Completing the Classification of Representations of SLn with Complete Intersection Invariant Ring

Lukas Braun

doi:10.1007/s00031-020-09605-0

Abstract

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 Gröbner basis with respect to this order, in between 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.

Highlights

Let A = C[x1, . . . , xn]/I be an affine algebra
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
We say that A is regular if it is isomorphic to some polynomial ring, a hypersurface if I is a principal ideal, a complete intersection if I is minimally generated by hd(A) = n − dim(A) elements, where dim(A) is the Krull and hd(A) the homological dimension of A

Summary

Introduction

A representation of SLn has a complete intersection invariant ring of homological dimension ≥ 2 if and only if it or its dual is contained in the following table. This is done with the help of covariant rings of subrepresentations and toric invariant rings thereof. The resulting algebras A are ’almost’ toric — the ideal of syzygies being generated by binomials as well as trinomials, such that a torus of dimension dim(A) − 1 acts on them, compare [3]. See [17] for a related approach that uses both a graphical method for the invariants and a toric degeneration of the invariant ring

The basic techniques

The Crosshair-sieve-algorithm

Hilbert series by MacMahon partition analysis

The single cases

The serial cases

Rings of Covariants