Related papers: Arithmetic Invariant Theory of Reductive Groups

We investigate the transfer of the Cohen-Macaulay property from a commutative ring to a subring of invariants under the action of a finite group. Our point of view is ring theoretic and not a priori tailored to a particular type of group…

Over a field of characteristic 0, every ring of invariants of a finite group is Cohen-Macaulay. This is not true for fields of positive characteristic. We consider permutation representations and their invariant rings over fields…

This thesis addresses questions in representation and invariant theory of finite groups. The first concerns singularities of quotient spaces under actions of finite groups. We introduce a class of finite groups such that the quotients have…

In this paper we investigate a property for commutative rings with identity which is possessed by every coherent regular ring and is equivalent to Cohen-Macaulay for Noetherian rings. We study the behavior of this property in the context of…

We study initial algebras of determinantal rings, defined by minors of generic matrices, with respect to their classical generic point. This approach leads to very short proofs for the structural properties of determinantal rings. Moreover,…

Let K be an algebraically closed field. For a finitely generated graded K algebra R, let cmdef R := dim R - depth R denote the Cohen-Macaulay-defect of R. Let G be a linear algebraic group over K that is reductive but not linearly…

For G=SL_n or GL_n we construct representations V such that the invariant ring K[V]^G is not Cohen-Macaulay.

Consider the diagonal action of the special orthogonal group on the direct sum of a finite number of copies of the standard representation--the underlying field is assumed to be algebraically closed and of characteristic not equal to two.…

We study rings of integral modular forms for congruence subgroups as modules over the ring of integral modular forms for the full modular group. In many cases these modules are free or decompose at least into well-understood pieces. We…

We describe the ring of invariants for the finite orthogonal groups in odd dimension and even characteristic acting on the defining representation. We construct a minimal algebra generating set and describe the relations among the…

We introduce and study some families of groups whose irreducible characters take values on quadratic extensions of the rationals. We focus mostly on a generalization of inverse semi-rational groups, which we call uniformly semi-rational…

Our goal is to determine when the trivial extensions of commutative rings by modules are Cohen-Macaulay in the sense of Hamilton and Marley. For this purpose, we provide a generalization of the concept of Cohen-Macaulayness of rings to…

We consider actions of reductive groups on a varieties with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox…

We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group $G$. By definition, these are $G$-actions on Laurent polynomial algebras that stabilize the multiplicative group consisting of…

Let $A$ be the ring of integers of global field $K$. Let $G \subseteq GL_2(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y]$ (fixing $A$). Let $R^G$ be the ring of invariants. In the equi-characteristic case we prove $R^G$ is…

We determine the rings of invariants in the symmetric algebra on the dual of a vector space V over the field of two elements, for the group G of orthogonal transformations preserving a non-singular quadratic form on V. The invariant ring is…

This paper surveys and develops links between polynomial invariants of finite groups, factorization theory of Krull domains, and product-one sequences over finite groups. The goal is to gain a better understanding of the multiplicative…

The purpose of this paper is to introduce new invariants of Cohen-Macaulay local rings. Our focus is the class of Cohen-Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are…

We present a generalized version of classical geometric invariant theory \`a la Mumford where we consider an affine algebraic group $G$ acting on a specific affine algebraic variety $X$. We define the notions of linearly reductive and of…

Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every…