Varieties in finite transformation groups

Weights for symmetric and general linear groups

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

Let G denote the finite cyclic group of order n. The problem of determining necessary and sufficient conditions for F to be the fixed point set of a smooth cyclic group action on some sphere has been solved when n is a prime power. P. A. Smith proved that F must be a Zn-homology sphere. If n is odd it is also known that F is unitary. L. Jones has shown that these conditions are also sufficient to realize F as the fixed point set of smooth cyclic group action on some sphere when n is a prime power Pl. In the general case it is known that F is a union of smooth manifolds, unitary if n is odd. And if Zn acts on some even dimensional sphere, then the Euler characteristic number of F is 2 if action is orientation-preserving and 0 if action is orientation-reversing. We may ask about possible restrictions on Pontryagin numbers of components of the fixed set F. If n is a prime power, the Pontryagin numbers of the fixed point set all vanish by the P. A. Smith theorem and the Hirzebruch signature theorem. It is natural to ask whether such restrictions hold for other types of smooth cyclic groups acting on spheres. In case where n is not a prime power, by work of R. Oliver [01] and its extention of A. Assadi [Asd] and K. Pawalowski [Pa 1-4], we can construct exotic actions on spheres such that Pontryagin classes of fixed point sets do not vanish. But in these examples the Pontryagin numbers all vanish because these actions bound group actions on disks. R. Schultz has shown in [S2] that if G is a cyclic group whose order is not a prime power, then there are smooth actions of G on spheres such that the fixed point sets have nonzero Pontryagin numbers provided the dimension of the fixed point set is greater than 16. Testing to see if the lower bound in dimensions is necessary. There are two possibilities. First, more sophisticated computations might make it possible to remove the restriction on dimensions. Second, there might be some unusual things happening in low dimensions (compare Ewing's result for Zp actions on spheres). Main Theorem is evidence for the first one.

p ∗-Theory and modular representation theory

Finite symmetry transformation groups and exact solutions of Lax integrable systems

In this paper, the finite symmetry transformation group of the (2+1)-dimensional coupled Burgers equation is studied by the modified direct method, and with the help of the truncated Painlevé expansion approach, some special localized structures for the (2+1)-dimensional coupled Burgers equation are obtained, in particular, the dromion-like and solitoff-like structures.

Symmetric local algebras and small blocks of finite groups

Let G be a finite group and X a space on which G acts. In this chapter we will describe a cohomological analysis of X which involves H*(G; \({\Bbb F}\) p) in a fundamental way. First developed by Borel and then by Quillen, this approach is the natural generalization of classical Smith Theory. After reviewing the basic constructions and a few examples, we will apply these techniques to certain complexes defined from subgroups of a group G, first introduced by K. Brown and D. Quillen. Using results due to Brown and P. Webb we will show how these complexes provide a systematic method for approaching the cohomology of simple groups which will be discussed later on. One of the aims of this chapter is to expose the reader to the important part played by group cohomology in the theory of finite transformation groups. By no means is this a complete account; in addition it requires a different background than the previous chapters have. We recommend the texts [AP], [Brel] as excellent sources for those wanting to learn more about group actions. For a more recent survey, we suggest [AD].KeywordsFinite GroupSpectral SequenceSimplicial ComplexEquivariant CohomologyHomotopy EquivalentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we shall study semi-free finite group actions on homotopy spheres. In [8], M. Sebastiani studied semi-free finite group actions on homotopy spheres with two points as the fixed point set. He showed that the collection of equivariant diffeomorphism classes of these semi-free finite group actions is an abelian group under the equivariant connected sum about a fixed point. By the methods analogous to Kervaire-Milnor [4], he proved that this abelian group is a finite group in the case of cyclic group actions.

