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 Zap. Nauchn. Sem. POMI, 2011, Volume 388, Pages 17–47 (Mi znsl4104)

Unitriangular factorisations of Chevalley groups

N. A. Vavilovab, A. V. Smolenskyab, B. Suryab

a Saint-Petersburg State University, Saint-Petersburg, Russia
b Indian Statistics Institute, Bangalore

Abstract: Lately, the following problem attracted a lot of attention in various contexts: find the shortest factorisation $G=UU^-UU^-…U^\pm$ of a Chevalley group $G=G(\Phi,R)$ in terms of the unipotent radical $U=U(\Phi,R)$ of the standard Borel subgroup $B=B(\Phi,R)$ and the unipotent radical $U^-=U^-(\Phi,R)$ of the opposite Borel subgroup $B^-=B^-(\Phi,R)$. So far, the record over a finite field was established in a 2010 paper by Babai, Nikolov, and Pyber, where they prove that a group of Lie type admits unitriangular factorisation $G=UU^-UU^-U$ of length 5. Their proof invokes deep analytic and combinatorial tools. In the present paper we notice that from the work of Bass and Tavgen one immediately gets a much more general result, asserting that over any ring of stable rank 1 one has unitriangular factorisation $G=UU^-UU^-$ of length 4. Moreover, we give a detailed survey of triangular factorisations, prove some related results, discuss prospects of generalisation to other classes of rings, and state several unsolved problems. Another main result of the present paper asserts that, in the assumption of the Generalised Riemann's Hypothesis, Chevalley groups over the ring $\mathbb Z[\frac1p]$ admit unitriangular factorisation $G=UU^-UU^-UU^-$ of length 6. Otherwise, the best length estimate for Hasse domains with infinite multiplicative groups that follows from the work of Cooke and Weinberger, gives 9 factors.

Key words and phrases: Chevalley groups, unitriangular factorisations, unipotent factorisations, rings of stable rank 1, Dedekind rings of arithmetic type, parabolic subgroups, bounded generation, Gauss decomposition, LULU-decomposition.

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English version:
Journal of Mathematical Sciences (New York), 2012, 183:5, 584–599

Document Type: Article
UDC: 512.5

Citation: N. A. Vavilov, A. V. Smolensky, B. Sury, “Unitriangular factorisations of Chevalley groups”, Problems in the theory of representations of algebras and groups. Part 21, Zap. Nauchn. Sem. POMI, 388, POMI, St. Petersburg, 2011, 17–47; J. Math. Sci. (N. Y.), 183:5 (2012), 584–599

Citation in format AMSBIB
\Bibitem{VavSmoñóð11} \by N.~A.~Vavilov, A.~V.~Smolensky, B.~Sury \paper Unitriangular factorisations of Chevalley groups \inbook Problems in the theory of representations of algebras and groups. Part~21 \serial Zap. Nauchn. Sem. POMI \yr 2011 \vol 388 \pages 17--47 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl4104} \transl \jour J. Math. Sci. (N. Y.) \yr 2012 \vol 183 \issue 5 \pages 584--599 \crossref{https://doi.org/10.1007/s10958-012-0826-z} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84862268867} 

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This publication is cited in the following articles:
1. N. A. Vavilov, E. I. Kovach, “$\mathrm{SL}_2$-factorisations of Chevalley groups”, J. Math. Sci. (N. Y.), 188:5 (2013), 483–489
2. Smolensky A., “Unitriangular Factorization of Twisted Chevalley Groups”, Int. J. Algebr. Comput., 23:6 (2013), 1497–1502
3. Vsemirnov M., “Short Unitriangular Factorizations of Sl2(Z[1/P])”, Q. J. Math., 65:1 (2014), 279–290
4. Garonzi M., Levy D., Maroti A., Simion I.I., “Minimal Length Factorizations of Finite Simple Groups of Lie Type By Unipotent Sylow Subgroups”, J. Group Theory, 19:2 (2016), 337–346
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