Characterizations and representations of the group inverse involving idempotents

This paper is to present some results on the group invertibility of products and differences of idempotents. In addition, some formulae for the group inverse of sums, differences and products of idempotents are established by using some given idempotents.     

On the invertibility of Hilbert space idempotents

This note is to present some results on the group invertibility of linear combinations of idempotents when the difference of two idempotents is group invertible.     

Idempotents in $$F_+(G)$$F+(G)

In this paper, we compute the idempotents of the ring $$F_+(G)$$ as defined by Boltje (J Algebra 206:293–343, 1998) in the particular case when the Green biset functor F is such that for all subgroups H of a finite group G, F(H) is a torsion-free ring, finitely-generated as an Abelian group, and has only the trivial idempotents. In this case, the only idempotents in $$F_+(G)$$ are those arising from the Burnside ring B(G).     

有足够幂等元的群分次环上的Morita对偶

首先定义了具有足够幂等元的群分次环的Ｓｍ ａｓｈ 积并讨论其性质,其次,借助Ｓｍ ａｓｈ 积给出了这类分环上的分次Ｍｏｒｉｔａ对偶的刻划．     

Explicit determination of certain minimal constabelian codes

Constabelian codes can be viewed as ideals in twisted group algebras over finite fields. In this paper we study decomposition of semisimple twisted group algebras of finite abelian groups and prove results regarding complete determination of a full set of primitive orthogonal idempotents in such algebras. We also explicitly determine complete sets of primitive orthogonal idempotents of twisted group algebras of finite cyclic and abelian p-groups. We also describe methods of determining complete set of primitive idempotents of abelian groups whose orders are divisible by more than one prime and give concrete (numerical) examples of minimal constabelian codes, illustrating the above mentioned results.     

Primitive Idempotents of Schur Rings

In this paper, we explore the nature of central idempotents of Schur rings over finite groups. We introduce the concept of a lattice Schur ring and explore properties of these kinds of Schur rings. In particular, the primitive, central idempotents of lattice Schur rings are completely determined. For a general Schur ring S, S contains a maximal lattice Schur ring, whose central, primitive idempotents form a system of pairwise orthogonal, central idempotents in S. We show that if S is a Schur ring with rational coefficients over a cyclic group, then these idempotents are always primitive and are spanned by the normal subgroups contained in S. Furthermore, a Wedderburn decomposition of Schur rings over cyclic groups is given. Some examples of Schur rings over non-cyclic groups will also be explored.     

Fusion Procedure for Wreath Products of Finite Groups by the Symmetric Group

Let G be a finite group. A complete system of pairwise orthogonal idempotents is constructed for the wreath product of G by the symmetric group by means of a fusion procedure, that is by consecutive evaluations of a rational function with values in the group ring. This complete system of idempotents is indexed by standard Young multi-tableaux. Associated to the wreath product of G by the symmetric group, a Baxterized form for the Artin generators of the symmetric group is defined and appears in the rational function used in the fusion procedure.     

The p-Blocks of the Mackey Algebra

Let p be a prime number. This paper describes the primitive idempotents and prime spectrum of the crossed Burnside algebra of a finite group over a p-local ring. The main application is a formula for the block idempotents of the p-local Mackey algebra of the group, in terms of the corresponding bloks of the group algebra.     

Inverse Near Permutation Semigroups

A transformation semigroup over a set X with N elements is said to be a near permutation semigroup if it is generated by a group G of permutations on N elements and by a set H of transformations of rank N - 1. In this paper we give necessary and sufficient conditions for a near permutation semigroup S = ‹G,H›, where H is a group, to be inverse. Moreover, we obtain conditions which guarantee that its semilattice of idempotents is generated by the idempotents of S of rank greater than N - 2 or N - 3.     

Extending Sets of Idempotents to Ring Extensions

In this paper the idea of an intrinsic extension of a ring, first proposed by Faith and Utumi, is generalized and studied in its own right. For these types of ring extensions, it is shown that, with relatively mild conditions on the base ring, R, a complete set of primitive idempotents (a complete set of left triangulating idempotents, a complete set of centrally primitive idempotents) can be constructed for an intrinsic extension, T, from a corresponding set in the base ring R. Examples and applications are given for rings that occur in functional analysis and group ring theory.     

