The Smith Normal Form of a Matrix Associated with Young's Lattice

TL;DR

This paper proves a conjecture regarding the Smith normal form of a specific operator linked to Young's lattice, connecting combinatorial structures with algebraic properties of symmetric functions.

Contribution

It establishes the Smith normal form for the operator associated with Young's lattice, confirming a conjecture by Miller and Reiner for this case.

Findings

Confirmed the Smith normal form for the operator in Young's lattice

Connected the operator to differential operators on symmetric functions

Provided algebraic insights into the structure of differential posets

Abstract

We prove a conjecture of Miller and Reiner on the Smith normal form of the operator $DU$ associated with a differential poset for the special case of Young's lattice. Equivalently, this operator can be described as $\frac{\partial}{\partial p_1}p_1$ acting on homogeneous symmetric functions of degree $n$.