# The Algebra of Schur Operators

## TL;DR

This paper explores the algebraic structure of Schur operators acting on partitions, providing a comprehensive list of relations that define their algebraic behavior, which enhances understanding of their combinatorial and algebraic properties.

## Contribution

It offers a complete characterization of the relations in the algebra generated by Schur operators, advancing the algebraic understanding of the plactic monoid representation.

## Key findings

- Complete list of relations in the algebra of Schur operators
- Clarification of how Schur operators act on partitions
- Insights into the algebraic structure of the plactic monoid

## Abstract

We study a representation of the (local) plactic monoid given by Schur operators $u_i$, which act on partitions by adding a box in column $i$ (if possible). In particular, we give a complete list of the relations that hold in the algebra of Schur operators.