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Degree bounds for modular covariants

Jonathan Elmer ORCID logo and Müfit Sezer
From the journal Forum Mathematicum


Let V,W be representations of a cyclic group G of prime order p over a field 𝕜 of characteristic p. The module of covariants 𝕜[V,W]G is the set of G-equivariant polynomial maps VW, and is a module over 𝕜[V]G. We give a formula for the Noether bound β(𝕜[V,W]G,𝕜[V]G), i.e. the minimal degree d such that 𝕜[V,W]G is generated over 𝕜[V]G by elements of degree at most d.

MSC 2010: 13A50

Communicated by Frederick R. Cohen

Funding statement: The second author is supported by a grant from TÜBITAK:119F181.


[1] A. Broer and J. Chuai, Modules of covariants in modular invariant theory, Proc. Lond. Math. Soc. (3) 100 (2010), no. 3, 705–735. 10.1112/plms/pdp044Search in Google Scholar

[2] H. E. A. Campbell and I. P. Hughes, Vector invariants of U2(𝐅p): A proof of a conjecture of Richman, Adv. Math. 126 (1997), no. 1, 1–20. 10.1006/aima.1996.1590Search in Google Scholar

[3] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. 10.2307/2372597Search in Google Scholar

[4] L. E. Dickson, On Invariants and the Theory of Numbers, Dover Publications, New York, 1966. Search in Google Scholar

[5] J. Elmer, Modular covariants of cyclic groups of order p, preprint (2019), 10.1016/j.jalgebra.2022.01.015Search in Google Scholar

[6] P. Fleischmann, M. Sezer, R. J. Shank and C. F. Woodcock, The Noether numbers for cyclic groups of prime order, Adv. Math. 207 (2006), no. 1, 149–155. 10.1016/j.aim.2005.11.009Search in Google Scholar

[7] M. Hochster and J. A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020–1058. 10.2307/2373744Search in Google Scholar

[8] I. Hughes and G. Kemper, Symmetric powers of modular representations, Hilbert series and degree bounds, Comm. Algebra 28 (2000), no. 4, 2059–2088. 10.1080/00927870008826944Search in Google Scholar

[9] M. Sezer and R. J. Shank, On the coinvariants of modular representations of cyclic groups of prime order, J. Pure Appl. Algebra 205 (2006), no. 1, 210–225. 10.1016/j.jpaa.2005.07.003Search in Google Scholar

[10] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. 10.4153/CJM-1954-028-3Search in Google Scholar

Received: 2019-07-25
Revised: 2020-01-30
Published Online: 2020-03-20
Published in Print: 2020-07-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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