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.

Jonathan Elmer

Suppose $\mathbb{F}$ is a field of prime characteristic $p$ and $E$ is a finite subgroup of the additive group $(\mathbb{F},+)$. Then $E$ is an elementary abelian $p$-group. We consider two such subgroups, say $E$ and $E'$, to be equivalent if there is an $\alpha\in\mathbb{F}^*:=\mathbb{F}\setminus\{0\}$ such that $E=\alpha E'$. In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants.

H. E. A. Campbell J. Chuai R. J. Shank D. L. Wehlau

We describe the graded characters and Hilbert functions of certain graded artinian Gorenstein quotients of the polynomial ring which are also representations of the symmetric group. Specifically, we look at those algebras whose socles are trivial representations and whose principal apolar submodules are generated by the sum of the orbit of a power of a linear form.

Anthony V. Geramita Andrew H. Hoefel David L. Wehlau

Consider a finite dimensional vector space $V$ over a finite field $\mathbb{F}_q$. We give a minimal generating set for the ring of invariants $\mathbb{F}_q[V \oplus V^*]^{\text{GL}(V)}$, and show that this ring is a Gorenstein ring but is not a complete intersection. These results confirm a conjecture of Bonnaf\'e and Kemper.

Yin Chen David L. Wehlau

We prove that the coindex of the box complex $\mathrm{B}(H)$ of a graph $H$ can be measured by the generalised Mycielski graphs which admit a homomorphism to it. As a consequence, we exhibit for every graph $H$ a system of linear equations solvable in polynomial time, with the following properties: If the system has no solutions, then $\mathrm{coind}(\mathrm{B}(H)) + 2 \leq 3$; if the system has solutions, then $\chi(H) \geq 4$. We generalise the method to other bounds on chromatic numbers using linear algebra.

Gord Simons Claude Tardif David Wehlau

Over a field of characteristic 0, every ring of invariants of any finite group is Cohen-Macaulay. This is not true for fields of positive characteristic. We consider permutation representations and their invariant rings over fields $\mathbb{F}_p$ of prime order. We give an efficient algorithm which for any given permutation representation, determines those primes $p$ for which the invariant ring over $\mathbb{F}_p$ is Cohen-Macaulay. Extensions to subgroups of reflection groups other than the symmetric group are indicated.

H. E. A. Campbell David L. Wehlau

We present a complete solution of the constraints for two-dimensional, N=2 supergravity in N=2 superspace. We obtain explicit expressions for the covariant derivatives in terms of the vector superfield $H^m$ and, for the two versions of minimal (2,2) supergravity, a chiral or twisted chiral scalar superfield $\phi$.

Marcus T. Grisaru Marcia E. Wehlau

Starting with the prepotential description of two-dimensional $(2,2)$ supergravity we use local supersymmetry transformations to go to light-cone gauge. We discuss properties of the theory in this gauge and derive Ward identities for correlation functions defined with respect to the induced supergravity action.

M. T. Grisaru M. E. Wehlau

We study two-dimensional N=2 supersymmetric actions describing general models of scalar and vector multiplets coupled to supergravity.

S. J. Gates, Jr. M. T. Grisaru M. E. Wehlau

The 16-year old Blaise Pascal found a way to determine if 6 points lie on a conic using a straightedge. Nearly 400 years later, we develop a method that uses a straightedge to check whether 10 points lie on a plane cubic curve.

Will Traves David Wehlau