Action Of Finite Group Research Articles

Odds and Ends on Finite Group Actions and Traces

In this article, we study several problems related to virtual traces for finite group actions on schemes of finite type over an algebraically closed field. We also discuss applications to fixed point sets. Our results generalize previous results obtained by Deligne, Laumon, Serre and others.

The p.q.-Baer Property of Fixed Rings under Finite Group Action

A ring R is called right principally quasi-Baer (simply, right p.q.-Baer) if the right annihilator of every principal right ideal of R is generated by an idempotent. For a ring R, let G be a finite group of ring automorphisms of R. We denote the fixed ring of R under G by RG. In this work, we investigated the right p.q.-Baer property of fixed rings under finite group action. Assume that R is a semiprime ring with a finite group G of X-outer ring automorphisms of R. Then we show that: 1) If R is G-p.q.-Baer, then RG is p.q.-Baer; 2) If R is p.q.-Baer, then RG are p.q.-Baer.

The Tracial Class Property for Crossed Products by Finite Group Actions

We define the concept of tracial 𝒞‐algebra of C*‐algebras, which generalize the concept of local 𝒞‐algebra of C*‐algebras given by H. Osaka and N. C. Phillips. Let 𝒞 be any class of separable unital C*‐algebras. Let A be an infinite dimensional simple unital tracial 𝒞‐algebra with the (SP)‐property, and let α : G → Aut(A) be an action of a finite group G on A which has the tracial Rokhlin property. Then A ×α G is a simple unital tracial 𝒞‐algebra.

Argumentwise invariant kernels for the approximation of invariant functions

We consider the problem of designing adapted kernels for approximating functions invariant under a known finite group action. We introduce the class of argumentwise invariant kernels, and show that they characterize centered square-integrable random fields with invariant paths, as well as Reproducing Kernel Hilbert Spaces of invariant functions. Two subclasses of argumentwise kernels are considered, involving a fundamental domain or a double sum over orbits. We then derive invariance properties for Kriging and conditional simulation based on argumentwise invariant kernels. The applicability and advantages of argumentwise invariant kernels are demonstrated on several examples, including a symmetric function from the reliability literature.

Lattice of Finite Group Actions on Prism Manifolds

The set of finite group actions (up to equivalence) which operate on a prism manifold M, preserve a Heegaard Klein bottle and have a fixed orbifold quotient type, form a partially ordered set. We describe the partial ordering of these actions by relating them to certain sets of ordered pairs of integers. There are seven possible orbifold quotient types, and for any fixed quotient type we show that the partially ordered set is isomorphic to a union of distributive lattices of a certain type. We give necessary and sufficent conditions, for these partially ordered sets to be isomorphic and to be a union of Boolean algebras.

Cohomology of line bundles: Applications

Massless modes of both heterotic and Type II string compactifications on compact manifolds are determined by vector bundle valued cohomology classes. Various applications of our recent algorithm for the computation of line bundle valued cohomology classes over toric varieties are presented. For the heterotic string, the prime examples are so-called monad constructions on Calabi-Yau manifolds. In the context of Type II orientifolds, one often needs to compute cohomology for line bundles on finite group action coset spaces, necessitating us to generalize our algorithm to this case. Moreover, we exemplify that the different terms in Batyrev's formula and its generalizations can be given a one-to-one cohomological interpretation. Furthermore, we derive a combinatorial closed form expression for two Hodge numbers of a codimension two Calabi-Yau fourfold.

Darboux Integrability For Polynomial Vector Fields Invariant Under Action of Finite Group

In this paper we present a Darboux–Jouanolou integrability type theorem for polynomial vector fields invariant under action of finite group. As corollary, we obtain a generalization of Darboux–Jouanolou integrability theorem for affine algebraic varieties with quotient singularities.

Crossed product $C^*$-algebras by finite group actions with the tracial Rokhlin property

Let $A$ be a stably finite simple unital $C^*$-algebra and suppose $\alpha $ is an action of a finite group $G$ with the tracial Rokhlin property. Suppose further $A$ has real rank zero and the order on projections over $A$ is determined by traces. Then the crossed product $C^*$-algebra $C^*(G,A, \alpha)$ also has real rank zero and order on projections over $A$ is determined by traces. Moreover, if $A$ also has stable rank one, then $C^*(G,A, \alpha)$ also has stable rank one.

The rationality problem for purely monomial group actions

We investigate the rationality problem for purely monomial actions of finite groups. We solve it affirmatively in the following case: K is a field with char K ≠ 2 and G is a subgroup of GL(n; ℤ) isomorphic to (C 2 ), where n > 0. Then the fixed field of K(x 1, ... x n ) under the purely monomial action of G is rational over K.

Borsuk-Ulam type theorems for manifolds

This paper establishes a Borsuk-Ulam type theorem for PL- manifolds with a finite group action, depending on the free equivariant cobordism class of a manifold. In particular, necessary and sufficient conditions are considered for a manifold with a free involution to be of Borsuk-Ulam type.

Relative invariants, ideal classes and quasi-canonical modules of modular rings of invariants

We describe “quasi-canonical modules” for modular invariant rings R of finite group actions on factorial Gorenstein domains. From this we derive a general “quasi-Gorenstein criterion” in terms of certain 1-cocycles. This generalizes a recent result of A. Braun for linear group actions on polynomial rings, which itself generalizes a classical result of Watanabe for non-modular invariant rings. We use an explicit classification of all reflexive rank one R-modules, which is given in terms of the class group of R, or in terms of R-semi-invariants. This result is implicitly contained in a paper of Nakajima (1982) [15].

