A Study of the Number of Roots of xk =g in a Finite Group via Its Frobenius-Schur Indicators

Let G be a finite group. For k ∈ N, X ∈ Irr（G）, define c（k）x ：= 1/G ∑g∈G x（gk）. This is called the k-th Frobenius-Schur indicator of X. In this article we study the Probenius-Schur indicators for Frobenius groups, p-groups, semidihedral groups and mod- ular p-groups. Further, we use this to study the function ζ kG（g） which counts the number of roots of xk = g in a finite group G.

Function spaces on local fields

We study the function spaces on local fields in this paper, such as Triebel Btype and F-type spaces, Holder type spaces, Sobolev type spaces, and so on, moreover,study the relationship between the p-type derivatives and the Holder type spaces. Our obtained results show that there exists quite difference between the functions defined on Euclidean spaces and local fields, respectively. Furthermore, many properties of functions defined on local fields motivate the new idea of solving some important topics on fractal analysis.

Grade zero part of forced graded algebras

This paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arises from the semisimple modules for the corresponding quantum group at a p-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for G when the Lusztig character formula does not hold.

On the character of certain tilting modules

Let G be a semisimple group over an algebraically closed field of characteristic p > 0. We give a(partly conjectural) closed formula for the character of many indecomposable tilting rational G-modules assuming that p is large.

A Virtual Character and Nonzero Kronecker Coefficients for Self-conjugate Partitions

We generalize Regev's results on a virtual character of the symmetric group S_(n),that is,an integer-combination of some irreducible characters.Suppose that λ and μ are integer partitions of n.For the associated irreducible character χ^(λ) of S_(n),when χ^(λ)(μ)≠0,we find another partition τ related to λ such that χ^(τ)(μ)≠0 by the virtual character.Applying this result,we obtain a class of nonzero Kronecker coefficients by Pak et al.'s character criterion.Moreover,we discuss the effectiveness of Pak et al.'s character criterion bya concreteexample.

A Chronicle and Panorama:An Intensive Analysis of Ordinary World

Though Lu Yao's Ordinary World(Pingfan de Shijie)has enjoyed considerable sales volume and reading quantity similar to his other novel Life(Rensheng),it is not accepted by the literary circle as Life and failed to spark the kind of nationwide discussion once created by Life.One explanation for this is the sweeping desire for innovation that was present in literary circles in the 1980s,but the creation method of the Ordinary World was too conservative to stimulate the interpretive impulse of the new critics;another reason is that the incredible length and sheer complexity of Ordinary World prevented scholars from recognizing how difficult it was conceived and what innovation it made.This paper embarks on an intensive reading of Ordinary World in terms of its characterization,the character groups of urban and rural youths,senior cadres,and rural grassroots cadres,and a re-elucidation of its“"overlapping areas,"in an attempt to extensively analyze the content of the panoramic chronicle of Chinese society at the beginning of reform and opening up,and the writer's profound thinking and artistic innovation in his description,trying to clarify the many vague understandings of this masterpiece in the literarycircle.