Computation of invariants for reductive groups
Let \(G\) be a linear algebraic group defined over the field \(k\), \(\rho: G\to GL(V)\) a representation of \(G\) on a \(k\)-vector space of finite dimension \(n\) and \(A:=k[X_1, \dots, X_n]\) the coordinate ring of \(V\) with natural linear action of \(G\). Due to Hilbert, the invariant ring \(A^G: =\{a\in A\mid {}^ga=a\}\) is finitely generated if \(G\) is linearly reductive, i.e. if every finite dimensional \(kG\)-module is completely reducible. In the present paper the author presents an algorithm for computing generators of the invariant ring for arbitrary linearly reductive groups. His algorithm is based on Hilbert's original proof of 1890, which was considered non-constructive at that time. Under the hypotheses of certain conjectures in commutative algebra, the presented algorithm also gives improved upper bounds for the degrees of generators: In the case of \(k\) being of characteristic 0 and \(G\) linearly reductive of positive dimension, the degree bound significantly improves Popov's degree bound, but it depends on a conjecture of Eisenbud-Goto about degree bounds for syzygies of homogeneous prime ideals in \(A\). In the case of a finite group of order \(| G|\) not divisible by char \(k\), the author proves that Noether's degree bound holds, i.e. \(A^G\) can be generated by invariants of degree \(\leq| G|\), provided a so-called `subspaces conjecture' is true. Notice that Noether's bound is well known to be true for finite groups and \(\text{char }k=0\) of larger than \(| G|\). It is also known to be false for \(\text{char } k\mid| G|\). Recently the reviewer has proved with different methods, that Noether's bound holds whenever \(| G|\) is invertible in \(k\) [cf. \textit{P. Fleischmann}, Trans. Am. Math. Soc. 350, No. 4, 1703-1712 (1998; Zbl 0891.13002)].
computing generators of the invariant ring
scientific article; zbMATH DE number 1271738