Translational hull of an inverse semigroup

The theory of ideal extensions of semigroups is dependent upon adequate knowledge of the translational hull. We characterize here certain basic structures in the translational hull, Ω(S), of an arbitrary inverse semigroup S: in particular, the semilattice of idempotents of Ω(S) and the group of units of Ω(S). An isomorphic copy of each of these structures is given in terms of the semilattice of idempotents of S.     

On a Class of Lattice Ordered Inverse Semigroups

It is well known that the free group on a non-empty set can be totally ordered and, further, that each compatible latttice ordering on a free group is a total ordering. On the other hand, Saitô has shown that no non-trivial free inverse semigroup can be totally ordered. In this note we show, however, that every free inverse monoid admits compatible lattice orderings which are closely related to the total orderings on free groups.These orderings are natural in the sense that the imposed partial ordering on the idempotents coincides with the natural partial ordering. For this to happen in a lattice ordered inverse semigroup, the idempotents must form a distributive lattice. The method of construction of the lattice orderings on free inverse monoids can be applied to show that naturally lattice ordered inverse semigroups with a given distributive lattice E of idempotents can have arbitrary Green's relation structure. Analogous results hold for naturally -semilatticed inverse semigroups. In this case, there is no restriction on the semilattice E of idempotents.We also show that every compatible lattice ordering on the free monogenic inverse monoid is of the type considered here. This permits us to prove that there are precisely eight distinct compatible lattice orderings on this semigroup. They belong to two families, each of which contains four members, of conjuguate lattice orderings.     

Characteristic maps for the Brauer algebra

The classical characteristic map associates symmetric functions to characters of the symmetric groups. There are two natural analogues of this map involving the Brauer algebra. The first of them relies on the action of the orthogonal or symplectic group on a space of tensors, while the second is provided by the action of this group on the symmetric algebra of the corresponding Lie algebra. We consider the second characteristic map both in the orthogonal and symplectic case, and calculate the images of central idempotents of the Brauer algebra in terms of the Schur polynomials. The calculation is based on the Okounkov–Olshanski binomial formula for the classical Lie groups. We also reproduce the hook dimension formulas for representations of the classical groups by deriving them from the properties of the primitive idempotents of the symmetric group and the Brauer algebra.     

关于幂等元之差的可逆性

本文研究在一个有单位元的环中两个幂等元之差的可逆性问题,利用幂等元的性质,得到了两个幂等元之差可逆的几个充分必要条件,并给出了在矩阵环中的几个应用.     

具有弱中间幂等元的正则半群

本文在正则半群上引入弱中间幂等元和拟中间幂等元,着重探讨了这两类幂等元的性质特征.构造了若干具有弱(拟)中间幂等元的正则半群,确定了弱中间幂等元和拟中间幂等元之间的关系,给出了弱中间幂等元和拟中间幂等元各自的等价判定,利用拟中间幂等元刻画了纯正半群.最后,得到了构造具有拟中间幂等元的正则半群的一般途径,并在此基础上进一步给出了判定正则半群是否具有乘逆断面的方法.     

Group invariant weighing matrices

We investigate the existence problem of group invariant matrices using algebraic approaches. We extend the usual concept of multipliers to group rings with cyclotomic integers as coefficients. This concept is combined with the field descent method and rational idempotents to develop new non-existence results.     

具有弱正规幂等元的富足半群的结构

本文研究含弱正规幂等元的富足半群.在给出这类半群的若干特征后,建立了具有弱正规幂等元的富足半群的结构．作为应用,给出具有正规幂等元的富足半群和具有（弱）正规幂等元的拟适当半群的结构．     

A Compact Monothetic Semitopological Semigroup Whose Set of Idempotents Is Not Closed

For each monothetic unitary group without proper co-compact subgroups, we show the existence of a compact semitopological semigroup (affine) compactification in which the set of idempotents is not closed answering a question of Berglund.     

正则半群的局部化及应用

证明了正则半群S在其幂等元素E（S）所生成的子半群＜E（S）＞上的局部化在同构意义下存在惟一，其为其最大群同态象，同时也给出了其最小群同余。