Weighted branching formulas for the hook lengths

Matjaž Konvalinka,Igor Pak,Ionuţ Ciocan-Fontanine

doi:10.46298/dmtcs.2850

Matjaž Konvalinka, Igor Pak + Show 1 more

Open Access

https://doi.org/10.46298/dmtcs.2850

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Abstract

The famous hook-length formula is a simple consequence of the branching rule for the hook lengths. While the Greene-Nijenhuis-Wilf probabilistic proof is the most famous proof of the rule, it is not completely combinatorial, and a simple bijection was an open problem for a long time. In this extended abstract, we show an elegant bijective argument that proves a stronger, weighted analogue of the branching rule. Variants of the bijection prove seven other interesting formulas. Another important approach to the formulas is via weighted hook walks; we discuss some results in this area. We present another motivation for our work: $J$-functions of the Hilbert scheme of points. La formule bien connue de la longueur des crochets est une conséquence simple de la règle de branchement des longueurs des crochets. La preuve la plus répandue de cette règle est de nature probabiliste et est due à Greene-Njenhuis-Wilf. Elle n'est toutefois pas complètement combinatoire et une simple bijection a été pendant longtemps un problème ouvert. Dans ce résumé étendu, nous proposons un argument bijectif élégant qui démontre une version à poids plus forte de cette règle. Des variantes de cette bijection permettent d'obtenir sept autres formules intéressantes. Une autre approche importante de ces formules est via les marches des crochets à poids. Nous discutons certains résultats dans cette direction. Enfin, nous présentons aussi une autre motivation à l'origine de ce travail: les $J$-fonctions du schéma d'Hilbert des points.

Highlights

Introduction and main resultsThe classical hook-length formula gives an elegant product formula for the number of standard Young tableaux
The hook-length formula states that if λ is a partition of n, fλ =
In the last fifteen years, deep relations have been uncovered between representation theory and the geometry of the Hilbert scheme of points in the complex affine plane Hilbn(C2)

Summary

Introduction and main results

The classical hook-length formula gives an elegant product formula for the number of standard Young tableaux. One way to prove the hook-length formula is by induction on n. It is easy to see that this is equivalent to the following branching rule for the hook lengths:. A bijective proof is presented of the following more general identity, called the weighted branching formula. C [λ] is the set of outer corners of λ, squares (i, j) ∈/ [λ] satisfying i = 1 or (i−1, j) ∈ [λ], and j = 1 or (i, j − 1) ∈ [λ] The motivation for this formula is as follows, see [22]. This extended abstract is based on papers [6], [7] and [18]

Motivation

Bijective proofs of weighted branching formulas

Weighted hook walks

Final remarks