Maximally Symmetric Trees

We characterize the best model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct best model geometries in an appropriate sense – these are the maximally symmetric trees. The first theorem gives several equivalent conditions on a bounded valence, cocompact tree T without valence 1 vertices saying that T is maximally symmetric. The second theorem gives general constructions for maximally symmetric trees, showing for instance that every virtually free group has a maximally symmetric tree for a model geometry.     

关于有限群不可约特征标零点的一点注记

在这篇文章中,我们分类了每一个非线性不可约特征标的零点集最多由$3$个共轭类组成的有限亚交换群的结构.     

Rapidly Mixing Random Walks and Bounds on Characters of the Symmetric Group

We investigate mixing of random walks on S _n and A _n generated by permutations of a given cycle structure. The approach follows methods developed by Diaconis, which requires certain estimates on characters of the symmetric group and uses combinatorics of Young tableaux. We conclude with conjectures and open problems.     

Correspondences Between 2-Brauer Characters of Solvable Groups

Let G be a finite solvable group and let p be a prime. Let P ∈ Syl _p (G) and N = N _G (P). We prove that there exists a natural bijection between the 2-Brauer irreducible characters of p′-degree of G and those of N _G (P).     

On the decomposition map for symmetric groups

Let R ^d be the ℤ-module generated by the irreducible characters of the symmetric group . We determine bases for the kernel of the decomposition map. It is known that R ^d ⊗ _ℤ F is isomorphic to the radical quotient of the Solomon descent algebra when F is a field of characteristic zero. We show that when F has prime characteristic and I _br ^d is the kernel of the decomposition map for prime p then R ^d /I _br ^d ⊗ _ℤ F is isomorphic to the radical quotient of the p-modular Solomon descent algebra. To the memory of Manfred Schocker.     

On the construction of irreducible characters in p-blocks with normal metacyclic defect groups

The author carries out a detailed study of p-blocks with normal metacyclic defect groups provided that p is odd and the defect groups are splitting and nonabelian.In particular,the author constructs all the irreducible ordinary and modular characters in this type of blocks.As a by-product,k(B) and l(B) are obtained.     

Fuzzy symmetric groups

We study fuzzy symmetric subgroups and obtain some properties of fuzzy symmetric subgroups of symmetric groups.     

有限群的特征标的零点和可解φ-群

有两个对偶的问题如下：问题Ⅰ：将满足下述条件的有限群G分类：G的特征标表中，除一行外其余各行最多有一个零．问题Ⅱ：将满足下述条件的有限群G分类：G的特征标表中，除一列外其余各列最多有一个零．在这篇文章中，我们对于有限可解群解答上述两个问题，并确定和这两个问题密切相关的一类有限可解群的结构（这类可解群在本文中称之为可解φ-群）．附带我们还完全回答了[4]中的问题1，并说明[6，定理]的条件可以极大地减弱．     

On Asymptotic Dimension of Groups Acting on Trees

We prove the following.THEOREM. Let be the fundamental group of a finite graph of groups with finitely generated vertex groups G _v having asdim G _v n for all vertices v. Then asdim n+1.This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.     

On Natural Monomial Characters of S n

A class of natural linear characters for the centralizers of elements in the symmetric group is introduced. The character values of the corresponding monomial characters are calculated. They have a surprising combinatorial interpretation.     

Counting Overlattices in Automorphism Groups of Trees

We give an upper bound for the number u _Γ(n) of “overlattices” in the automorphism group of a tree, containing a fixed lattice Γ with index n. For an example of Γ in the automorphism group of a 2p-regular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well.     

关于Brauer指标乘积的一个注记

给出在一定条件下不可约Brauer指标乘积的分解形式,并与不可约常指标乘积相比较,由此说明Brauer指标乘积的不可约分解有其自身的特点.     

Characters and Equivalence Classes of Central Simple Group Graded Algebras

We introduce an equivalence between central simple strongly G-graded algebras. Such classes are associated in a natural way to absolutely irreducible characters of semisimple G-graded algebras. We study invariants of this equivalence relation, and also the structure of certain representatives of the equivalence classes.     

Partition graphs for finite symmetric groups

This paper outlines an investigation of a class of arc-transitive graphs admitting a finite symmetric group S_n acting primitively on vertices, with vertex-stabilizer isomorphic to the wreath product S_m wr S_r (preserving a partition of {1,2,…n} into r parts of equal size m). Several properties of these graphs are considered, including their correspondence with r × r matrices with constant row- and column-sums equal to m, their girth, and the local action of the vertex-stabilizer. Also, it is shown that the only instance where S_n acts transitively on 2-arcs occurs in the case m = r = 2. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 107–117, 1997     

Fully Ramified Characters and Clifford Codes

In this article we study relations between groups and quantum error correcting codes. Groups of central type are used to construct quantum error correcting codes. Both stabilizer and Clifford codes can be derived from a construction involving this kind of groups. A more general construction of Clifford codes will be given and their correcting properties will be examined using group theoretical techniques.