Action Of Finite Group Research Articles

Categorical measures for finite group actions

Given a variety with a finite group action, we compare its equivariant categorical measure, that is, the categorical measure of the corresponding quotient stack, and the categorical measure of the extended quotient. Using weak factorization for orbifolds, we show that for a wide range of cases that these two measures coincide. This implies, in particular, a conjecture of Galkin and Shinder on categorical and motivic zeta-functions of varieties. We provide examples showing that, in general, these two measures are not equal. We also give an example related to a conjecture of Polishchuk and Van den Bergh, showing that a certain condition in this conjecture is indeed necessary.

The dual actions, equivariant autoequivalences and stable tilting objects

Pour une catégorie linéaire munie d’une action d’un groupe abélien fini, on étudie l’action duale du groupe des caractères de ce groupe abélien sur la catégorie des objets équivariants. On montre que les groupes des auto-équivalences équivariantes de ces deux catégories sont isomorphes. Dans la situation où la catégorie linéaire est triangulée, les objets basculants stables dans ces deux catégories sont en bijection naturelle compatible avec l’isomorphisme ci-dessus. On applique ces résultats aux complexes basculants stables sur les droites projectives à poids de type tubulaire.

Smoothing finite group actions on three-manifolds

We show that every continuous action of a finite group on a smooth three-manifold is a uniform limit of smooth actions.

Holonomic modules for rings of invariant differential operators

We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings. Our main tool is the filter dimension introduced by Bavula. We extend the results for the invariants of the Weyl algebra with respect to the symplectic action of a finite group, for the rings of invariant differential operators on quotient varieties, and invariants of certain generalized Weyl algebras under the linear actions. We show that the filter dimension of all above mentioned algebras equals 1.

Nuclear dimension of crossed products associated to partial dynamical systems

We bound the nuclear dimension of crossed products associated to some partial actions of finite groups or Z on finite dimensional locally compact Hausdorff second countable spaces. Our results apply to globalizable partial actions, finite group partial actions, minimal partial automorphisms, and partial automorphisms acting on zero-dimensional spaces, or a class of one dimensional spaces, containing 1-dimensional CW complexes. This extends work on global systems by Hirshberg and Wu.

Tracially sequentially split $$^*$$-homomorphisms between $$C^*$$-algebras

We define a tracial analog of the notion called a sequentially split \(^*\)-homomorphism between \(C^*\)-algebras due to Barlak and Szabó and show that several important approximation properties related to the classification theory of \(C^*\)-algebras pass from the target algebra to the domain algebra. Then, we show that this framework arises from tracial Rokhlin properties of a finite group action and an inclusion of unital \(C^*\)-algebras.

KSBA compactification of the moduli space of K3 surfaces with a purely non-symplectic automorphism of order four

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and$U(2)\oplus D_4^{\oplus 2}$lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of$\mathbb {P}^{1}\times \mathbb {P}^{1}$branched along a specific$(4,\,4)$curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient$(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$with the symmetric linearization.

Rank conditions for finite group actions on 4-manifolds

AbstractLet M be a closed, connected, orientable topological $4$ -manifold, and G be a finite group acting topologically and locally linearly on M. In this paper, we investigate the spectral sequence for the Borel cohomology $H^*_G(M)$ and establish new bounds on the rank of G for homologically trivial actions with discrete singular set.

Counting Rational Curves on K3 Surfaces With Finite Group Actions

Abstract Göttsche gave a formula for the dimension of the cohomology of Hilbert schemes of points on a smooth projective surface $S$. When $S$ admits an action by a finite group $G$, we describe the action of $G$ on the Hodge structure. In the case that $S$ is a K3 surface, each element of $G$ gives a trace on $\sum _{n=0}^{\infty }\sum _{i=0}^{\infty }(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$. When $G$ acts faithfully and symplectically on $S$, the resulting generating function is of the form $q/f(q)$, where $f(q)$ is a cusp form. We relate the Hodge structure of Hilbert schemes of points to the Hodge structure of the compactified Jacobian of the tautological family of curves over an integral linear system on a K3 surface as $G$-representations. Finally, we give a sufficient condition for a $G$-orbit of curves with nodal singularities not to contribute to the representation.

On equivariant dendriform algebras

Dendriform algebras are certain associative algebras whose product splits into two binary operations and the associativity splits into three new identities. In this paper, we study finite group actions on dendriform algebras. We define equivariant cohomology for dendriform algebras equipped with finite group actions similar to the Bredon cohomology for topological $G$-spaces. We show that equivariant cohomology of such dendriform algebras controls equivariant one-parameter formal deformations.

Fukaya categories of surfaces, spherical objects and mapping class groups

Abstract We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $2$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $1$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.

Equivariant one-parameter formal deformations of Hom-Leibniz algebras

The aim of this paper is to define a new type of cohomology for multiplicative Hom-Leibniz algebras which control deformations of Hom-Leibniz algebras. The cohomology and the associated deformation theory for Hom-Leibniz algebras as developed here are also extended to equivariant context, under the presence of finite group actions on Hom-Leibniz algebras.

