Extended Affine Weyl Groups: Presentation by Conjugation via Integral Collection

We give several necessary and sufficient conditions for the existence of the presentation by conjugation for a non-simply laced extended affine Weyl group. We invent a computational tool by which one can determine simply the existence of the presentation by conjugation for an extended affine Weyl group. As an application, we determine the existence of the presentation by conjugation for a large class of extended affine Weyl groups.     

Nonreduced extended affine root systems of nullity 3

In 1985 K. Saito [S] introduced the concept of an extended affine root system (EARS). His study of these root systems was motivated by his interest in singularities. Later in [AABGP], this notion played an important role in the study of extended affine Lie algebras. Saito classified the EARS's of nullity ≤ 2 which have the further property that the quotient modulo a "marking" is reduced. In [AABGP], a construction was given of all EARS's, and this was used to give a classification of EARS's of reduced type. However, when the EARS was not reduced, the isomorphism problem for the construction is quite difficult and the classification was only done for EARS'S with nullity ≤ 2. The present paper extends this classification to nullity 3.     

3-维不定空间中晶体群的分类

设V是3-维不定空间,W是V中某个不可约根系的无限Weyl群,本文在仿射群A(V)中共轭的意义下,给出了点群为W的晶体群的分类。     

四维不定空间中晶体群的分类

设V是四维不定空间，W是V中某个不可约根系的无限Weyl群，本文给出了点群为W的晶体群在仿射群A（V）中的共轭类。     

REPRESENTATIONS OF AFFINE HECKEALGE BRAS OF TYPE G2

Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affne Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representations of Hq by using based rings of two-sided cells of an affne Weyl group W of type G2. We shall give the classification of irreducible representations of Hq. We also remark that a calculation in [11] actually shows that Theorem 2 in [1] needs a modification, a fact is known to Grojnowski and Tanisaki long time ago. In this paper we also show an interesting relation between Hq and an Hecke algebra corresponding to a certain Coxeter group. Apparently the idea in this paper works for all affne Weyl groups, but that is the theme of another paper.     

Special matchings and Coxeter groups

In the recent paper [Adv. Applied Math., 38 (2007), 210–226] it is proved that the special matchings of permutations generate a Coxeter group. In this paper we generalize this result to a class of Coxeter groups which includes many Weyl and affine Weyl groups. Our proofs are simpler, and shorter, than those in [loc. cit.] All authors are partially supported by EU grant HPRN-CT-2001-00272. Received: 30 October 2006     

双曲型5-维不定空间中晶体群的分类

设V是双曲型5-维不定空间，W是V中某个不可约根系的无限Weyl群。本文中，我们在仿射群A(V)中共轭的意义下，给出了点群为W的晶体群的分类。     

Minimal reduction type and the Kazhdan–Lusztig map

We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusztig in Lfromto, (2011) and that the Kazhdan–Lusztig map in Kazhdan and Lusztig, (1998) is a section of our map. This settles several conjectures in the literature. For classical groups, we prove more refined results by introducing and studying the “skeleta” of affine Springer fibers.     

单圈图的零度

图的零度是指在图的谱中特征值0的重数.在文献[2]中作者给出了刻画非奇异单圈图的充分条件,并提出了一个问题,即这个条件是否也是必要的.在本文中,我们先对这个问题作出肯定回答,然后介绍一个新的概念:保留点,最后通过最大匹配数给出公式计算单圈图的零度.     

A categorical approach to Weyl modules

Global and local Weyl modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained. In this paper we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that, unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.     

A note on the units of semigroup rings

In [2] Coleman and Enochs obtained results about the units of the polynomial ring R[x] for rings R satisfying a condi-tion which is, in some sense, a generalization of commutativity. In [3] some of these results were extended to group rings over an ordered group. In this note a class of rings larger than the class considered in [2] is used to extend the results in [2] and 3] to the semigroup ring RG, G an u.p, semigroup.

In the last section we give a necessary and sufficient condi-tion for an element to be a divisor of zero in RG where G is an u.p. semigroup.     

Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy

Following the approach of Carlet et al. (2011) [9], we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type A defined by Dubrovin and Zhang.     

A criterion of smoothness at infinity for an arithmetic quotient of the future tube

Let Γ be an arithmetic group of affine automorphisms of the n-dimensional future tube T. It is proved that the quotient space T/Γ is smooth at infinity if and only if the group Γ is generated by reflections and the fundamental polyhedral cone (“Weyl chamber”) of the group dΓ in the future cone is a simplicial cone (which is possible only for n ≤ 10). As a consequence of this result, a smoothness criterion for the Satake–Baily–Borel compactification of an arithmetic quotient of a symmetric domain of type IV is obtained.     

On the Lax Pairs of the Symmetric Painlevé Equations

The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N + 1 variables that admit the action of an extended affine Weyl group of type     , as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of ( N + 1) × ( N + 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type     on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group.     

A1型扩张仿射Lie代数的分次自同构群

文献[1]从Euclid空间R^v（v≥1）的一个半格S出发,定义了一个Jordan代数J（S）：然后通过Tits—Kantor-Koecher方法由J（S）构造出Lie代数G（J（S））．最后利用G（J（S））得到A1型扩张仿射Lie代数L（J（S））．本文给出v=2,S为格时。A1型扩张仿射Lie代数L（J（S））的Z^2一分次自同构群．     

A Characterization for a (2, 0)-Geodesic Affine Immersion

We give a characterization for a (2, 0)-geodesic affine immersion to an affine space by using its index of relative nullity. Especially, we prove a cylinder theorem for such a hypersurface. We also show a cylinder theorem for a (2, 0)-geodesic isometric immersion from a Kähler manifold and anti-Kähler manifold as a corollary.