Vector invariants for the two-dimensional modular representation of a cyclic group of prime order
Vector invariants for the two-dimensional modular representation of a cyclic group of prime order
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DOI：
10.1016/j.aim.2010.03.018
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发表时间：
2009-01
影响因子：
1.7
通讯作者：
H. Campbell;R. Shank;D. Wehlau
中科院分区：
文献类型：
--
作者：
H. Campbell;R. Shank;D. Wehlau
In this paper, we study the vector invariants of the 2-dimensional indecomposable representation V2of the cyclic group, Cp, of order p over a field F of characteristic p, [Formula: see text] . This ring of invariants was first studied by David Richman (1990) [20] who showed that the 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 (1997) in [3]. Later, Shank and Wehlau (2002) in [24] 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 [Formula: see text] . In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for [Formula: see text] . Further, our results provide a procedure for finding an explicit decomposition of F[mV2] into a direct sum of indecomposable Cp-modules. Finally, noting that our representation of Cpon V2is as the p-Sylow subgroup of SL2(Fp), we describe a generating set for the ring of invariants [Formula: see text] and show that (p+m−2)(p−1) is an upper bound for the Noether number, for m>2.
