# Connected Operators for the Totally Asymmetric Exclusion Process

## TL;DR

This paper provides a complete algebraic characterization of the connected operators commuting with the TASEP Markov matrix, confirming a previously conjectured combinatorial formula and advancing the understanding of its algebraic structure.

## Contribution

It introduces a purely algebraic derivation of the connected operators' structure and proves a conjectured combinatorial formula for these operators in TASEP.

## Key findings

- Proved the combinatorial formula for connected operators.
- Established an algebraic framework for TASEP operators.
- Enhanced understanding of TASEP's algebraic structure.

## Abstract

We fully elucidate the structure of the hierarchy of the connected operators that commute with the Markov matrix of the Totally Asymmetric Exclusion Process (TASEP). We prove for the connected operators a combinatorial formula that was conjectured in a previous work. Our derivation is purely algebraic and relies on the algebra generated by the local jump operators involved in the TASEP. Keywords: Non-Equilibrium Statistical Mechanics, ASEP, Exact Results, Algebraic Bethe Ansatz.