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Jonathan Elmer

Publications and source records attributed to Jonathan Elmer.

26 records · Page 2Linked to original sources
arXiv2017

Symmetric powers and modular invariants of elementary abelian p-groups

Let $E$ be a elementary abelian $p$-group of order $q=p^n$. Let $W$ be a faithful indecomposable representation of $E$ with dimension 2 over a field $k$ of characteristic $p$, and let $V= S^m(W)$ with $m<q$. We prove that the rings of invariants $k[V]^E$ are generated by elements of degree at most $q$ and relative transfers. This extends recent work of Wehlau on modular invariants of cyclic groups of order $p$. If $m<p$ we prove that $k[V]^E$ is generated by invariants of degree at most $2q-3$, extending a result of Fleischmann, Sezer, Shank and Woodcock for cyclic groups of order $p$. Our methods are primarily representation-theoretic, and along the way we prove that for any $d<q$ with $d+m \geq q$, $S^d(V^*)$ is projective relative to the set of subgroups of $E$ with order at most $m$, and that the sequence $S^d(V^*)_{d \in \mathbb{N}}$ is periodic with period $q$, modulo summands which are projective relative to the same set of subgroups. These results extend results of Almkvist and Fossum on cyclic groups of prime order.

math.RT↗
arXiv2016

Locally finite derivations and modular coinvariants

We consider a finite dimensional $\kk G$-module $V$ of a $p$-group $G$ over a field $\kk$ of characteristic $p$. We describe a generating set for the corresponding Hilbert Ideal. In case $G$ is cyclic this yields that the algebra $\kk[V]_G$ of coinvariants is a free module over its subalgebra generated by $\kk G$-module generators of $V^*$. This subalgebra is a quotient of a polynomial ring by pure powers of its variables. The coinvariant ring was known to have this property only when $G$ was cyclic of prime order, \cite{SezerCoinv}. In addition, we show that if $G$ is the Klein 4-group and $V$ does not contain an indecomposable summand isomorphic to the regular module, then the Hilbert Ideal is a complete intersection, extending a result of the second author and R. J. Shank \cite{SezerShank}.

math.AC↗
arXiv2015

On separating fixed points from zero by invariants

Assume a fixed point v in a G-module V can be separated from zero by a homogeneous invariant of degree dp^r where p>0 is the characteristic of the ground field k and p, d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r , which we obtain explicitly from f. It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power.

math.AC↗
arXiv2014

Zero-separating invariants for linear algebraic groups

Let $G$ be a linear algebraic group over a field $k$, and let $V$ be a $G$-module. Recall that the nullcone of $(G,V)$ is the set of points $v$ in $V$ with the property that $f(v)=0$ for every positive degree homogeneous invariant $f$ in $k[V]^G$. We define numbers $δ(G,V)$ and $σ(G,V)$ associated with a given representation as follows: $δ(G,V)$ is the smallest number $d$ such that, for any point $v$ in $V^G$ outside the nullcone, there exists an invariant $f$ of degree at most $d$ such that $f(v)$ is not zero; $σ(G,V)$ is the same thing with $V^G$ replaced by $V$. If k has positive characteristic, we show that $δ(G,V)$ is infinite for all subgroups of $GL_2(k)$ containing a unipotent subgroup, and that $σ(G,V)$ is finite if and only if $G$ is finite. If $k$ has characteristic zero we show that $δ(G,V)=1$ for all linear algebraic groups and that if $σ(G,V)$ is finite then the connected component of $G$ is unipotent.

math.AC↗
arXiv2013

Zero-separating invariants for finite groups

We fix a field $\kk$ of characteristic $p$. For a finite group $G$ denote by $δ(G)$ and $σ(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\kk$ and any $v\in V^{G}\setminus\{0\}$ or $v\in V\setminus\{0\}$ respectively, there exists a homogeneous invariant $f\in\kk[V]^{G}$ of positive degree at most $d$ such that $f(v)\ne 0$. Let $P$ be a Sylow-$p$-subgroup of $G$ (which we take to be trivial if the group order is not divisble by $p$). We show that $δ(G)=|P|$. If $N_{G}(P)/P$ is cyclic, we show $σ(G)\ge|N_{G}(P)|$. If $G$ is $p$-nilpotent and $P$ is not normal in $G$, we show $σ(G)\le \frac{|G|}{l}$, where $l$ is the smallest prime divisor of $|G|$. These results extend known results in the non-modular case to the modular case.

math.AC↗
arXiv2010

Separating invariants for the basic G_a-actions

We explicitly construct a finite set of separating invariants for the basic $\Ga$-actions. These are the finite dimensional indecomposable rational linear representations of the additive group $\Ga$ of a field of characteristic zero, and their invariants are the kernel of the Weitzenböck derivation $D_{n}=x_{0}\frac{\partial}{\partial{x_{1}}}+...+ x_{n-1}\frac{\partial}{\partial{x_{n}}}$.

math.AC↗
arXiv2009

The Cohen-Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply that the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Furthermore, we show that, for a $p$-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.

math.AC↗