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

We present a full list of all representations of the special linear group SLn over the complex numbers with complete intersection invariant ring of homological dimension greater than or equal to two, 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 Grobner basis with respect to this order, in between usual Grobner basis computation and computation of the Grobner fan. Lastly, a modification of an algorithm by Xin for MacMahon partition analysis to compute Hilbert series.