Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-29T22:18:33.205Z Has data issue: false hasContentIssue false
  • English
  • Français

Prenilpotent pairs in the E10 root lattice

Published online by Cambridge University Press:  08 May 2017

DANIEL ALLCOCK
DANIEL ALLCOCK*
Affiliation:
University of Texas at Austin, Department of Mathematics, RLM 8.100 2515 Speedway Stop C1200 Austin, Texas 78712-1202, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Tits has defined Kac–Moody groups for all root systems, over all commutative rings with unit. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his group presentation explicitly would require knowing all the Weyl-group orbits of such pairs. We show that for the hyperbolic root system E10 there are so many orbits that any attempt at direct enumeration is impractical. Namely, the number of orbits of prenilpotent pairs having inner product k grows at least as fast as (constant) ⋅ k7 as k → ∞. Our purpose is to motivate alternate approaches to Tits' groups, such as the one in [2].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 3 , May 2018 , pp. 473 - 483
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Abramenko, P. and Mühlherr, B. Preśentations de certaines BN-paires jumelées comme sommes amalgamées. C.R.A.S. Série I 325 (1997), 701–706.Google Scholar
[2] Allcock, D. Steinberg groups as amalgams. To appear in Algebra and Number Theory. arXiv:1307.2689.Google Scholar
[3] Allcock, D. Presentation of affine Kac–Moody groups over rings. Algebra and Number Theory 10 (2016), No. 3, 533–556. arXiv:1409.0176.Google Scholar
[4] Allcock, D. and Carbone, L. Presentation of hyperbolic Kac-Moody groups over rings. J. Algebra 445 (2016), 232–243. arXiv:1409.5918.Google Scholar
[5] Caprace, P.-E. On 2-spherical Kac–Moody groups and their central extensions. Forum Math. 19 (2007), 763–781.Google Scholar
[6] Carbone, L., Chung, S., Cobbs, L., McRae, R., Nandi, D., Naqvi, Y. and Penta, D. Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits. J. Physics A: Mathematical and Theoretical 43 (2010), 1–30.Google Scholar
[7] Carbone, L. and Murray, S. Computation in hyperbolic Kac–Moody algebras. In preparation (2014).Google Scholar
[8] Hée, J.-Y. Sur les p-morphisms, unpublished manuscript (1993).Google Scholar
[9] Conway, J. H. and Sloane, N. J. A. Sphere packings, lattices and groups. Grundlehren der Mathematischen Wissenschaften 290 (Springer–Verlag, New York, 1999).Google Scholar
[10] Conway, J. H. and Sloane, N. J. A. Low-dimensional lattices II. Subgroups of GL(n, ℤ). Proc. Roy. Soc. London Ser. A 419 (1988), no. 1856, 29–68.Google Scholar
[11] Conway, J. H. and Sloane, N. J. A. Low-dimensional lattices IV. The mass formula. Proc. Roy. Soc. London Ser. A 419 (1988), no. 1857, 259–286.Google Scholar
[12] Morita, J. Commutator relations in Kac–Moody groups. Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), 21–22.Google Scholar
[13] Muhlherr, B. On the simple connectedness of a chamber system associated to a twin building. Unpublished (1999).Google Scholar
[14] Serre, J.-P. A Course in Arithmetic. Graduate Texts in Math. no. 7 (Springer–Verlag, New York–Heidelberg, 1973).Google Scholar
[15] Tits, J. Uniqueness and presentation of Kac–Moody groups over fields. J. Algebra 105 (1987), 542–573.Google Scholar
[16] Vinberg, È. B. Some arithmetical discrete groups in Lobaevski spaces. Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) (Oxford University Press, Bombay, 1975), pp. 323–348.Google Scholar