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• W2008436722 abstract "In this paper, we study the vector invariants of the 2-dimensional indecomposable representation V 2 of the cyclic group, C p , of order p over a field F of characteristic p , F [ m V 2 ] C p . 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 F [ m V 2 ] C p . In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for F [ m V 2 ] C p . Further, our results provide a procedure for finding an explicit decomposition of F [ m V 2 ] into a direct sum of indecomposable C p -modules. Finally, noting that our representation of C p on V 2 is as the p -Sylow subgroup of SL 2 ( F p ) , we describe a generating set for the ring of invariants F [ m V 2 ] SL 2 ( F p ) and show that ( p + m − 2 ) ( p − 1 ) is an upper bound for the Noether number, for m > 2 ." @default.
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• W2008436722 date "2010-10-01" @default.
• W2008436722 modified "2023-09-29" @default.
• W2008436722 title "Vector invariants for the two-dimensional modular representation of a cyclic group of prime order" @default.
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• W2008436722 doi "https://doi.org/10.1016/j.aim.2010.03.018" @default.
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