Kazhdan-Lusztig polynomials and canonical basis

International audience We show that dual canonical basis elements of the quantum polynomial ring in $n^2$ variables can be expressed as specializations of dual canonical basis elements of $0$-weight spaces of other quantum polynomial rings. Our results rely upon the natural appearance in the quantum polynomial ring of Kazhdan-Lusztig polynomials, $R$-polynomials, and certain single and double parabolic generalizations of these. Nous démontrons que des éléments de la base canonique duale de l'anneau quantique des polynômes en $n^2$ variables peuvent s'exprimer en termes des spécialisations d'éléments de la base canonique duale des espaces de poids $0$ d'autres anneaux quantiques. Nos résultats dépendent fortement de l'apparition naturelle des polynômes de Kazhdan-Lusztig, des $R$-polynômes, et de certaines généralisations simplement et doublement paraboliques de ces polynômes dans l'anneau quantique.

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The Einstein–Maxwell-particle system in the York canonical basis of ADM tetrad gravity. Part 2. The weak field approximation in the 3-orthogonal gauges and Hamiltonian post-minkowskian gravity: the N-body problem and gravitational waves with asymptotic background 1This paper is one of three companion papers published in the same issue of Can. J. Phys.

Canadian Journal of Physics ◽

10.1139/p11-101 ◽

2012 ◽

Vol 90 (11) ◽

pp. 1077-1130 ◽

Cited By ~ 11

Author(s):

David Alba ◽

Luca Lusanna

Keyword(s):

Electromagnetic Field ◽

Gravitational Waves ◽

Particle System ◽

Canonical Basis ◽

Field Approximation ◽

Weak Field ◽

N Body Problem ◽

Hamilton Equations ◽

Weak Field Approximation ◽

Tetrad Gravity

In this second paper we define a post-minkowskian (PM) weak field approximation leading to a linearization of the Hamilton equations of Arnowitt–Deser–Misner (ADM) tetrad gravity in the York canonical basis in a family of nonharmonic 3-orthogonal Schwinger time gauges. The York time 3K (the relativistic inertial gauge variable, not existing in newtonian gravity, parametrizing the family, and connected to the freedom in clock synchronization, i.e., to the definition of the the shape of the instantaneous 3-spaces) is set equal to an arbitrary numerical function. The matter are considered point particles, with a Grassmann regularization of self-energies, and the electromagnetic field in the radiation gauge: an ultraviolet cutoff allows a consistent linearization, which is shown to be the lowest order of a hamiltonian PM expansion. We solve the constraints and the Hamilton equations for the tidal variables and we find PM gravitational waves with asymptotic background (and the correct quadrupole emission formula) propagating on dynamically determined non-euclidean 3-spaces. The conserved ADM energy and the Grassmann regularization of self-energies imply the correct energy balance. A generalized transverse–traceless gauge can be identified and the main tools for the detection of gravitational waves are reproduced in these nonharmonic gauges. In conclusion, we get a PM solution for the gravitational field and we identify a class of PM Einstein space–times, which will be studied in more detail in a third paper together with the PM equations of motion for the particles and their post-newtonian expansion (but in the absence of the electromagnetic field). Finally we make a discussion on the gauge problem in general relativity to understand which type of experimental observations may lead to a preferred choice for the inertial gauge variable 3K in PM space–times. In the third paper we will show that this choice is connected with the problem of dark matter.

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Fuzzy decision implication canonical basis

International Journal of Machine Learning and Cybernetics ◽

10.1007/s13042-017-0780-7 ◽

2018 ◽

Vol 9 (11) ◽

pp. 1909-1917 ◽

Cited By ~ 3

Author(s):

Yanhui Zhai ◽

Deyu Li ◽

Kaishe Qu

Keyword(s):

Canonical Basis ◽

Fuzzy Decision

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Quantum groups via cyclic quiver varieties I

Compositio Mathematica ◽

10.1112/s0010437x15007551 ◽

2015 ◽

Vol 152 (2) ◽

pp. 299-326 ◽

Cited By ~ 6

Author(s):

Fan Qin

Keyword(s):

Quantum Groups ◽

Bilinear Form ◽

Canonical Basis ◽

Natural Basis ◽

Grothendieck Ring ◽

Quiver Varieties ◽

Enveloping Algebra ◽

Dual Canonical Basis ◽

Quantized Enveloping Algebra ◽

Cyclic Quiver

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.

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The Canonical Basis for the Quantum Group of Type B2

Acta Mathematica Scientia ◽

10.1016/s0252-9602(13)60099-5 ◽

2013 ◽

Vol 33 (5) ◽

pp. 1499-1506

Author(s):

Jie ZHANG

Keyword(s):

Quantum Group ◽

Canonical Basis

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The Hecke Algebra and Kazhdan–Lusztig Polynomials

Introduction to Soergel Bimodules - RSME Springer Series ◽

10.1007/978-3-030-48826-0_3 ◽

2020 ◽

pp. 39-58

Author(s):

Ben Elias ◽

Shotaro Makisumi ◽

Ulrich Thiel ◽

Geordie Williamson

Keyword(s):

Hecke Algebra ◽

Lusztig Polynomials

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Stages of formation of religious and philosophical doctrine of L. Silenko

Ukrainian Religious Studies ◽

10.32420/2005.34.1588 ◽

2005 ◽

pp. 133-145

Author(s):

T.R. Bednarchyk

Keyword(s):

Canonical Basis ◽

Religious Organizations ◽

Religious Movement ◽

Religious Organization ◽

Philosophical Doctrine

The religious and philosophical doctrine of L. Silenko is the canonical basis of the doctrine of the religious organization of the Sons and Daughters of Ukraine of the Native Ukrainian National Faith (OSCE RUNVira) in the Western Diaspora and religious organizations that emerged on the basis of RUNVira in independent Ukraine. This doctrine became final in the early 1980's and is in this form the most famous in modern Ukraine. RUNVira is the most studied denomination of the Ukrainian denominational religious movement in the diaspora.

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Incremental Method of Generating Decision Implication Canonical Basis

10.21203/rs.3.rs-524907/v1 ◽

2021 ◽

Author(s):

Shaoxia Zhang ◽

Deyu Li ◽

Yanhui Zhai

Keyword(s):

Formal Concept Analysis ◽

Canonical Basis ◽

Concept Analysis ◽

Experimental Results ◽

Incremental Algorithm ◽

Compact Representation ◽

Formal Concept ◽

Practical Applications ◽

Incremental Method ◽

Elementary Representation

Abstract Decision implication is an elementary representation of decision knowledge in formal concept analysis. Decision implication canonical basis (DICB), a set of decision implications with completeness and nonredundancy, is the most compact representation of decision implications. The method based on true premises (MBTP) for DICB generation is the most efficient one at present. In practical applications, however, data is always changing dynamically, and MBTP has to re-generate inefficiently the whole DICB. This paper proposes an incremental algorithm for DICB generation, which obtains a new DICB just by modifying and updating the existing one. Experimental results verify that when the samples in data are much more than condition attributes, which is actually a general case in practical applications, the incremental algorithm is significantly superior to MBTP. Furthermore, we conclude that, even for the data in which samples is less than condition attributes, when new samples are continually added into data, the incremental algorithm must be also more efficient than MBTP, because the incremental algorithm just needs to modify the existing DICB, which is only a part of work of MBTP.

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