On the combinatorics of tableaux -- Classification of lattices underlying Schensted correspondences
Dale R. Worley
TL;DR
This work develops a rigorous framework to classify Fomin lattices—modular lattices that support Robinson–Schensted type growth diagrams via weighted-differential posets—under a stringent but natural set of criteria. By combining local (distributive, unique-cover-modular, admissible weight) and global (two-dimensional grid) analyses, the authors reduce the problem to analyzing the join-irreducible poset $J$ and its weight function $w$, deriving a complete classification: the only lattices meeting the criteria correspond to Young's lattice $\,\mathbb{Y}$, the shifted Young's lattice $\,\mathbb{SY}$, the $k$-row strip $\mathbb{Y}_k$, and a quadrant-like case; none yield new Fomin lattices. A key outcome is the explicit relation between the diagram size, the differential degree $r$, and weights through $f(x)$ and $c(x)$, notably $n!\,r^n = \sum_{\text{diagrams }x\text{ of size }n} f(x)^2\,c(x)$. The results illuminate why known Fomin lattices occupy a tightly constrained landscape and set a foundation for future exploration when relaxing admissibility or modularity assumptions. The work thus advances understanding of growth diagrams, RS variants, and the lattice-theoretic underpinnings of combinatorial insertion procedures.
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. Disappointingly, we discover no new Fomin lattices.