.
• The results support the coincidence of the present maps with the Henriques-Kamnitzer commutor, as verified through comparison with the Schützenberger involution and octahedral map in a detailed example.