# Completing the classification of representations of $\mathrm{SL}_n$ with complete intersection invariant ring

## TL;DR

This paper completes the classification of all representations of the special linear group $ ext{SL}_n$ over complex numbers that have a complete intersection invariant ring, using a combination of advanced algebraic and computational techniques.

## Contribution

The authors provide a complete list of $ ext{SL}_n$ representations with complete intersection invariant rings, extending Shmelkin's classification with new algorithms and methods.

## Key findings

- Full classification of $ ext{SL}_n$ representations with complete intersection invariant rings.
- Development of a new algorithm for monomial order selection in Gr"obner basis computation.
- Implementation of a modified MacMahon partition analysis algorithm for Hilbert series.

## Abstract

We present a full list of all representations of the special linear group $\mathrm{SL}_n$ over the complex numbers with complete intersection invariant ring, completing the classification of Shmelkin. For this task, we combine three techniques. Firstly, the graph method for invariants of $\mathrm{SL}_n$ developed by the author to compute invariants, covariants and explicit forms of syzygies. Secondly, a new algorithm for finding a monomial order such that a certain basis of an ideal is a Gr\"obner basis with respect to this order, inbetween usual Gr\"obner basis computation and computation of the Gr\"obner fan. Lastly, a modification of an algorithm by Xin for MacMahon partition analysis to compute Hilbert series.