On the cohomology of Coxeter groups and their finite parabolic subgroups. II by T. Akita Linear source modules and trivial source modules by R. Boltje Resolutions de foncteurs de Mackey by S. Bouc Homotopy uniqueness of $BPU(3)$ by C. Broto and A. Viruel Lowering operators for $GL(n)$ and quantum $GL(n)$ by J. Brundan Homomorphisms in higher complexity quotient categories by J. F. Carlson and W. W. Wheeler On the homotopy theory of $p$-completed classifying spaces by F. R. Cohen and R. Levi $H^*V$ is of bounded type over $\mathcal {A} (p)$ by M. D. Crossley Classifying spaces of central group extensions by J. Dietz and T. Ratliff Sharp homology decompositions for classifying spaces of finite groups by W. G. Dwyer On the geometry and cohomology of the simple groups $G_2(q)$ and $^3D_4(q)$ by P. Fong and R. J. Milgram Transfer and Chern classes for extraspecial $p$-groups by D. J. Green and P. A. Minh The infinite order of the symplectic classes in the cohomology of the stable mapping class group by R. Grieder A variant of the proof of the Landweber Stong conjecture by H.-W. Henn Unstable modules over the Steenrod algebra and cohomology of groups by H.-W. Henn The category of Mackey functors for a compact Lie group by L. G. Lewis, Jr. Comparison of extensions of modules for algebraic groups and quantum groups by Z. Lin The defect of the completion of a Green functor by F. Luca Simple modules of Ariki-Koike algebras by A. Mathas On the geometry and cohomology of the simple groups $G_2(q)$ and $^3D_4(q)$: II by R. J. Milgram On the relation between simple groups and homotopy theory by R. J. Milgram Varieties for $G_rT$-modules by D. K. Nakano On defect groups of the Mackey algebras for finite groups by F. Oda Applications of stable classifying space theory to group cohomology by S. Priddy The spectra $k$ and $kO$ are not Thom spectra by Yu. B. Rudyak The code of a regular generalized quadrangle of even order by N. S. N. Sastry and P. Sin Quivers of blocks with normal defect group by M. Schaps, D. Shapira, and O. Shlomo Cohomology of a homogeneous space $E_6/T^1\cdot SU(6)$ by T. Watanabe Problem session, Seattle 1996 by D. Benson.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $$S^3.$$ S 3 . Such involutions are called hyperelliptic as the manifolds admitting such an action. We prove that the sectional 2-rank of a finite group acting on a 3-manifold and containing a hyperelliptic involution whose fixed-point set has two components has sectional 2-rank at most four; this upper bound is sharp. The cases where the hyperelliptic involution has a fixed-point set with a number of components different from 2 have been already considered in the literature. Our result completes the analysis and we obtain general results where the number of the components of the fixed-point set is not fixed. In particular, we obtain that a finite group acting on a 3-manifold and containing a hyperelliptic involution has 2-rank at most four, and four is the best possible upper bound. Finally, we restrict to the basic case of simple groups acting on hyperelliptic 3-manifolds: we use our result about the sectional 2-rank to prove that a simple group containing a hyperelliptic involution is isomorphic to PSL(2, q) for some odd prime power q, or to one of four other small simple groups.

Working with tensor triangulated categories, we prove two theorems relative to supports and then discuss their incarnations in algebraic geometry and in modular representation theory. First, we show that an indecomposable object has connected support. Then we consider the filtrations by dimension or codimension of the support and prove that the associated subquotients decompose as sums of local terms.

A characterization of blocks with vertices

We prove a version of Quillen’s stratification theorem in equivariant homotopy theory for a finite group G, generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as coefficients, and secondly, we categorify it to a result about equivariant modules. Our general stratification theorem is formulated in the language of equivariant tensor-triangular geometry, which we show to be tightly controlled by the non-equivariant tensor-triangular geometry of the geometric fixed points. We then apply our methods to the case of Borel-equivariant Lubin–Tate E-theory En_, for any finite height n and any finite group G, where we obtain a sharper theorem in the form of cohomological stratification. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing ⊗-ideals of the category of equivariant modules over En_, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.