A Note on Restricted Representations of the Witt Superalgebras

Let F be an algebraically closed field of prime characteristic p 〉 3, and W（n） the Witt superalgebra over F, which is the Lie superalgebra of superderivations of the Grassmann algebra in n indeterminates. The dimensions of simple atypical modules in the restricted supermodule category for W（n） are precisely calculated in this paper, and thereby the dimensions of all simple modules can be precisely given. Moreover, the restricted supermodule category for W（n） is proved to have one block.     

Root Systems for Asymmetric Geometric Representations of Coxeter Groups

Results are obtained concerning root systems for asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include certain restrictions of all Kac–Moody Weyl groups. In particular, a characterization of when a nontrivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.     

Poincaré Series of the Weyl Groups of the Elliptic Root Systems A 1 (1,1), A 1 (1,1)* and A 2 (1,1)

We calculate the Poincaré series of the elliptic Weyl group W(A ₂ ^(1,1)), which is the Weyl group of the elliptic root system of type A ₂ ^(1,1). The generators and relations of W(A ₂ ^(1,1)) have been already given by K. Saito and the author.     

局限李超代数的局限表示

本文对如下结果给出一个构造性、初等的、简洁的证明:有限维局限李超代数的有限维局限表示均完全可约当且仅当它的奇部为0而偶部是一个环面.这使得李代数中相应结果成为特例.     

Spin Representations and Powers of 2

The elementary divisors of the reduced spin 2–decomposition matrix for the double covers of the finite symmetric groups are described. In addition, the maximal power of 2 dividing all spin character values on a fixed conjugacy class corresponding to a cycle type with odd parts only is determined.     

Maximal Strings in the Crystal Graph of Spin Representations of the Symmetric and Alternating Groups

We define a block-reduced version of the crystal graph of spin representations of the symmetric and alternating groups, and separate it into layers, each obtained by translating the previous layer and, possibly, adding new defect zero blocks. We demonstrate that each layer has weight-preserving central symmetry, and study the sequence of weights occurring in the maximal strings.

The Broué conjecture, that a block with abelian defect group is derived equivalent to its Brauer correspondent, has been proven for blocks of cyclic defect group and verified for many other blocks. This article is part of a study of the spin block case.     

On a Construction of the Basic Spin Representations of Symmetric Groups

We present an inductive method for constructing the basic spin representations of the double covers of the symmetric groups over fields of any characteristic.     

A Weighted Enumeration of Maximal Chains in the Bruhat Order

Given a finite Weyl group W with root system , assign the weight to each covering pair in the Bruhat order related by the reflection corresponding to . Extending this multiplicatively to chains, we prove that the sum of the weights of all maximal chains in the Bruhat order has an explicit product formula, and prove a similar result for a weighted sum over maximal chains in the Bruhat ordering of any parabolic quotient of W. Several variations and open problems are discussed.     

Root Systems and Length Functions

We introduce root systems for those imprimitive complex reflection groups which are generated by involutory reflections, and study the associated length functions. These have many properties in common with the usual length functions for finite Weyl groups.     

Bi-Element Representations of Ternary Groups#

General properties of ternary semigroups and groups are consi-dered. The bi-element representation theory in which every representation matrix corresponds to a pair of elements is built up. Connections with the standard theory are considered and several concrete examples are constructed. For the sake of clarity and completeness the shortened versions of classical Gluskin-Hosszú and Post theorems are also presented.     

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．     

RESTRICTED FLOWS OF A HIERARCHYOF INTEGRABLE DISCRETE SYSTEMS

1.IntroductionTherestrictedflowsofsolitonhierarchyhavebeenextensivelystudied(see,forexample,[1--7]).Theapproachforconstructingrestrictedflowsofsolitonhierarchycanalsobeappliedtoobtainrestrictedflows(discretemaps)ofahierarchyofdiscreteintegrablesystems(nonlineardifferential-differenceequations)IS,9].TheserestrictedflowshavetheformofLagrangeequationsandthereforecanmodelphysicallyinterestingprocesses.Wesupposethatthehierarchyofdiscreteintegrablesystems(DIS)isassociatedwithadiscreteisospectralp…     

Weil Representations and Some Nonreductive Dual Pairs in Symplectic and Unitary Groups

We study the restriction of the Weil representations of symplectic and unitary groups to subgroups that are the centralizers of certain elements, and show that these are multiplicity free. This work is along the line of the Howe philosophy for so-called dual pairs in the symplectic and unitary groups, but our dual pairs are not reductive. We also obtain an explicit formula for the character of the restriction to the centralizer of a regular unipotent element.     

典型根系中不可约子根系的数目

Let Ф be an irreducible root system of classical type. In this short note, we study the irreducible subsystems of Ф and compute the number of irreducible subsystems of any rank k in Ф.     

一类特殊广义Kac-Moody代数虚根决定的反射与Weyl群间的关系(英)

对有限型李代数g（A）,相应于每个根α的反射r_α均在g（A）的Weyl群W中．当g（A）为可对称化的不定型Kac－Moody代数时,若α为一虚根且（α,α）＜0,则亦可定义反射r_α并有r_α∈－W或r_α是－W中元与一个图自同构之积（见[3]）．本文给出了一类秩为3的广义Kac－Moody代数的虚根系,然后讨论了一类特殊的广义Kac－Moody代数的虚根决定的反射与Weyl群之间的关系．     

On Minimal Permutation Representations of Classical Simple Groups

We describe minimal permutation representations, i.e., faithful permutation representations of least degree, of classical finite simple groups as groups of automorphisms of simple Lie algebras.     

A Residue Calculus for Root Systems

Let V be a finite-dimensional real vector space on which a root system is given. Consider a meromorphic function on V =V+iV, the singular locus of which is a locally finite union of hyperplanes of the form V , = s, , s . Assume is of suitable decay in the imaginary directions, so that integrals of the form _+iV , d make sense for generic V. A residue calculus is developed that allows shifting . This residue calculus can be used to obtain Plancherel and Paley–Wiener theorems on semisimple symmetric spaces.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

On Blocks in Restricted Representations of Lie Superalgebras of Cartan Type

Let g be a restricted Lie superalgebra of Cartan type W (n), S(n) or H(n) over an algebraically closed field k of prime characteristic p > 3, in the sense of modular version of Kac's definition in 1977. In this note, we show that the restricted representation category over g has only one block (reckoning parities in). This phenomenon is very different from the case of characteristic zero.