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




This article is cited in 1 scientific paper (total in 1 paper)
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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UDC:
512.5 Received: 01.12.2014
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
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 3252
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6081}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3486760}
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This publication is cited in the following articles:

V. A. Petrov, “Razlozhenie transvektsii: algebrogeometricheskii podkhod”, Algebra i analiz, 28:1 (2016), 150–157 ; V. A. Petrov, “Decomposition of transvections: an algebrogeometric approach”, St. Petersburg Math. J., 28:1 (2017), 109–114

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