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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Numerology of square equations


Author: N. A. Vavilov
Translated by: the author
Original publication: Algebra i Analiz, tom 20 (2008), nomer 5.
Journal: St. Petersburg Math. J. 20 (2009), 687-707
MSC (2000): Primary 20G15, 20G35
DOI: https://doi.org/10.1090/S1061-0022-09-01068-1
Published electronically: July 21, 2009
MathSciNet review: 2492358
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Abstract: In the present work, which is a sequel of the paper ``Can one see the signs of structure constants?'', we describe how one can see the form and the signs of the senior Weyl orbit of equations on the highest weight orbit directly in the weight diagram of microweight representations and adjoint representations for the simply-laced case. As special cases, the square equations we consider include the vanishing of second order minors, Plücker equations in polyvector and adjoint representations of classical groups, Cartan equations in spin and half-spin representations, Borel-Freudenthal equations defining the projective octave plane $ \operatorname{E}_6/P_1$, and most of the equations defining Freudenthal's variety $ \operatorname{E}_7/P_7$. In view of forthcoming applications to the construction of decomposition of unipotents in the adjoint case, special emphasis is placed on the senior Weyl orbit of equations for the adjoint representations of groups of types $ \operatorname{E}_6$, $ \operatorname{E}_7$, and $ \operatorname{E}_8$. This orbit consists of 270, 756, or 2160 equations, respectively, and we minutely discuss their form and signs. This generalizes Theorem 3 of the preceding paper ``A third look at weight diagrams'', where we considered microweight representations of $ \operatorname{E}_6$ and $ \operatorname{E}_7$.


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Additional Information

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, 198504 St. Petersburg, Russia
Email: nikolai-vavilov@yandex.ru

DOI: https://doi.org/10.1090/S1061-0022-09-01068-1
Keywords: Chevalley groups, polyvector representations, spin representations, minimal modules, highest weight orbit, weight diagram, crystal graph, standard monomial theory, decomposition of unipotents, Pl\"ucker equations, Cartan equations, Borel--Freudenthal equations, projective octave plane
Received by editor(s): April 1, 2007
Published electronically: July 21, 2009
Additional Notes: The basic ideas, which eventually led to the present paper, as also to \cite{47, 4} — a third look — were developed by the author in 1995 at the Universität Bielefeld (with the support of AvH-Stiftung, SFB-343 and INTAS 93-436). Preliminary versions of this text were taking shape since 1997, at the Universitá Milano I (with the support of the Cariplo Foundation for Fundamental Research), at the Newton Institute for Mathematical Sciences at Cambridge University, and at Bar Ilan University. At the final stage of this work the author was supported by RFBR 03-01-00349 (POMI RAN), by INTAS 00-566, and INTAS 03-51-3251, and by an express grant of the Russian Ministry of Education ‘Overgroups of semi-simple groups’ E02-1.0-61.
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