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 Zap. Nauchn. Sem. POMI, 2014, Volume 430, Pages 32–52 (Mi znsl6081)

Decomposition of unipotents for $\mathrm E_6$ and $\mathrm E_7$: 25 years after

N. A. Vavilov

St. Petersburg State University

Abstract: In this paper I sketch two new variations of the method of decomposition of unipotents in the microweight representations $(\mathrm E_6,\varpi_1)$ and $(\mathrm E_7,\varpi_7)$. To put them in context, I first very briefly recall the two previous stages of the method, an $\mathrm A_5$-proof for $\mathrm E_6$ and an $\mathrm A_7$-proof for $\mathrm E_7$, first developed some 25 years ago by Alexei Stepanov, Eugene Plotkin and myself (a definitive exposition was given in my paper “A thirdlook at weight diagrams”), and an $\mathrm A_2$-proof for $\mathrm E_6$ and $\mathrm E_7$ developed by Mikhail Gavrilovich and myself in early 2000. The first new twist outlined in this paper is an observation that the $\mathrm A_2$-proof actually effectuates reduction to small parabolics, of corank 3 in $\mathrm E_6$ and of corank 5 in $\mathrm E_7$. This allows to revamp proofs and sharpen existing bounds in many applications. The second new variation is a $\mathrm D_5$-proof for $\mathrm E_6$, based on stabilisation of columns with one zero. [I devised also a similar $\mathrm D_6$-proof for $\mathrm E_7$, based on stabilisation of columns with two adjacent zeroes, but it is too abstruse to be included in a casual exposition.] Also, I list several further variations. Actual detailed calculations will appear in my paper “A closer look at weight diagrams of types $(\mathrm E_6,\varpi_1)$ and $(\mathrm E_7,\varpi_7)$”.

Key words and phrases: Chevalley groups, elementary subgroups, exceptional groups, microweight representation, decomposition of unipotents, parabolic subgroups, highest weight orbit.

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English version:
Journal of Mathematical Sciences (New York), 2016, 219:3, 355–369

Bibliographic databases:

Document Type: Article
UDC: 512.5
Language: English

Citation: N. A. Vavilov, “Decomposition of unipotents for $\mathrm E_6$ and $\mathrm E_7$: 25 years after”, Problems in the theory of representations of algebras and groups. Part 27, Zap. Nauchn. Sem. POMI, 430, POMI, St. Petersburg, 2014, 32–52; J. Math. Sci. (N. Y.), 219:3 (2016), 355–369

Citation in format AMSBIB
\Bibitem{Vav14} \by N.~A.~Vavilov \paper Decomposition of unipotents for $\mathrm E_6$ and $\mathrm E_7$: 25~years after \inbook Problems in the theory of representations of algebras and groups. Part~27 \serial Zap. Nauchn. Sem. POMI \yr 2014 \vol 430 \pages 32--52 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6081} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3486760} \transl \jour J. Math. Sci. (N. Y.) \yr 2016 \vol 219 \issue 3 \pages 355--369 \crossref{https://doi.org/10.1007/s10958-016-3111-8} 

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This publication is cited in the following articles:
1. V. A. Petrov, “Decomposition of transvections: an algebro-geometric approach”, St. Petersburg Math. J., 28:1 (2017), 109–114
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