Unitarizable Representations of Quivers

We investigate representations of *-algebras associated with posets. Unitarizable representations of the corresponding (bound) quivers (which are polystable representations for some appropriately chosen slope function) give rise to representations of these algebras. Considering posets which correspond to unbound quivers this leads to an ADE-classification which describes the unitarization behaviour of their representations. Considering posets which correspond to bound quivers, it is possible to construct unitarizable representations starting with polystable representations of related unbound quivers which can be glued together with a suitable direct sum of simple representations. Finally, we estimate the number of complex parameters parametrizing irreducible unitary non-equivalent representations of the corresponding algebras.     

动力学群微分表示的计算:以Lorentz群 SO(3,1) 为例

该文给出了动力学群在群参数空间以及陪集空间上的右、左微分表示和伴随微分表示的符号计算方法.作为例子, 计算了Lorentz 群SO(3,1)的6 -参数和3 -参数的右、左及伴随微分表示,这些表示是旋转群SO(3)关于欧拉角和极角的微分表示的相对论性推广.特别,作者给出了伴随微分表示的两种不同的3 -参数形式,同时也得到了Wigner小群SO(2,1) 和 SO(3)$的6 -参数和3 -参数的相应表示.这些表示在相对论性量子陀螺的研究中可得到应用.     

THE INFLUENCE OF TEACHERS’ PRESENTATIONS ON PUPILS’ MENTAL REPRESENTATIONS

Teachers use a variety of external representations to communicate mathematical ideas to their pupils. This paper reports a preliminary study of the internal mental representations that 6- and 7- year-old pupils form as a result of their interactions with the teacher's verbal, written, pictorial and concrete material representations, involving two-digit numbers and operations on them. The results presented here concern the picture-like mental representations that pupils use in performing two-digit calculations mentally. The evidence suggests that pupils seldom spontaneously visualise teachers’ representations or attempt mental manipulation of visual images to help with calculation. Pupils can, however, have mental representations which reproduce some aspects of the teachers’ representations.     

Pointed representations of Virasoro algebra

In this paper we study the pointed representations of the Virasoro algebra. We show that unitary irreducible pointed representations of the Virasoro algebra are Harish-Chandra representations, thus they either are of highest or lowest weights or have all weight spaces of dimension 1. Further, we prove that unitary irreducible weight representations of Virasoro superalgebras are either of highest weights or of lowest weights, hence they are also Harish-Chandra representations. This work was supported by CNSF     

Orthoscalar quiver representations corresponding to extended Dynkin graphs in the category of Hilbert spaces

It is known that finitely representable quivers correspond to Dynkin graphs and tame quivers correspond to extended Dynkin graphs. In an earlier paper, the authors generalized some of these results to locally scalar (later renamed to orthoscalar) quiver representations in Hilbert spaces; in particular, an analog of the Gabriel theorem was proved. In this paper, we study the relationships between indecomposable representations in the category of orthoscalar representations and indecomposable representations in the category of all quiver representations. For the quivers corresponding to extended Dynkin graphs, the indecomposable orthoscalar representations are classified up to unitary equivalence.     

PSL（2,13）的最小级连通3度弧传递陪集图表示

通过寻找给定群G的图表示,对PSL（2,13）的连通3度弧传递陪集图表示进行研究,得到了如下结果：PSL（2,13）的最小级连通3度弧传递陪集图表示的级是182.并且给出了该陪集图表示的例子.     

On the Existence of Auslander–Reiten Sequences of Group Representations II

This is the second part of our study of the existence ofAuslander–Reiten sequences of group representations. In Part I weconsidered representations of group schemes in characteristic 0; in thispart we consider representations of group schemes in characteristicp; and in Part III we give applications to representations ofabstract groups and Lie algebras.     

Multiplier representations of abelian groups

The multiplier representations of a locally compact abelian group are classified, in the case there are only type I representations, and a simple criterion is obtained for determining when only type I representations occur. It is also shown that all the irreducible multiplier representations belonging to a fixed multiplier have the same dimension. To obtain these results we have to extend the little group method of Mackey and Blattner to handle multiplier representations of nonseparable groups.     