On the permutation modules for orthogonal groups [formula omitted] acting on nonsingular points of their standard modules

We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups O m ± ( 3 ) with m ≥ 6 on nonsingular points of their standard modules. These actions, together with those studied in Hall and Nguyen (2011) [2] are all examples of primitive rank 3 actions of finite classical groups on nonsingular points.

Homotopy classification of actions of finite groups on aspherical spaces

In this paper, actions of finite groups on aspherical spaces are studied. A complete homotopy classification of free actions of finite groups on aspherical spaces is obtained, including some results in nonfree actions. For example, a relation between cohomologies of finite groups and the lattice structure of their subgroups is obtained by this method; this relation is formulated in terms of spectral sequences.

Zeros of Brauer characters and linear actions of finite groups

Let G be a finite group, and p a prime number greater than 3. It is known that, if every irreducible p-Brauer character of G does not vanish on any p ′ -element of G, then G is solvable. The primary aim of this work is to describe the structure of groups satisfying the above condition; among other more specific properties, we show that the p ′ -length of G is at most 2 (the bound being the best possible). The structural results are obtained as an application of the main theorem in this paper, that deals with particular linear actions of solvable groups on finite vector spaces.

Finite group actions on homologically peripheral 3-manifolds

AbstractWe generalize the notion of totally peripheral 3-manifolds to define homologically peripheral 3-manifolds. The homologically peripheral property survives canonical decompositions of 3-manifolds as well as it defines a sufficiently large class of 3-manifolds containing link exteriors. The aim of this paper is to study finite group actions on a homologically peripheral 3-manifold, which agree on the boundary, up to equivalence relative to the boundary. As an application, we generalize Sakuma's theorems on the uniqueness of symmetries of knots to the case of symmetries of links.

The tracial Rokhlin property for actions of finite groups on C*-algebras

We define ``tracial'' analogs of the Rokhlin property for actions of finite groups, approximate representability of actions of finite abelian groups, and approximate innerness. We prove the following four analogs of related ``nontracial'' results. $\bullet$ The crossed product of an infinite dimensional simple separable unital C*-algebra with tracial rank zero by an action of a finite group with the tracial Rokhlin property again has tracial rank zero. $\bullet$ An outer action of a finite abelian group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property if and only if its dual is tracially approximately representable, and is tracially approximately representable if and only if its dual has the tracial Rokhlin property. $\bullet$ If a strongly tracially approximately inner action of a finite cyclic group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property, then it is tracially approximately representable. $\bullet$ An automorphism of an infinite dimensional simple separable unital C*-algebra $A$ with tracial rank zero is tracially approximately inner if and only if it is the identity on $K_0 (A)$ mod infinitesimals.

Conjugations on 6-manifolds with free integral cohomology

In this article, we show the existence of conjugations on many smooth simply-connected spin 6-manifolds with free integral cohomology. In a certain class the only condition on X 6 to admit a conjugation with fixed point set M 3 is the obvious one: the existence of a degree-halving ring isomorphism between the $${\mathbb Z_2}$$ -cohomologies of X and M. As a consequence certain 6-manifolds, for which Puppe (J Fixed Point Theory Appl 2(1):85–96, 2007) proved the non-existence of non-trivial orientation-preserving finite group actions, do admit many involutions.

Crossed products by finite group actions with the Rokhlin property

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (a) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (b) Simple unital AH algebras with slow dimension growth and real rank zero. (c) C*-algebras with real rank zero or stable rank one. (d) Simple C*-algebras for which the order on projections is determined by traces. (e) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. (f) C*-algebras with a unique tracial state. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective.

$G$-invariant positive solutions for a quasilinear Schrödinger equation

We are concerned with a quasilinear elliptic equation of the form \begin{equation*} -\Delta u+a(x) u-\Delta(|u|^\alpha)|u|^{\alpha-2}u=h(u)\quad \hbox{in }\mathbb R^N, \end{equation*} where $\alpha> 1$ and $N\geq 1$. By using variational approaches, we prove the existence of at least one positive solution of the above equation under suitable conditions on $a(x)$ and $h$. In particular, we are interested in the situation that $a(x)$ is invariant under the finite group action $G$.

Exact Sequences of Tensor Categories

We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as defined by Schneider, and in particular, exact sequences of (finite) groups. We classify exact sequences of tensor categories C' -> C -> C'' (such that C' is finite) in terms of normal faithful Hopf monads on C'' and also, in terms of self-trivializing commutative algebras in the center of C. More generally, we show that, given any dominant tensor functor C -> D admitting an exact (right or left) adjoint there exists a canonical commutative algebra A in the center of C such that F is tensor equivalent to the free module functor C -> mod_C A, where mod_C A denotes the category of A-modules in C endowed with a monoidal structure defined using the half-braiding of A. We re-interpret equivariantization under a finite group action on a tensor category and, in particular, the modularization construction, in terms of exact sequences, Hopf monads and commutative central algebras. As an application, we prove that a braided fusion category whose dimension is odd and square-free is equivalent, as a fusion category, to the representation category of a group.