Modular covariants of cyclic groups of order p

Article


Elmer, J. 2022. Modular covariants of cyclic groups of order p. Journal of Algebra. 598, pp. 134-155. https://doi.org/10.1016/j.jalgebra.2022.01.015
TypeArticle
TitleModular covariants of cyclic groups of order p
AuthorsElmer, J.
Abstract

Let G be a cyclic group of order p and let V, W be kG-modules. We study the modules of covariants k[V,W]^G = (S(V^∗) ⊗ W)^G . Recall that G has exactly p inequivalent indecomposable kG-modules, denoted V_n (n = 1, . . . , p) and V_n has dimension n. For any n, we show that k[V_2,V_n]^G is a free k[V_2]^G- module (recovering a result of Broer and Chuai) and we give an explicit set of covariants generating k[V_2,V_n]^G freely over k[V_2]^G . For any n, we show that k[V_3,V_n]^G is a Cohen-Macaulay k[V_3]^G -module (again recovering a result of Broer and Chuai) and we give an explicit set of covariants which generate k[V 3 , V n ] G freely over a homogeneous system of parameters for k[V_3]^G . We also use our results to compute a minimal generating set for the transfer ideal of k[V_3]^G over a homogeneous system of parameters.

KeywordsModular invariant theory; Covariants; Free module; Cohen-Macaulay; Hilbert series
PublisherElsevier
JournalJournal of Algebra
ISSN0021-8693
Electronic1090-266X
Publication dates
Online03 Feb 2022
Print15 May 2022
Publication process dates
Deposited03 Feb 2022
Submitted14 Nov 2019
Accepted29 Jan 2022
Output statusPublished
Publisher's version
License
File Access Level
Open
Accepted author manuscript
License
File Access Level
Restricted
Copyright Statement

© 2022 The Author. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Digital Object Identifier (DOI)https://doi.org/10.1016/j.jalgebra.2022.01.015
Web of Science identifierWOS:000793251800008
LanguageEnglish
Permalink -

https://repository.mdx.ac.uk/item/89q79

Download files


Publisher's version
1-s2.0-S002186932200031X-main.pdf
License: CC BY-NC-ND 4.0
File access level: Open

  • 67
    total views
  • 11
    total downloads
  • 0
    views this month
  • 0
    downloads this month

Export as

Related outputs

The separating variety for 2 x 2 matrix invariants
Elmer, J. 2024. The separating variety for 2 x 2 matrix invariants. Linear and Multilinear Algebra. 72 (3), pp. 389-411. https://doi.org/10.1080/03081087.2022.2158300
The separating variety for matrix semi-invariants
Elmer, J. 2023. The separating variety for matrix semi-invariants. Linear Algebra and its Applications. 674, pp. 466-492. https://doi.org/10.1016/j.laa.2023.06.012
The relative Heller operator and relative cohomology for the Klein 4-group
Elmer, J. 2022. The relative Heller operator and relative cohomology for the Klein 4-group. Communications in Algebra. 50 (4), pp. 1518-1534. https://doi.org/10.1080/00927872.2021.1984496
Degree bounds for modular covariants
Elmer, J. and Sezer, M. 2020. Degree bounds for modular covariants. Forum Mathematicum. 32 (4), pp. 905-910. https://doi.org/10.1515/forum-2019-0196
Locally finite derivations and modular coinvariants
Elmer, J. and Sezer, M. 2018. Locally finite derivations and modular coinvariants. Quarterly Journal of Mathematics. 69 (3), pp. 1053-1062. https://doi.org/10.1093/qmath/hay013
Symmetric powers and modular invariants of elementary abelian p-groups
Elmer, J. 2017. Symmetric powers and modular invariants of elementary abelian p-groups. Journal of Algebra. 492, pp. 157-184. https://doi.org/10.1016/j.jalgebra.2017.07.020
On separating a fixed point from zero by invariants
Elmer, J. and Kohls, M. 2017. On separating a fixed point from zero by invariants. Communications in Algebra. 45 (1), pp. 371-375. https://doi.org/10.1080/00927872.2016.1175465
Zero-separating invariants for finite groups
Elmer, J. and Kohls, M. 2014. Zero-separating invariants for finite groups. Journal of Algebra. 411, pp. 92-113. https://doi.org/10.1016/j.jalgebra.2014.03.044
Separating invariants for arbitrary linear actions of the additive group
Dufresne, E., Elmer, J. and Sezer, M. 2014. Separating invariants for arbitrary linear actions of the additive group. Manuscripta Mathematica. 143 (1), pp. 207-219. https://doi.org/10.1007/s00229-013-0625-y
Separating Invariants for the Basic G_a actions
Elmer, J. and Kohls, M. 2012. Separating Invariants for the Basic G_a actions. Proceedings of the American Mathematical Society. 140 (1), pp. 135-146.
On the depth of separating invariants for finite groups
Elmer, J. 2012. On the depth of separating invariants for finite groups. Beitrage zur Algebra und Geometrie. 53 (1), pp. 31-39.
The Cohen-Macaulay property of separating invariants of finite groups
Dufresne, E., Elmer, J. and Kohls, M. 2009. The Cohen-Macaulay property of separating invariants of finite groups. Transformation Groups. 14 (4), pp. 771-785.
Depth and detection in modular invariant theory
Elmer, J. 2009. Depth and detection in modular invariant theory. Journal of Algebra. 322 (5), pp. 1653-1666. https://doi.org/10.1016/j.jalgebra.2009.04.036
On the depth of modular invariant rings for the groups C_p x C_p
Elmer, J. and Fleischmann, P. 2009. On the depth of modular invariant rings for the groups C_p x C_p. in: Symmetry and Spaces: in honour of Gerry Schwarz Birkhauser Boston.
Associated primes for cohomology modules
Elmer, J. 2008. Associated primes for cohomology modules. Archiv der Mathematik. 91 (6), pp. 481-485. https://doi.org/10.1007/s00013-008-2902-7
Zero-separating invariants for linear algebraic groups
Elmer, J. and Kohls, M. 2016. Zero-separating invariants for linear algebraic groups. Proceedings of the Edinburgh Mathematical Society. 59 (4), pp. 911-924. https://doi.org/10.1017/S0013091515000322