# Another proof of the Nowicki conjecture

## TL;DR

This paper provides a new proof of the Nowicki conjecture, which describes the generators of the algebra of constants for a specific derivation on a polynomial algebra, using representation theory of GL_2(K).

## Contribution

It offers an alternative proof of the Nowicki conjecture leveraging representation theory, expanding the methods used in previous proofs.

## Key findings

- Confirmed that the algebra of constants is generated by specified elements.
- Demonstrated the effectiveness of representation theory in invariant algebra problems.
- Provided a new perspective on the structure of derivation invariants.

## Abstract

Let $K[X_d,Y_d]=K[x_1,\ldots,x_d,y_1,\ldots,y_d]$ be the polynomial algebra in $2d$ variables over a field $K$ of characteristic 0 and let $\delta$ be the derivation of $K[X_d,Y_d]$ defined by $\delta(y_i)=x_i$, $\delta(x_i)=0$, $i=1,\ldots,d$. In 1994 Nowicki conjectured that the algebra $K[X_d,Y_d]^{\delta}$ of constants of $\delta$ is generated by $X_d$ and $x_iy_j-y_ix_j$ for all $1\leq i<j\leq d$. The affirmative answer was given by several authors using different ideas. In the present paper we give another proof of the conjecture based on representation theory of the general linear group $GL_2(K)$.