7,460 research outputs found

    The anti-spherical category

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    We study a diagrammatic categorification (the "anti-spherical category") of the anti-spherical module for any Coxeter group. We deduce that Deodhar's (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients, and that a monotonicity conjecture of Brenti's holds. The main technical observation is a localisation procedure for the anti-spherical category, from which we construct a "light leaves" basis of morphisms. Our techniques may be used to calculate many new elements of the pp-canonical basis in the anti-spherical module.Comment: Best viewed in colo

    Approximate subgroups of linear groups

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    We establish various results on the structure of approximate subgroups in linear groups such as SL_n(k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of SL_n(F_q) which generates the group must be either very small or else nearly all of SL_n(F_q). The argument generalises to other absolutely almost simple connected (and non-commutative) algebraic groups G over a finite field k. In a subsequent paper, we will give applications of this result to the expansion properties of Cayley graphs.Comment: 48 page

    Linear Approximate Groups

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    This is an informal announcement of results to be described and proved in detail in a paper to appear. We give various results on the structure of approximate subgroups in linear groups such as \SL_n(k). For example, generalising a result of Helfgott (who handled the cases n=2n = 2 and 3), we show that any approximate subgroup of \SL_n(\F_q) which generates the group must be either very small or else nearly all of \SL_n(\F_q). The argument is valid for all Chevalley groups G(\F_q).Comment: 11 pages. Submitted, Electronic Research Announcements. Small change

    The Dimensions of Field Theory : From Particles to Strings

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    This is an editorial summary of the contents of a Book comprising a set of Articles by acknowledged experts dealing with the impact of Field Theory on major areas of physics (from elementary particles through condensed matter to strings), arranged subjectwise under six broad heads. The Book which emphasizes the conceptual, logical and formal aspects of the state of the art in these respective fields, carries a Foreword by Freeman Dyson, and is to be published by the Indian National Science Academy on the occasion of the International Mathematical Year 2000. The authors and full titles of all the Articles (33) are listed sequentially (in the order of their first appearance in the narration) under the bibliography at the end of this Summary, while a few of the individual articles to appear in the Book are already available on the LANL internet.Comment: LaTex file, 24 page

    Pascual Jordan, his contributions to quantum mechanics and his legacy in contemporary local quantum physics

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    After recalling episodes from Pascual Jordan's biography including his pivotal role in the shaping of quantum field theory and his much criticized conduct during the NS regime, I draw attention to his presentation of the first phase of development of quantum field theory in a talk presented at the 1929 Kharkov conference. He starts by giving a comprehensive account of the beginnings of quantum theory, emphasising that particle-like properties arise as a consequence of treating wave-motions quantum-mechanically. He then goes on to his recent discovery of quantization of ``wave fields'' and problems of gauge invariance. The most surprising aspect of Jordan's presentation is however his strong belief that his field quantization is a transitory not yet optimal formulation of the principles underlying causal, local quantum physics. The expectation of a future more radical change coming from the main architect of field quantization already shortly after his discovery is certainly quite startling. I try to answer the question to what extent Jordan's 1929 expectations have been vindicated. The larger part of the present essay consists in arguing that Jordan's plea for a formulation without ``classical correspondence crutches'', i.e. for an intrinsic approach (which avoids classical fields altogether), is successfully addressed in past and recent publications on local quantum physics.Comment: More biographical detail, expansion of the part referring to Jordan's legacy in quantum field theory, 37 pages late
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