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Mat. Zametki, 2015, Volume 97, Issue 1, Pages 35–47 (Mi mz10384)  

This article is cited in 7 scientific papers (total in 7 papers)

Essential Signatures and Canonical Bases of Irreducible Representations of the Group $G_{2}$

A. A. Gornitskii

M. V. Lomonosov Moscow State University

Abstract: We consider representations of simple Lie algebras and the problem of constructing a “canonical” weight basis in an arbitrary irreducible finite-dimensional highest-weight module. Vinberg suggested a method for constructing such bases by applying the lowering operators corresponding to all negative roots to the highest-weight vector and put forward a number of conjectures on the parametrization and structure of such bases. It follows from papers by Feigin, Fourier, and Littelmann that these conjectures are true for the cases of $A_n$ and $C_{n}$. In the present paper, we prove these conjectures for the case of $G_2$ by using a different approach suggested by Vinberg.

Keywords: simple Lie algebra, group $G_{2}$, irreducible representation, canonical base, essential signature, weight basis.

DOI: https://doi.org/10.4213/mzm10384

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English version:
Mathematical Notes, 2015, 97:1, 30–41

Bibliographic databases:

Document Type: Article
UDC: 512.54
Received: 30.06.2013
Revised: 18.02.2014

Citation: A. A. Gornitskii, “Essential Signatures and Canonical Bases of Irreducible Representations of the Group $G_{2}$”, Mat. Zametki, 97:1 (2015), 35–47; Math. Notes, 97:1 (2015), 30–41

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. T. Backhaus, X. Fang, G. Fourier, “Degree cones and monomial bases of Lie algebras and quantum groups”, Glasg. Math. J., 59:3, 3 (2017), 595–621  crossref  mathscinet  zmath  isi  scopus
    2. X. Fang, G. Fourier, P. Littelmann, “Essential bases and toric degenerations arising from birational sequences”, Adv. Math., 312 (2017), 107–149  crossref  mathscinet  zmath  isi  scopus
    3. X. Fang, G. Fourier, P. Littelmann, “On toric degenerations of flag varieties”, Representation Theory - Current Trends and Perspectives, EMS Ser. Congr. Rep., eds. H. Krause, P. Littelmann, G. Malle, KH. Neeb, C. Schweigert, Eur. Math. Soc., 2017, 187–232  mathscinet  zmath  isi
    4. X. Fang, P. Littelmann, M. Pabiniak, “Simplices in Newton-Okounkov bodies and the Gromov width of coadjoint orbits”, Bull. London Math. Soc., 50:2 (2018), 202–218  crossref  zmath  isi  scopus
    5. A. A. Gornitskii, “Essential signatures and canonical bases of irreducible representations of simple Lie algebras”, Russian Math. Surveys, 73:5 (2018), 925–927  mathnet  crossref  crossref  adsnasa  isi  elib
    6. Backhaus T., Kus D., “The Pbw Filtration and Convex Polytopes in Type B”, J. Pure Appl. Algebr., 223:1 (2019), 245–276  crossref  mathscinet  zmath  isi  scopus
    7. Gornitskii A.A., “Essential Signatures and Monomial Bases For B-N and D-N”, J. Lie Theory, 29:1 (2019), 277–302  isi
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