Summary
For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p 2, and show that the Noether number for the representation is p 2 + p−3. We then use the constructed invariants to explicitly describe the decomposition of the symmetric algebra as a module over the group ring, confirming the Periodicity Conjecture of Ian Hughes and Gregor Kemper for this case. In the final section, we use our results to compute the Hilbert series for the corresponding ring of invariants together with some other related generating functions.
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Shank, R.J., Wehlau, D.L. (2010). Decomposing Symmetric Powers of Certain Modular Representations of Cyclic Groups. In: Campbell, H., Helminck, A., Kraft, H., Wehlau, D. (eds) Symmetry and Spaces. Progress in Mathematics, vol 278. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4875-6_9
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DOI: https://doi.org/10.1007/978-0-8176-4875-6_9
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