On the depth of the invariants of the symmetric power representations of \(SL_2(\mathbb{F}_p)\)
The authors study the depth of the invariant ring of \(\text{SL}_2(p)\) acting on symmetric powers of the natural representation. In other words, the topic of this paper is the modular version of ``invariants of binary forms'', which is a recurrent theme in classical invariant theory. The main result is that the depth is exactly 3 if \(\text{SL}_2(p)\) acts on the \(n\)-th symmetric power, where \(2 \leq n < p\) and gcd\((n,p-1) \leq 2\). The authors also determine the depth for \(n = 3\) or 4 and \(p\) arbitrary. The result is always 3, except if \((n,p) \in \{(3,7),(4,5)\}\), where the invariant ring is Cohen-Macaulay. To obtain their results, the authors first reduce the question to the Borel subgroup \(B\) of \(\text{SL}_2(p)\), using a result of the reviewer [\textit{G. Kemper}, J. Algebra 215, No. 1, 330-351 (1999; Zbl 0934.13003)]. Then they explicitly give a homogeneous system of parameters for the invariants of \(B\) and prove that the first three polynomials of this system form a maximal regular sequence. The proof is achieved by using Gröbner bases, among other methods. In view of the fact that previously the cyclic groups were essentially the only ones for which the depth of the modular invariant ring was known [see \textit{G. Ellingsrud} and \textit{T. Skjelbred}, Compos. Math. 41, 233-244 (1980; Zbl 0438.13007)], this paper constitutes a very significant progress.
scientific article; zbMATH DE number 1348176
scientific article