Indecomposable representations for real roots of a wild quiver

For a given quiver and dimension vector, Kac has shown that there is exactly one indecomposable representation up to isomorphism if and only if this dimension vector is a positive real root. However, it is not clear how to compute these indecomposable representations in an explicit and minimal way, and the properties of these representations are mostly unknown. In this note we study representations of a particular wild quiver. We define operations which act on representations of this quiver, and using these operations we construct indecomposable representations for positive real roots, compute their endomorphism rings and show that these representations are tree representations. The operations correspond to the fundamental reflections in the Weyl group of the quiver. Our results are independent of the characteristic of the field.     

On the Existence of Auslander–Reiten Sequences of Group Representations. I

This is the first part of our study of the existence ofAuslander–Reiten sequences of group representations. In this part weconsider representations of group schemes in characteristic 0; in Part II weconsider representations of group schemes in characteristic p; andin Part III we give applications to representations of groups and Liealgebras.     

The multiplicative meaning conveyed by visual representations

Research and practitioner articles advocate the use of visual representations in scaffolding elementary students’ learning of multiplication and division. Prior research suggests students use different strategies when provided with different visualized representations of multiplication and division. However, there is relatively little study examining how children’s multiplicative reasoning corresponds with different representations. The present study collected data from 182 elementary students responding to set, area, and length representations of multiplication/division. Rasch modeling was used to estimate item difficulty statistics to measure differences between visual representations. Results suggest that visual representations differed primarily in how unit was represented and quantified, and not regarding the form of representation (set, area, length).     

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.     

On quasifree representations of infinite dimensional symplectic group

We consider an infinite dimensional generalization of metaplectic representations (Weil representations) for the (double covering of) symplectic group. Given quasifree states of an infinite dimensional CCR algebra, projective unitary representations of the infinite dimensional symplectic group are constructed via unitary implementors of Bogoliubov automorphisms. Complete classification of these representations up to quasi-equivalence is obtained.     

On the Existence of Auslander–Reiten Sequences of Group Representations. III

This is the third and final part of our study of the existence of Auslander–Reiten sequences ofgroup representations. In Part I we considered representations of group schemes in characteristic 0.In Part II we considered representations of group schemes in characteristic p. In this partwe give applications to representations of abstract groups and Lie algebras.     

Gaussian random fields

Nonanticipative representations of Gaussian random fields equivalent to the two-parameter Wiener process are defined, and necessary and sufficient conditions for their existence derived. When such representations exist they provide examples of canonical representations of multiplicity one. In contrast to the one-parameter case, examples are given where nonanticipative representations do not exist. Nonanticipative representations along increasing paths are also studied.University of Minnesota and University of North Carolina. Presented at theWorkshop on Stochastic Processes in Infinite Dimensional Spaces and Random Fields, held at UCLA in April 1979.     

Measures in the space of planes and convex bodies

The present paper gives two representations (so-called flag representations) for the measure of planes intersecting a convex body. These representations are obtained by means of a stochastic approximation of a convex body depending on a distribution P that are defined on the surface of the body. Also it is proved that the flag representations do not depend on the density of P.     

Representations of efficient score for coarse data problems based on Neumann series expansion

We derive new representations of the efficient score for the coarse data problems based on Neumann series expansion. The representations can be applied to both ignorable and nonignorable coarse data. Approximations to the new representations may be used for computing locally efficient scores in such problems. We show that many of the successive approximation approaches to the computation of the locally efficient score proposed in the literature for such problems can be derived as special cases of the representations. In addition, the representations lead to new algorithms for computing the locally efficient scores for the coarse data problems.     

Representations, convolution square roots and spherical vectors

We extend results on square integrable representations of a locally compact group to subrepresentations of a representations induced from a character of a closed subgroup. We do so working in the framework of quotient representations of ∗-algebras.     

Dual pair correspondences for non-linear covers of orthogonal groups

In this paper we study compact dual pair correspondences arising from smallest representations of non-linear covers of odd orthogonal groups. We identify representations appearing in these correspondences with subquotients of cohomologically induced representations.