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

Abstract

We present a full list of all representations of the special linear group SLn 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 SLn 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 Gröbner basis with respect to this order, inbetween usual Gröbner basis computation and computation of the Gröbner fan. Lastly, a modification of an algorithm by Xin for MacMahon partition analysis to compute Hilbert series.