Publications

Degree bounds for modular covariants (with Müfit Sezer)

Preprint submitted: pdf

Using some ideas developed in the paper below, we determine the Noether bound (minimum degree d for which there is a generating set in degrees at most d) for the module of covariants k[V,W]^G over k[V]^G when G is a cyclic group of order p.

Modular covariants of cyclic groups of order p

Preprint submitted: pdf

For G a group and V,W kG-modules, the module of covariants is the set of G-equivariant polynomial maps from V to W. This is k[V]^G-module. We give a generating set for this module when G is a cyclic group of order p, V is an indecomposable module of dimension 2 or 3, and W is an arbitrary indecomposable G-module.

On the depth of quotients of modular invariant rings by transfer ideals (with Müfit Sezer)

Preprint submitted: pdf

We prove that for a p-group, the depth of K[V]^G/I where I is a sum of images of transfers is bounded below by dim(V^G). We compute the depth of K[V]^G/I^G_1 for every indecomposable representation of the Klein four-group.

Symmetric powers and modular invariants of elementary abelian p-groups

Published version: Journal of Algebra 492,pages 157-184 (2017).

Preprint version: pdf

Let G be an elementary abelian p-group of order q and V a faithful modular representation of dimension 2. We describe how to decompose S(S^m(V)*) into indecomposable summands. We use this decomposition to prove, among other things, that the rings of invariants K[S^m(V)]^G are generated by invariants of degree at most 2q-3.

Locally finite derivations and modular coinvariants (with Müfit Sezer)

Published version: Quarterly Journal of Mathematics 69 issue 3, pages 1053-1062 (2018).

Preprint version: pdf

We prove that the algebra of coinvariants of a modular representation of a cyclic-p-group is free as a module over the subalgebra generated by so-called “terminal” variables. We also prove some weaker results for arbitrary p-groups.

On separating fixed points from zero by invariants (with Martin Kohls)

Published version: Communications in Algebra 45 issue 1, pages 371–375 (2017).

Preprint version: pdf

Let G be any group and V a representation of G over a field of characteristic p. We prove that if v is a G-fixed point, then the invariant f of smallest degree such that f(v)≠0 has degree a power of p.

Zero-separating invariants for linear algebraic groups (with Martin Kohls)

Preprint version: pdf

We extend the investigation of the paper below to infinite algebraic groups.

Zero-separating invariants for finite groups (with Martin Kohls)

Published version: Journal of Algebra 411, pages 92-113 (2014).

Preprint version: pdf

Let G be a finite group and K a field of characteristic p dividing |G|. We investigate the numbers δ(G) and σ(G). Here, for a kG-module V, δ(G,V) is the minimum degree such that every fixed point in V can be separated from zero by an invariant of degree at most δ(G,V), and δ(G) is the supremum taken over all finitely-generated kG-module, while σ(G,V) is the minimum degree such that every point in V (outside the nullcone) can be separated from zero by an invariant of degree at most σ(G,V), and σ(G) is defined analogously.

Separating invariants for arbitrary linear actions of the additive group (with Emilie Dufresne and Müfit Sezer)

Published version: Manuscripta Mathematica 143, pages 207-219 (2014).

Preprint version: pdf

We extend the results of the paper below to decomposable representations.

Separating invariants for the basic G_a actions (with Martin Kohls)

Preprint version: pdf

We describe an explicit separating algebra for the classical covariants of binary forms in characteristic zero.

On the depth of separating algebras of finite groups

Published version: Contributions to Algebra and Geometry 53, issue 1, pages 31-39 (2012).

Preprint version: pdf

Extending ideas in the paper below, we show that, under certain conditions, the depth of a separating subalgebra is bounded above by the depth of the ring of invariants. In particular this always holds for a cyclic group of order p.

On the Cohen-Macaulay property of separating algebras (with Emilie Dufresne and Martin Kohls)

Published version: Transformation Groups 14, pages 771-785 (2009).

Preprint version: pdf

We show that in the modular case, there are representations of of finite groups which not only have non-Cohen-Macaulay rings of invariants, but whose ring of invariants do not contain a Cohen-Macaulay separating subalgebra, answering negatively a question of G. Kemper.

Depth and detection in modular invariant theory

Published version: Journal of Algebra 322, issue 5, pages 1653-1666 (2009).

Preprint version: pdf

We find conditions for the depth of a modular ring of invariants to attain its minimum which are both necessary and sufficient. We apply these to classify the representations of C_2 x C_2 whose rings of invariants have minimal depth.

Associated primes for cohomology modules

Published version: Archiv der Math 91, issue 6, pages 481-485 (2008).

Preprint version: pdf

We show that the associated primes of the modules H*(G,R), where G is a finite group and R=S(V*) for V a modular representation of G, are radicals of certain transfer ideals.

On the depth of modular invariant rings for C_p x C_p (with Peter Fleischmann)

Published version: In Symmetry and Spaces: In honour of Gerry Schwarz, pages 45-63.

Preprint version: pdf

Contains the most significant results of my thesis. We apply known sufficient conditions (due to Fleischmann, Kemper and Shank) for the depth of a modular ring of invariants to attain its minimum value to groups of the form C_p x C_p. In particular we show that the minimum is attained for all but one indecomposable representation of C_2 x C_2.