Geodesic
Invariants for the modular cyclic group of prime order via classical invariant theory
Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 775-803

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DOI

Let F be any field of characteristic p. It is well-known that there are exactly p inequivalent indecomposable representations V1​,V2​,...,Vp​ of Cp​ defined over F. Thus if V is any finite dimensional Cp​-representation there are non-negative integers 0≤n1​,n2​,...,nk​≤p−1 such that V≅⊕i=1k​Vni​+1​. It is also well-known there is a unique (up to equivalence) d+1 dimensional irreducible complex representation of SL2​(C) given by its action on the space Rd​ of d forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring of Cp​-invariants F[⊕i=1k​Vni​+1​]Cp​ to the computation of the classical ring of invariants (or covariants) C[R1​⊕(⊕i=1k​Rni​​)]SL2​(C). This shows that the problem of computing modular Cp​ invariants is equivalent to the problem of computing classical SL2​(C) invariants. This allows us to compute for the first time the ring of invariants for many representations of Cp​. In particular, we easily obtain from this generators for the rings of vector invariants F[mV2​]Cp​, F[mV3​]Cp​ and F[mV4​]Cp​for all m∈N. This is the first computation of the latter two families of rings of invariants.
DOI : 10.4171/jems/376
Classification : 13-XX, 20-XX, 00-XX
Keywords: Modular invariant theory, cyclic group, classical invariant theory, Roberts' isomorphism 
David L. Wehlau. Invariants for the modular cyclic group of prime order via classical invariant theory. Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 775-803. doi: 10.4171/jems/376
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     title = {Invariants for the modular cyclic group of prime order via classical invariant theory},
     journal = {Journal of the European Mathematical Society},
     pages = {775--803},
     year = {2013},
     volume = {15},
     number = {3},
     doi = {10.4171/jems/376},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/376/}
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