Up- and Down-Operators on Young's Lattice

TL;DR

This paper characterizes the algebra generated by up- and down-operators on Young's lattice, showing it can be described with only quadratic relations, advancing understanding of their algebraic structure.

Contribution

It provides a quadratic presentation of the algebra generated by both up- and down-operators on Young's lattice, extending previous work on relations among these operators.

Findings

The algebra can be presented with only quadratic relations.

Relations of degree at most 4 suffice for up-operators alone.

The combined algebra's structure is fully characterized by quadratic relations.

Abstract

The up-operators $u_i$ and down-operators $d_i$ (introduced as Schur operators by Fomin) act on partitions by adding/removing a box to/from the $i$th column if possible. It is well known that the $u_i$ alone satisfy the relations of the (local) plactic monoid, and the present authors recently showed that relations of degree at most 4 suffice to describe all relations between the up-operators. Here we characterize the algebra generated by the up- and down-operators together, showing that it can be presented using only quadratic relations.