Semidirect Products of Association Schemes

In his 1996 work developing the theory of association schemes as a generalized group theory, Zieschang introduced the concept of the semidirect product as a possible product operation of certain association schemes. In this paper we extend the semidirect product operation into the entire set of association schemes. We then derive a way to decompose certain association schemes into smaller association schemes. We also investigate to what extent this product helps us to understand and characterize the structure of association schemes. We give some examples to show that the semidirect product produces many schemes that cannot be described as neither the direct product nor the wreath product of smaller schemes.This research was supported by Com²MaC-KOSEF, Korea.     

Commutative Association Schemes Whose Symmetrizations Have Two Classes

If a symmetric association scheme of class two is realized as the symmetrization of a commutative association scheme, then it either admits a unique symmetrizable fission scheme of class three or four, or admits three fission schemes, two of which are class three and one is of class four. We investigate the classification problem for symmetrizable (commutative) association schemes of two-class symmetric association schemes. In particular, we give a classification of association schemes whose symmetrizations are obtained from completely multipartite strongly regular graphs in the notion of wreath product of two schemes. Also the cyclotomic schemes associated to Paley graphs and their symmetrizable fission schemes are discussed in terms of their character tables.     

一些(双)代数的不可约表示

本文完全刻画了下列代数k,k,k(x,Y,Z|ZX-qXZ=1-θY2,ZY=qYZ,YX= qXY>和M(p,q)=k的不可约表示．     

Two-Arc Transitive Graphs and Twisted Wreath Products

The paper addresses a part of the problem of classifying all 2-arc transitive graphs: namely, that of finding all groups acting 2-arc transitively on finite connected graphs such that there exists a minimal normal subgroup that is nonabelian and regular on vertices. A construction is given for such groups, together with the associated graphs, in terms of the following ingredients: a nonabelian simple group T, a permutation group P acting 2-transitively on a set , and a map F : Tsuch that x = x ^–1 for all x F() and such that Tis generated by F(). Conversely we show that all such groups and graphs arise in this way. Necessary and sufficient conditions are found for the construction to yield groups that are permutation equivalent in their action on the vertices of the associated graphs (which are consequently isomorphic). The different types of groups arising are discussed and various examples given.     

Semicomplete Permutational Wreath Products

A group is called semicomplete if every automorphism which induces the identity on the factor commutator group is inner. In this paper, we study the connection of the semicompleteness of the permutational wreath product W of two groups with the semicompleteness of these groups. We give necessary conditions under which the group W is semicomplete.2000 Mathematics Subject Classification: 20E22, 20E36     

右（左）逆半群的半直积及圈积

给出了两个幺半群的半直积及圈积为右（左）逆半群的充分必要条件，从而推广了[2]中两个幺半群的半幺直积和圈积为逆半群的充分必要条件．     

Weil Representations as Globally Irreducible Representations

The notion of globally irreducible representations of finite groups was introduced by B.H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell–Weil lattices of elliptic curves. It has been observed by R. Gow and Gross that irreducible Weil representations of certain finite classical groups lead to globally irreducible representations. In this paper we classify all globally irreducible representations coming from Weil representations of finite classical groups.     

Computation of Irreducible Decompositions of Permutation Representations of Wreath Products of Finite Groups

Computational Mathematics and Mathematical Physics - An algorithm for the computation of the complete set of primitive orthogonal idempotents of the centralizer ring of the permutation...     

Globally Irreducible Representations of Finite Groups and Integral Lattices

The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell--Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L ₂(q) and ²B₂(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (pⁿ-1)/2 of the symplectic group Sp_2n(p) (p 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pⁿ–1). A summary of currently known globally irreducible representations is given.     

The Automaton Realization of Iterated Wreath Products of Cyclic Groups

We study profinite groups which are infinitely iterated wreath products W _∞ = …?C _{n 2} ?C _{n 1} of finite cyclic groups via combinatorial language of transducers. Namely, we provide a naturally defined automaton realization of the group W _∞ by an automaton over a changing alphabet. Our construction gives a characterization of these profinite groups as automaton groups, i.e. as groups generated by a single automaton.     

Topologically Irreducible Representations and Radicals in Banach Algebras

It is shown that the topologically irreducible representationsof a normed algebra define a certain topological radical inthe same way that the strictly irreducible representations definethe Jacobson radical and that this radical can be strictly smallerthan the Jacobson radical. An abstract theory of ‘topologicalradicals’ in topological algebras is developed and usedto relate this radical to the Baer radical (prime radical).The relations with topologically transitive representationsand standard representations in the sense of Meyer are alsoexplored. 1991 Mathematics Subject Classification: 46H15, 46H25,16Nxx.     

Three-Class Association Schemes

We study (symmetric) three-class association schemes. The graphs with four distinct eigenvalues which are one of the relations of such a scheme are characterized. We also give an overview of most known constructions, and obtain necessary conditions for existence. A list of feasible parameter sets on at most 100 vertices is generated.     

Random Walks on Wreath Products of Groups

We bound the rate of convergence to uniformity for a certain random walk on the complete monomial groups GS _n for any group G. Specifically, we determine that n log n+ n log (|G|–1|) steps are both necessary and sufficient for ² distance to become small. We also determine that n log n steps are both necessary and sufficient for total variation distance to become small. These results provide rates of convergence for random walks on a number of groups of interest: the hyperoctahedral group ₂S _n , the generalized symmetric group _m S _n , and S _m S _n . In the special case of the hyperoctahedral group, our random walk exhibits the cutoff phenomenon.     

Finite Dimensional Representations of Symplectic Reflection Algebras Associated to Wreath Products II

In this note we refine the methods of Etingof and Montarani (Represent. Theory, 9: 457–467, 2005) in order to extend the main result of that article to a wider class of finite dimensional representations of wreath product symplectic reflection algebras. Presented by Claus M. Ringel.     

The Numbers of Permutation Groups Orbits and of the Wreath Product Orbits

ＴｈｅＮｕｍｂｅｒｓｏｆＰｅｒｍｕｔａｔｉｏｎＧｒｏｕｐｓＯｒｂｉｔｓａｎｄｏｆｔｈｅＷｒｅａｔｈＰｒｏｄｕｃｔＯｒｂｉｔｓ￥ＬｉＸｉａｎｚｈｏｎｇ；ＬｉｕＧｕａｎｇｌｉａｎｇ（ＰｕｙａｎｇＥｄｕｃａｔｉｏｎａｌＣｏｌｌｅｇｅ，Ｈｅｎａｎ，４５７００...     

Quasi-Invariant Measures and Irreducible Representations of the Inductive Limit of Special Linear Groups

Unitary representations of the group are constructed. The construction uses G-quasi-invariant measures on some G-spaces that are subspaces of the space of two-way infinite real matrices. We give a criterion for the irreducibility of these representations.     

Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues

We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X–E[X])/(Var(X))^1/2. We show that for a fixed set of arcs I ₁,...,I _N , the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles.     

Free Wreath Product by the Quantum Permutation Group

Let A be a compact quantum group, let nN ^* and let A _aut(X _n ) be the quantum permutation group on n letters. A free wreath product construction A*_w A _aut(X _n ) is introduced. This construction provides new examples of quantum groups, and is useful to describe the quantum automorphism group of the n-times disjoint union of a finite connected graph.     

The Finite Dimension Property of the Irreducible Representations of Noetherian p-Lie Algebras

We prove that the irreducible representations are finite-dimensional of the almost solvable restricted Lie algebras with the ascending chain condition for p-subalgebras over a perfect field.     

辫子群的全部二维不可约表示

通过直接解矩阵方程给出了Bn群的全部二维不可约表示．