Graph Calculus and the Disconnected-Boundary Schwinger-Dyson Equations of Quartic Tensor Field Theories

Tensor field theory (TFT) focuses on quantum field theory aspects of random tensor models, a quantum-gravity-motivated generalisation of random matrix models. The TFT correlation functions have been shown to be classified by graphs that describe the geometry of the boundary states, the so-called boundary graphs. These graphs can be disconnected, although the correlation functions are themselves connected. In a recent work, the Schwinger-Dyson equations for an arbitrary albeit connected boundary were obtained. Here, we introduce the multivariable graph calculus in order to derive the missing equations for all correlation functions with disconnected boundary, thus completing the Schwinger-Dyson pyramid for quartic melonic (‘pillow’-vertices) models in arbitrary rank. We first study finite group actions that are parametrised by graphs and build the graph calculus on a suitable quotient of the monoid algebra mathcal {A}[G] corresponding to a certain function space mathcal {A} and to the free monoid G in finitely many graph variables; a derivative of an element of mathcal {A}[G] with respect to a graph yields its corresponding group action on mathcal {A}. The present result and the graph calculus have three potential applications: the non-perturbative large-N limit of tensor field theories, the solvability of the theory by using methods that generalise the topological recursion to the TFT setting and the study of ‘higher dimensional maps’ via Tutte-like equations. In fact, we also offer a term-by-term comparison between Tutte equations and the present Schwinger-Dyson equations.

Some Results on Inner Quasidiagonal C*-algebras

In the current article, we prove the crossed product C*-algebra by a Rokhlin action of finite group on a strongly quasidiagonal C*-algebra is strongly quasidiagonal again. We also show that a just-infinite C*-algebra is quasidiagonal if and only if it is inner quasidiagonal. Finally, we compute the topological free entropy dimension in just-infinite C*-algebras.

Plethysm and cohomology representations of external and symmetric products

We prove refined generating series formulae for characters of (virtual) cohomology representations of external products of suitable coefficients, e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules on spaces such as (possibly singular) complex quasi-projective varieties. These formulae generalize our previous results for symmetric and alternating powers of such coefficients, and apply also to other Schur functors. The proofs of these results are reduced via an equivariant Künneth formula to a more general generating series identity for abstract characters of tensor powers V⊗n of an element V in a suitable symmetric monoidal category A. This abstract approach applies directly also in the equivariant context for spaces with additional symmetries (e.g., finite group actions, finite order automorphisms, resp., endomorphisms), as well as for introducing an abstract plethysm calculus for symmetric sequences of objects in A.

Homologically trivial group actions on elliptic surfaces

In this paper, we study the minimal symplectic elliptic surfaces [Formula: see text] with homologically trivial symplectic finite group actions, and get a rigidity theorem under some restriction.

Cartan subalgebras and the UCT problem, II

We study the connection between the UCT problem and Cartan subalgebras in C*-algebras. The UCT problem asks whether every separable nuclear C*-algebra satisfies the UCT, i.e., a noncommutative analogue of the classical universal coefficient theorem from algebraic topology. This UCT problem is one of the remaining major open questions in the structure and classification theory of simple nuclear C*-algebras. Since the class of separable nuclear C*-algebras is closed under crossed products by finite groups, it is a natural and important task to understand the behaviour of the UCT under such crossed products. We make a contribution towards a better understanding by showing that for certain approximately inner actions of finite cyclic groups on UCT Kirchberg algebras, the crossed products satisfy the UCT if and only if we can find Cartan subalgebras which are invariant under the actions of our finite cyclic groups. We also show that the class of actions we are able to treat is big enough to characterize the UCT problem, in the sense that every such action (even on a particular Kirchberg algebra, namely the Cuntz algebra mathcal O_2) leads to a crossed product satisfying the UCT if and only if every separable nuclear C*-algebra satisfies the UCT. Our results rely on a new construction of Cartan subalgebras in certain inductive limit C*-algebras. This new tool turns out to be of independent interest. For instance, among other things, the second author has used it to construct Cartan subalgebras in all classifiable unital stably finite C*-algebras.

Hochschild cohomology and group actions

Given a finite group action on a (suitably enhanced) triangulated category linear over a field, we establish a formula for the Hochschild cohomology of the category of invariants, assuming the order of the group is coprime to the characteristic of the base field. The formula shows that the cohomology splits canonically with one summand given by the invariant subspace of the Hochschild cohomology of the original category. We also prove that Serre functors act trivially on Hochschild cohomology, and combine this with our formula to give a useful mechanism for computing the Hochschild cohomology of fractional Calabi–Yau categories.

Noncommutative rational functions invariant under the action of a finite solvable group

This paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in d generators. These invariant skew fields are always finitely generated, which contrasts with the free algebra case. For abelian groups or solvable groups G with a well-behaved representation theory it is shown that the invariant skew fields are free on |G|(d−1)+1 generators. Finally, positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented.

Large orbit sizes in finite group actions

In this paper, we study the relations of the sizes of various sections of finite linear groups and the largest orbit size of the linear group actions. We also study various applications of those orbit theorems.