2009/01/19 by H. E. A. Campbell, Campbell, H. E. A., R. James Shank +3
Computer Science · Mathematics · #13A50 #Algebraic Geometry and Number Theory #Coding theory and cryptography #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.0901.2811
openalex publication_date 2009/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the vector invariants, \bfF[m V2]Cp, of the 2-dimensional indecomposable representation V2 of the cylic group, Cp, of order p over a field \bfF of characteristic p. This ring of invariants was first studied by David Richman \citerichman who showed that this ring required a generator of degree m(p-1), thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p=2. This conjecture was proved by Campbell and Hughes in \citecampbell-hughes. Later, Shank and Wehlau in \citecmipg determined which elements in Richman's generating set were redundant thereby producing a minimal generating set. We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants \bfF[m V2]Cp. In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for \bfF[m V2]Cp. Further, our techniques also serve to give an explicit decomposition of \bfF[m V2] into a direct sum of indecomposable Cp-modules. Finally, noting that our representation of Cp on V2 is as the p-Sylow subgroup of SL2(\bf Fp), we are able to determine a generating set for the ring of invariants of \bfF[m V2]^SL2(\bf Fp).