An isomorphism problem such as the graph isomorphism problem can be formulated as an orbit problem: Given a linear algebraic group G over a field k, a representation V, and two elements v,w ∈ V, do v,w lie in the same G-orbit? For every d we will construct a k-category C_d(V) such that elements in V are objects of C_d(V). If two elements v,w ∈ V have the same G-orbit, then v and w are isomorphic objects in C_d(V). There exists an efficient algorithm to test whether two objects in C_d(V) are isomorphic. Applied to graphs, this yields a polynomial time algorithm which is often able to distinguish non-isomorphic graphs. The algorithm is at least as good as the higher-dimensional Weisfeiler-Lehman algorithm. Cai, Fürer and Immerman constructed graphs that cannot be distinguished in polynomial time with the Weisfeiler-Lehman method. For k = F₂, our algorithm can distinguish these kind of graphs.

Roberts, Freudenburg, and Daigle and Freudenburg have given the smallest counterexamples to Hilbert's fourteenth problem. Each arises as the ring of invariants of an additive group action on a polynomial ring over a field of characteristic zero, and thus, each corresponds to the kernel of a locally nilpotent derivation. In positive characteristic, additive group actions correspond to locally finite iterative higher derivations, a more restrictive notion. We set up characteristic-free analogs of the three examples mentioned above, and show that, contrary to characteristic zero, in every positive characteristic, the invariant rings are finitely generated.

Joint work with Andreas Maurischat.

Suppose G is a finite group acting linearly on a k-vector space V and consider the ring of invariants k[V]^G. A separating algebra is a subalgebra A of k[V]^G with the same orbit separation properties.

There has been much recent interest in separating algebras, as it is often easier to compute generating sets for separating algebras rather than their parent invariant rings. In the non-modular case, k[V]^G is a Cohen-Macaulay ring, and so the inequality depth(A) ≤ depth(k[V]^G) holds trivially for all separating algebras A ⊂ k[V]^G. In this talk we will give examples in the modular case where this inequality also holds.

We study G_a-actions on certain affine varieties V which admit a dominant morphism to affine space Aⁿ. In particular, V is a variety whose coordinate ring B is the simple algebraic extension of the polynomial ring R defined by B=R[z], where zⁿ ∈ R for n ≥ 2. In many cases, our methods show either that V admits no non-trivial G_a-actions, or that the only Z_n-homogeneous actions are those which lift from Aⁿ. Important examples include the Russell cubic threefold X ⊂ A⁴, defined by x+x²y+z²+t³=0, and the Pham-Brieskorn varieties V ⊂ Aⁿ, defined by x₁^e₁+…+x_n^e_n=0 for integers e_i ≥ 2. By studying the locally nilpotent derivations (LNDs) of B=R[z] relative to the affine ring R, we reveal a surprisingly effective general approach to understanding large classes of these varieties. In particular, our work features:

1. criteria to determine that certain rings are rigid;
2. a concise new proof that the Russell cubic X is not isomorphic to affine space;
3. proofs for rigidity for a large family of Pham-Brieskorn threefolds and related varieties; and
4. numerical criteria to determine when a homogeneous LND of the polynomial ring A[x₁,...,x_n] over A contains a variable x_i in its kernel.

This talk will discuss joint work with Lucy Moser-Jauslin.

Since Lagrange's seminal work in 1770, resolvent sextics have been used to study the quintic equation. One of the most interesting is the "noteworthy covariant resolvent discovered by Perrin and independently by McClintock" [L. E. Dickson]. We shall discuss the remarkable properties of this resolvent in light of the invariant theory of F₂₀, the Frobenius group of order 20 acting as a group of permutations on five variables, and SL₂ acting on binary forms.

Consider the representations of an algebraic group G. In general, polynomial invariant functions may fail to separate orbits. The invariant subring may not be finitely generated, or the number and complexity of the generators may grow rapidly with the size of the representation. We instead study "constructible" functions defined by straight line programs in the polynomial ring, with a new "quasi-inverse" that computes the inverse of a function where defined.

We write straight line programs defining constructible functions that separate the orbits of G. The number of these programs and their length have polynomial bounds in the parameters of the representation.

For an action of a group G on an F-vector space V, we consider the invariant ring F[V ⊕V^*]^G. We are particularly interested in the case where V = F_qⁿ and G is the group U_n of all upper unipotent matrices or the group B_n of all invertible upper triangular matrices.

In fact, we determine F[V ⊕V^*]^G for G = U_n and G = B_n. The result is a complete intersection for all values of n and q. We get explicit lists of generating invariants and the relations between them. This makes an addition to the rather short list of "doubly parametrized" series of group actions whose invariant rings are known to have a uniform description.

This talk is about joint work with Cédric Bonnafé.

Degree bounds for invariants are an old topic. The Noether bound says that in the non-modular case, invariants up to degree at most the group order generate the invariant ring. This bound has been sharpened in many ways. Today, many people prefer to consider separating invariants, instead of a full set of generators. This has many advantages. For example, the Noether bound always holds for separating invariants, even in the modular case. The goal of this talk is to give similar sharpenings for the Noether bound for separating invariants, as they exist for generating invariants. One result is that for p-groups in characteristic p, the Noether bound for separating invariants is sharp.

Separating invariants have been studied extensively in the last decade. They have nice properties that generating sets (as an algebra) do not have, especially in the modular case. Nevertheless, many results on separating invariants require that the underlying field should have sufficiently many elements, which seems to be a very natural condition. In this talk, we will restrict our attention only to the case where the underlying field is prime and discuss the problem of finding separating invariants.

We consider a finite dimensional indecomposable modular representation of a cyclic p-group and we give a recursive description of an associated separating set: We show that a separating set for a representation can be obtained by adding, to a separating set for any subrepresentation, some explicitly defined invariant polynomials.

Suppose that G is a finite group, F is a field and V is finite dimensional representation of G over F. The action of G on V induces an action on the dual V^* which extends to an action by algebra automorphisms on the symmetric algebra S : = S(V^*). The subring of fixed points, S^G, is known as the ring of invariants of V. For fixed G, F, and dim(V), the representations of G can be parameterised by an algebraic variety. I will discuss the resulting parameterisation of invariant rings, using modular representations of elementary abelian p-groups as illustrative examples.