7,460 research outputs found
The anti-spherical category
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 -canonical basis in the
anti-spherical module.Comment: Best viewed in colo
Approximate subgroups of linear groups
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
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 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
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
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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