78 research outputs found
Counting filter restricted paths in lattice
Get PDFWe derive a path counting formula for two-dimensional lattice path model on a
plane with filter restrictions. A filter is a line that restricts the path
passing it to one of possible directions. Moreover, each path that touches this
line is assigned a special weight. The periodic filter restrictions are
motivated by the problem of tensor power decomposition for representations of
quantum at roots of unity. Our main result is the explicit
formula for the weighted number of paths from the origin to a fixed point
between two filters in this model.Comment: 32 page
Limit shape of probability measure on tensor product of algebra modules
Get PDFWe study a probability measure on integral dominant weights in the
decomposition of -th tensor power of spinor representation of the Lie
algebra . The probability of the dominant weight is defined
as the ratio of the dimension of the irreducible component of divided
by the total dimension of the tensor power. We prove that as the measure weakly converges to the radial part of the
-invariant measure on induced by the Killing form. Thus,
we generalize Kerov's theorem for to .Comment: Submitted to Zapiski Nauchnykh Seminarov POM
Skew Howe duality and limit shapes of Young diagrams
Get PDFWe consider the skew Howe duality for the action of certain dual pairs of Lie
groups on the exterior algebra as a probability measure on Young diagrams by the
decomposition into the sum of irreducible representations. We prove a
combinatorial version of this skew Howe for the pairs , ,
, and using crystal bases, which allows us to interpret the skew
Howe duality as a natural consequence of lattice paths on lozenge tilings of
certain partial hexagonal domains. The -representation multiplicity is
given as a determinant formula using the Lindstr\"om-Gessel-Viennot lemma and
as a product formula using Dodgson condensation. These admit natural
-analogs that we show equals the -dimension of a -representation (up
to an overall factor of ), giving a refined version of the combinatorial
skew Howe duality. Using these product formulas (at ), we take the
infinite rank limit and prove the diagrams converge uniformly to the limit
shape.Comment: 57 pages, 15 figures, 2 tables; v3 fixed typos, added comparison to
Biane's result, updated references, fixed typos; v2 fixed typos in Theorem
4.10, 4.14, shorter proof of Theorem 4.6 (thanks to C. Krattenthaler), proved
of Conjecture 4.17 in v
Limit shape for infinite rank limit of tensor power decomposition for Lie algebras of series so(2n+1)
Get PDFWe consider the Plancherel measure on irreducible components of tensor powers
of the spinor representation of so(2n+1). The irreducible representations
correspond to the generalized Young diagrams. With respect to this measure the
probability of an irreducible representation is the product of its multiplicity
and dimension, divided by the total dimension of the tensor product. We study
the limit shape of the generalized Young diagram when the tensor power N and
the rank n of the algebra tend to infinity with N/n fixed. We derive an
explicit formula for the limit shape and prove convergence to it in
probability. We prove central limit theorem for global fluctuations around the
limit shape.Comment: 36 pages, 7 figures. In version4 we have added proof of central limit
theorem for global fluctuations around the limit shape, that relies on
Christoffel transformation of Krawtchouk orthogonal polynomials. Discussion
of relation to Berele insertion and skew Howe duality is added. The code that
was used to produce the Figures is available at
https://github.com/naa/young-diagrams
Центральные меры в графах, связанных с графом Юнга
Get PDFПусть G = (V, E) - ориентированный граф.
Рассмотрим градуированный натуральными числами граф, каждый уровень которого -
копия множества V, а ребро с i-го уровня на (i+1)-й проводится в случае,
если между соответствующими вершинами есть путь в G. Применяя эту конструкцию
к графу диаграмм Юнга получаем градуированный граф, пути в котором соответствуют
цепочкам вложенных диаграмм Юнга. С помощью леммы Линдстрема-Гесселя-Вьенно
перечисление путей в таком графе сводится к вычислению определителей, причём это можно
делать разными способами. В ряде случаев эти определители вычисляются явно. В частности,
с помощью этого вычисления удаётся описать центральные меры, соответствующие двустрочечным диаграммам.Let G = (V, E) be a directed graph.
Consider a graph graded with positive integers, each level of which is a copy of the set V, and an edge from the i-th level to the (i + 1)-th level is drawn if there is a path in G between the corresponding vertices. Applying this construction to the graph of Young diagrams, we obtain a graded graph, the paths in which correspond to chains of nested Young diagrams. With the help of the Lindström - Gessel - Vienno lemma, the enumeration of paths in such a graph is reduced to the calculation of determinants, and this can be done in different ways. In some cases, these determinants are calculated explicitly. In particular, with the help of this calculation, it is possible to describe the central measures corresponding to two-row diagrams
Post-COVID junior physics lab: The new normal
Get PDFPhysics laboratory is the most challenging aspect of teaching physics in a pandemic environment: How can we teach experimental skills when students are not in the lab? How do we ensure that both on-campus and online students develop relevant experimental skills and enjoy labs? Finally, how do we ensure COVID safety when students work in groups? In junior physics labs there is an additional challenge of scaling-up any teaching approach to large student cohorts. At the School of Physics, we teach cohorts of ~800 students per semester over four units of study at different levels: fundamental, regular and advanced. In this presentation we will share our experience and lessons learned over the last three-four years moving from teaching labs in the pre-pandemic world to the current new normal that includes both on-campus and online labs with hundreds of students in each stream.
Back in 2019, our Junior Physics Labs were very traditional: printed lab manuals, hand-written logbooks, bench notes as supportive materials, crowded classes, hand-drawn graphs, in-person paper tests, etc. We just moved into a new beautiful lab space and had been working on modifying lab curriculum, as well as lab equipment which had been largely unchanged for 20 years.
However, in early 2020 the COVID-19 pandemic forced universities, including The University of Sydney (USYD), to move all classes online. For us this happened right at the start of semester, so it was necessary to quickly find a way to run labs in an online format. This included both running the experiments and managing all assignments, groupwork, and logbooks online. After some trial and error (including hybrid) over 2020-2021, we have set up completely independent online labs which now run in parallel with the campus labs and receive good feedback from remote students. They also provide a fallback plan for students who are in COVID-19 isolation and cannot attend the labs in-person.
This transition also required a new approach to labs navigation on Canvas (web-based learning management system used by USYD) so we designed and developed new pages for both on-campus and online streams so that students can easily find required information and materials for each week. We introduced e-Lab manuals, shared e-logbooks, online quizzes, practical online tests, videos, and simulators which are now used in both in-person and online labs, elevating student experience and simplifying lab coordination and management. The main software tools that we use are Canvas, Zoom, and MS Office 365 (or Google Docs/Sheets). In addition to these we use mobile apps, e.g. Phyphox, and simulators, e.g. MultiSim and Phet, for doing or simulating experiments at home.
Fast forward to 2022, physics labs at The University of Sydney have been returned to the fully face-to-face mode, though many students are still overseas. It is likely that many will also prefer to study remotely in the longer-term. In this presentation we discuss the rationale behind incorporating features of online labs into face-to-face labs and discuss how to run engaging and fun labs while maintaining appropriate social distancing and hygiene standards
On string functions and double-sum formulas
Get PDFString functions are important building blocks of characters of integrable
highest modules over affine Kac--Moody algebras. Kac and Peterson computed
string functions for affine Lie algebras of type in terms of
Dedekind eta functions. We produce new relations between string functions by
writing them as double-sums and then using certain symmetry relations. We
evaluate the series using special double-sum formulas that express Hecke-type
double-sums in terms of Appell--Lerch functions and theta functions, where we
point out that Appell--Lerch functions are the building blocks of Ramanujan's
classical mock theta functions.Comment: 24 page
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