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Let G be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, a configuration space of points in the affine Grassmannian is constructed. This configuration space gives a natural way of calculating the invariant vectors coming from webs. In the case of G = SL_3, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is CAT(0), is explained by the fact that affine buildings are CAT(0). In the case of G = SL_n, a sufficient condition for a set of webs to yield a basis is given. Using this condition and a generalization of a technique by Westbury, a basis is constructed for SL_n. Due to the geometric Satake correspondence there exists another natural basis of invariants, the Satake basis. This basis arises from the underlying geometry of the affine Grassmannian. There is an upper unitriangular change of basis from the basis constructed above to the Satake basis. An example is constructed showing that the Satake, web and dual canonical basis of the invariant space are all different. The natural action of rotation on tensor factors sends invariant space to invariant space. Since the rotation of web is still a web, the set of vectors coming from webs is fixed by this action. The Satake basis is also fixed, and an explicit geometric and combinatorial description of this action is developed.

Mental illness is an issue of concern to the general community, and is a major focus of professions such as psychology. Such professions demand extensive education and training, with the aim to develop a scientific understanding of mental illness that is portrayed in contrast to socially shared knowledge, or social representations. However, some aspects of these social representations may persist in conjunction with the development of scientific knowledge. The current study used a multimethodological approach to elucidate how such social representations may be transformed or modified by relevant education in psychology. Psychology students, non-psychology students and clinical psychologists participated in the current study to assess groups at different levels of psychology education and training. Four forms of data collection were used as part of a multimethodological approach. Intraindividual methods focused on the use of repertory grids and word associations to explore responses to the mentally ill as well as other relevant individuals such as the physically ill and mental health professionals. Inter-individual methods focused on social interaction in response to a case vignette of an individual with a mental disorder and group discussion within the educational setting. Several core themes described in previous research were identified consistently across different groups and different methodologies. Negative emotion, such as distress and sadness, impaired functioning, and the need for assistance were commonly used as indicators of mental health problems. One of the most prominent themes, however, was the notion of difference and distance. The mentally ill person was consistently differentiated, particularly from the self, even when the label of mentally ill was not imposed. The importance of the self was especially evident, acting as a means to define normality and difference. Several differences were also noticeable between different levels of education. A changing representation was evident from understandings of the mentally ill as crazy, viewed in a more stereotypical, negative and critical light, to representations of the mentally ill as sad, typified by greater sympathy. Social representations may therefore influence the social response to the mentally ill. Increasing education associated with scientific understandings was also characterised by exclusive technical discourse, a feature that may distance the psychologist from the general community. These findings are particularly relevant to how education affects social representations of mental illness and the mentally ill, as public campaigns seek to change community attitudes and understandings. In addition, there are particular implications for psychologists, in training or at work. While a primary goal for the psychologist is to empathise and connect with the individuals they are intending to assist, the emphasis on difference, in both social and scientific understandings of the mentally ill, may act as a barrier. The education and professional development of psychologists should incorporate an understanding of how such representations may influence professional practice.

Thesis (M. Sc.)--Carleton University, 2004. / Includes bibliographical references (p. 56-57). Also available in electronic format on the Internet.

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In the 1970's Serre conjectured a correspondence between modular forms and two-dimensional Galois representations. Ash, Doud, and Pollack have extended this conjecture to a correspondence between Hecke eigenclasses in arithmetic cohomology and n-dimensional Galois representations. We present some of the first examples of proven cases of this generalized conjecture.

A conflicted and contentious literature has emerged from the proposal that visuospatial information from static images can automatically trigger associated motor representations. Curiously, investigations into this visual-motor relationship have predominantly focused on images of manipulable objects, while relatively little work has made use of images of body parts- whose referents are represented directly in the motor system. Limited work has made use of hand images as task-irrelevant primes, in an effort to determine whether a hand image automatically evokes a motor representation of the viewer’s corresponding limb. The results of these studies have provided diverging evidence and have resulted in competing theoretical accounts. Here, I present results from a series of stimulus-response compatibility experiments that were designed to probe the nature of representations generated by static hand images, while also addressing potential methodological weaknesses of the previous works. The results show that both stimulus properties and task demands influence the way in which an image of a hand is coded. Notably, I provide clear evidence that motor representations can be evoked automatically by depictions of particular hand postures, but that these representations are not an automatic, ineluctable component of the general processing of any hand image. These results not only contribute to a more unified account of hand representations, but also have wider implications for our understanding of the conditions under which static displays can engage motor representations. / Graduate

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