Above the Glauberman correspondence

Let S be a finite solvable group, and suppose S acts on the finite group N, and they have coprime orders. Then, the celebrated Glauberman correspondence provides a natural bijection from the set IrrS(N) of irreducible characters of N which are invariant under the action of S to the set Irr(CN(S)) of all irreducible characters of the centralizer of S in N. Suppose, further, that the semidirect product SN is a normal subgroup of a finite group G. Let θ∈IrrS(N), and let ψ∈Irr(CN(S)) be its Glauberman correspondent. We prove that there is a bijection with good compatibility properties from the set Irr(G,θ) of the irreducible characters of G above θ to Irr(NG(S),ψ) such that, in the case when S is a p-group for some prime p, it preserves fields of values and Schur indices over Qp, the field of p-adic numbers. Using this result, we also prove a strengthening of the McKay Conjecture for all p-solvable groups.     

Glauberman对应诱导的Morita等价

假设G是一个p-可解群，A是一个可解群，它作用在G上且它的阶与G的阶互素，对适当的Dedekind整环R和RG的一个块B，A平凡地作用在它的某一个亏群上，我们证明B与它的Watanabe对应之间存在一个Morita等价．     

On the Brauer–Glauberman correspondence

The purpose of the present paper is two-fold: on the one hand, to show the existence of a correspondence unifying Brauer's and Glauberman's ones (see Theorem 4.6), and, on the other hand, to give an alternative proof of Watanabe's equivalence in [Atumi Watanabe, The Glauberman character correspondence and perfect isometries for blocks of finite groups, J. Algebra 216 (1999) 548–565]. By the way, we give a short proof of the coincidence of the Clifford extensions associated with a pair of Glauberman correspondent irreducible representations (see Corollary 4.16), a question that, surprisingly enough, has only been partially solved recently (see [Morton Harris, Markus Linckelmann, On the Glauberman and Watanabe correspondences for blocks of finite p-solvable groups, Trans. Amer. Math. Soc. 354 (2002) 3435–3453] and [Shigeo Koshitani, Gerhard Michler, Glauberman correspondence of p-blocks of finite groups, J. Algebra 243 (2001) 504–517]).     

McKay correspondence and orbifold equivalence

We prove that a pair of singularities related by a transformation arising from the McKay correspondence are orbifold equivalent. From this we deduce a McKay type category equivalence for the matrix factorization categories.     

Closure ordering and the Kostant-Sekiguchi correspondence

Let be a real semisimple Lie group with Lie algebra . The Kostant-Sekiguchi correspondence is a bijection between nilpotent orbits on and nilpotent orbits on . In this note we prove that the closure relations among nilpotent orbits are preserved under the Kostant-Sekiguchi correspondence. The techniques rely on work of M. Vergne and P. Kronheimer.

     

Note on Morita Equivalence in Ring Extensions

It seems that Morita invariance judges of the importance of classes of ring extensions concerned. Miyashita introduced the notion of Morita equivalence in ring extensions, and he showed that the classes of G-Galois extensions and Frobenius extensions are Morita invariant. After that, Ikehata showed that the classes of separable extensions, Hirata separable extensions, symmetric extensions, and QF-extensions are Morita invariant. In this article, we shall prove that the classes of several extensions are Morita invariant. Further, we will give an example of the class of ring extensions which is not Morita invariant.     

A remark on Mansfield's imprimitivity theorem

We show that the Morita equivalence part of Mansfield's Imprimitivity Theory can be obtained by Green's Imprimitivity Theorem (and duality theory).

     

Morita Equivalence of Sketches

Equivalence of sketches S and T means the equivalence of their categories Mod_S and Mod_T of all Set-valued models. E. Vitale and the second author have characterized equivalence of limit-sketches by means of bimodels, where a bimodel for limit sketches S and T is a model of S in the category Mod_T. For general sketches, we show that an analogous result holds provided that Mod_T is substituted by a more complex category; e.g., in case of limit-coproduct sketches, that category is (Mod_T), the free product completion of Mod_T.     

关于H?lder正则性的一个注记

Holder正则性在图像消噪、信号重构等方面都具有重要意义．本文对Holder空间的两种定义进行研究分析,利用归纳法和Littlewood—Paley分解方法证明了两种Ho1der定义在一定条件下还是等价的,为Holder正则性的应用提供了理论依据．     

On the Spaltenstein correspondence

In the article [Spa1], N. Spaltenstein has established a bijection between the irreducible components of _χ, the space of full flags fixed by a nilpotent element χ ? M(n, k), where k is an algebraically closed field, and the standard tableaux associated to the Young diagram of χ. In this present work we determine, when χ is of hook type, for each irreducible component X of _χ, the unique Schubert cell _X of the full flag manifold = (V) (where V is vector space of dimension n over k), such that X ∩ _X is a dense subspace in X. This result will allow us to optimize the computation of _χ and when k = is the complex field, to see that the graph resolution of the partition (2, 1, …, 1) of n is related to the Dynkin diagram of sl(n, ).     

A note on the fundamental groups of manifolds with almost nonnegative curvature

We show that given and , there exists a positive number such that if a closed -manifold satisfies and , then is almost abelian.

     

On the isotypy induced by the Glauberman–Dade correspondence between blocks of finite groups

We show that under some conditions the character correspondence introduced by Dade which partly generalizes the Glauberman correspondence induces isotypy between blocks of finite groups.     

A note on strongly π-regular rings

Let R be an associative ring with unit and let N(R) denote the set of nilpotent elements of R. R is said to be stronglyπ-regular if for each x∈R, there exist a positive integer n and an element y∈R such that x ⁿ=x ^{n +1} y and xy=yx. R is said to be periodic if for each x∈R there are integers m,n≥ 1 such that m≠n and x ^m=x ⁿ. Assume that the idempotents in R are central. It is shown in this paper that R is a strongly π-regular ring if and only if N(R) coincides with the Jacobson radical of R and R/N(R) is regular. Some similar conditions for periodic rings are also obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

Weak Equivalence of Internal Categories

Weak equivalence is defined as equivalence in the bicategory of modules between internal categories. It is known that two categories are weakly equivalent if and only if their Cauchy completions are equivalent. We prove that this condition can be generalized to a suitable notion of intermediate category, stable under composition with weak equivalences. Applications to categorical Morita theory are given.     

Blocks of central -group extensions

Let and be finite groups that have a common central -subgroup for a prime number , and let and respectively be -blocks of and induced by -blocks and respectively of and , both of which have the same defect group. We prove that if and are Morita equivalent via a certain special -bimodule, then such a Morita equivalence lifts to a Morita equivalence between and .