Differential posets and Smith normal forms

TL;DR

This paper explores a conjecture about the Smith normal forms of certain linear maps in differential posets, providing proofs for specific cases like the Young-Fibonacci lattice and its generalizations, revealing their integral structures.

Contribution

It introduces a conjecture on the Smith normal forms of up and down maps in differential posets and proves it for several important classes, advancing understanding of their algebraic properties.

Findings

Proved the conjecture for the Young-Fibonacci lattice YF.

Verified the conjecture for r-differential generalizations Z(r).

Explored consequences for Young's lattice Y and Cartesian products Y^r.

Abstract

We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice YF studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young's lattice Y and the r-differential Cartesian products Y^r.