Abstract
We consider the invariant ring for an indecomposable representation of a cyclic group of order p 2 over a field \({\mathbb F}\) of characteristic p. We describe a set of \({\mathbb F}\)-algebra generators of this ring of invariants, and thus derive an upper bound for the largest degree of an element in a minimal generating set for the ring of invariants. This bound, as a polynomial in p, is of degree two.
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References
Alperin J.L. (1986). Local Representation Theory, Cambridge Studies in Advanced Mathmatics 11. Cambridge University Press, Cambridge
Eisenbud D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics 150. Springer, New York
Fine N.J. (1947). Binomial coefficients modulo a prime. Am. Math. Monthly 54: 589–592
Fleischmann P., Kemper G. and Shank R.J. (2004). On the depth of cohomology modules. Q. J. Math. 55(2): 167–184
Fleischmann P., Sezer M., Shank R.J. and Woodcock C.F. (2006). The Noether numbers for cyclic groups of prime order. Adv. Math. 207: 149–155
Hardy G.H. and Wright E.M. (1979). An Introduction to the Theory of Numbers, 5th edn. Oxford Science Publications, Oxford University Press, Oxford
Hughes I. and Kemper G. (2000). Symmetric power of modular representations, Hilbert series and degree bounds. Commun. Algebra 28: 2059–2089
Neusel M.D. (2001). The transfer in the invariant theory of modular permutation representations. Pac. J. Math. 199: 121–136
Neusel M.D. (1997). Invariants of some Abelian p-groups in characteristic p. Proc. AMS 125: 1921–1931
Neusel M.D. (2007). Degree bounds. An invitation to postmodern invariant theory. Topol. Appl. 154: 792–814
Neusel, M.D., Smith, L.: Invariant theory of finite groups, Math. Surv. Monogr., vol. 94, Am. Math. Soc., Providence, RI (2002)
Richman D. (1996). Invariants of finite groups over fields of characteristic p. Adv. Math. 124: 25–48
Shank, R.J., Wehlau, D.L.: Decomposing symmetric powers of certain modular representations of cyclic groups. IMS Technical Report UKC/IMS/05/13. http://www.kent.ac.uk/IMS/personal/rjs/
Symonds P. (2007). Cyclic group actions on polynomial rings. Bull. Lond. Math. Soc. 39: 181–188
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Neusel, M.D., Sezer, M. The invariants of modular indecomposable representations of \({{\mathbb Z}_{p^2}}\) . Math. Ann. 341, 575–587 (2008). https://doi.org/10.1007/s00208-007-0203-2
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DOI: https://doi.org/10.1007/s00208-007-0203-2