一些特殊变换群的联合不变量和联合微分不变量

The theory of moving frames developed by Peter J Olver and M Fels has importaut applications to geometry,classical invariant theory.We will use this theory to classify joint invariants and joint differential invariants of some transformation groups.     

Homological Invariants and Quasi-Isometry

Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain that the cohomological dimension cd_R over a commutative ring R satisfies the inequality if Λ embeds uniformly into Γ and holds. Another consequence of our results is that the Hirsch ranks of quasi-isometric solvable groups coincide. Further, it is shown that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings. On the analytic side, we apply the induction technique to Novikov-Shubin invariants of amenable groups, which can be seen as homological invariants, and show their invariance under quasi-isometry. Received: November 2004 Revision: April 2004 Accepted: April 2004     

Invariants and Chebyshev polynomials

On different compact sets from ? ⁿ , new multidimensional analogs of algebraic polynomials least deviating from zero (Chebyshev polynomials) are constructed. A brief review of the analogs constructed earlier is given. Estimates of values of the best approximation obtained by using extremal signatures, lattices, and finite groups are presented.     

Dickson-Mui Invariants

Let V be a finite-dimensional vector space over a finite field.The general and special linear groups, GL(V) and SL(V), acton the exterior algebras ^*V and ^*V^* of V and its dual V^*, andon the symmetric algebra S^*V. The subring of SL(V)-invariantsof ^*VS^*V was determined by Dickson and Mui. This paper describesthe equivalent, but simpler, calculation of the invariant subringof ^*VS^*V as a representation of GL(V)/SL(V). 2000 MathematicsSubject Classification 13A50.     

Invariants and Chebyshev polynomials

On different compact sets from ℝ ⁿ , new multidimensional analogs of algebraic polynomials least deviating from zero (Chebyshev polynomials) are constructed. A brief review of the analogs constructed earlier is given. Estimates of values of the best approximation obtained by using extremal signatures, lattices, and finite groups are presented.     

Near-Rings of Invariants

For a group S acting on a group G we let I(S,G) = {?: G → G ? σ(x) = ? (x), ∈ G, σ ∈ S}, the near-ring of invariants of (S,G). In this paper we investigate the transfer of information between the ideal structure of I(S,G) and the group action (S, G).     

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.     

G a Invariants and Slices

Criteria for local triviality of algebraic actions of the additive group of complex numbers on complex affine space are extended to more general varieties. Finite generation of rings of invariants of locally trivial actions on factorial affine varieties is dicussed, giving some sufficient conditions for finite generation and examples where finite generation fails. A missing hypothesis in a theorem of Miyanishi is identified, and an example is given to demonstrate the necessity of the hypothesis. The corrected theorem is shown to hold for a class of triangular G _a actions on C⁴, with the consequence that all these actions are conjugate to translations. A new criterion for an action to be conjugate to a global translation is given.     

Invariants of Directed Spaces

Directed spaces are the objects of study within directed algebraic topology. They are characterised by spaces of directed paths associated to a source and a target, both elements of an underlying topological space. The algebraic topology of these path spaces and their connections are studied from a categorical perspective. In particular, we study the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors. Furthermore, we propose and study a new notion of directed homotopy equivalence between directed spaces.      

Invariants of knot diagrams

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams. J. Hass was partially supported by an NSF grant.     

Invariants of Conformai and Projective Structures

While in Euclidean, equiaffine or centroaffine differential geometry there exists a unique, distinguished normalization of a regular hypersurface immersion x: M ⁿ → Aⁿ⁺¹, in the geometry of the general affine transformation group, there only exists a distinguished class of such normalizations, the class of relative normalizations. Thus, the appropriate invariants for speaking about affine hypersurfaces are invariants of the induced classes, e.g. the conformai class of induced metrics and the projective class of induced conormal connections. The aim of this paper is to study such invariants. As an application, we reformulate the fundamental theorem of affine differential geometry.     

Invariants of three-dimensional manifolds

The work of the second author was partially supported by an NSF grant.     

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.     

Invariants of Algebraic Automorphisms

Let R be a prime ring with extended centroid C and let σ be a C-algebraic automorphism of R. We let $R^{(\sigma)}\mathop{=}\limits^{\rm def.}\{x\in R\mid \sigma(x)=x\}$ , the subring of invariants of σ in R, and let Out-deg(σ) and Inn-deg(σ) denote the outer and inner degrees of σ, respectively. In the paper we first prove the nilpotence of the prime radical of R ^(σ) with a bound and characterize the semiprimeness and primeness of R ^(σ). Moreover, we show that if R ^(σ) is a prime PI-ring, then PI-deg(R)?=?PI-deg(R ^(σ)) × Inn-deg(σ).     

Invariants and constructions of mendelsohn designs

In this paper we examine Mendelsohn designs and some connections to topology which lead to an easily described algorithm for computing invariants of these designs. The results are applied to designs which have natural group actions. We also use the topology to describe when ordinary two-fold triple systems with a group action lead to Mendelsohn designs with the same group action. Procedures for constructing Mendelsohn designs are also given. In particular, we give necessary and sufficient conditions for constructing 2-(v, 4, 1) Mendelsohn designs.