# Euler's difference table and decomposition of tensor powers of adjoint representation of $A_n$ Lie algebra

## TL;DR

This paper presents an explicit formula for decomposing tensor powers of the adjoint representation of the $A_n$ Lie algebra using Euler's difference table, applicable when $2k
leq n+1$.

## Contribution

It introduces a novel application of Euler's difference table to derive explicit decompositions of tensor powers of the adjoint representation of $A_n$ Lie algebra.

## Key findings

- Explicit formula for tensor power decomposition derived
- Applicable for $2k
leq n+1$ cases
- Simplifies understanding of representation structure

## Abstract

By using of Euler's difference table, we obtain simple explicit formula for the decomposition of $k$-th tensor power of adjoint representation of $A_n$ Lie algebra at $2 k \le{n+1}$.