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

Lukas Braun

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arXiv · 1812.01978

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

Lukas Braun

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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.

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Lukas Braun. 2018-12-05. Completing the classification of representations of $\mathrm{SL}_n$ with complete intersection invariant ring. https://arxiv.org/abs/1812.01978

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