On the combinatorics of tableaux -- Classification of lattices underlying Schensted correspondences

TL;DR

This paper classifies all distributive lattices suitable for constructing Robinson-Schensted algorithms via Fomin's growth diagram method, revealing a new class of lattices that cannot be used for such algorithms.

Contribution

It provides a complete classification of distributive lattices underlying Schensted correspondences and introduces a new class of Fomin lattices with limited applicability.

Findings

All known Fomin lattices satisfy the criteria, except for Young-Fibonacci lattices.

A new class of Fomin lattices is discovered.

The new lattices cannot be used to construct Robinson-Schensted algorithms.

Abstract

The celebrated Robinson-Schensted algorithm and each of its variants that have attracted substantial attention can be constructed using Fomin's "growth diagram" construction from a modular lattice that is also a weighted-differential poset. We classify all such lattices that meet certain criteria; the main criterion is that the lattice is distributive. Intuitively, these criteria seem excessively strict, but all known Fomin lattices satisfy all of these criteria, with the sole exception of one family that is not even distributive, the Young-Fibonacci lattices and cartesian products involving them. We discover a new class of Fomin lattices, but unfortunately they cannot be used to construct Robinson-Schensted algorithms.