Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-09T04:32:36.900Z Has data issue: false hasContentIssue false
  • English
  • Français

Modular Representations of Finite Groupswith Unsaturated Split (B,N)-Pairs

Published online by Cambridge University Press:  20 November 2018

N. B. Tinberg
N. B. Tinberg*
Affiliation:
Occidental College, Los Angeles, California
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

1. Introduction.Let p be a prime number. A finite group G = (G, B, N, R, U) is called a split(B, N)-pair of characteristic p and rank n if

(i) G has a (B, N)-pair (see [3, Definition 2.1, p. B-8]) where H= B ⋂ N and the Weyl group W= N/H is generated by the set R= ﹛ω 1,… , ω n) of “special generators.”

(ii) H= ⋂n∈N n-1Bn

(iii) There exists a p-subgroup U of G such that B = UH is a semidirect product, and H is abelian with order prime to p.

A (B, N)-pair satisfying (ii) is called a saturated (B, N)-pair. We call a finite group G which satisfies (i) and (iii) an unsaturated split (B, N)- pair. (Unsaturated means “not necessarily saturated”.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 714 - 733
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Bourbaki, N., Groupes et algèbres de Lie, Chapters 4, 5, 6 (Herman, Paris, 1968).Google Scholar
2. Carter, R. and Lusztig, G., Modular representations of finite groups of Lie type, Proc. London Math. Soc. (3. 32 (1976), 347–384.Google Scholar
3. Curtis, C. W., Modular representations of finite groups with split (B, N)-pairs, Lecture Notes in Mathematics No. 131, (B-l)-(B-39) (Springer, Berlin-Heidelberg-New York, 1970).Google Scholar
4. Fong, P. and Seitz, G., Groups with a ﹛B, N)-pair of rank 2.1, Inventiones Math. 21 (1973), 1–57.Google Scholar
5. Green, J. A., On a theorem of H. Sawada, To appear in Proc. London Math. Soc.Google Scholar
6. Kantor, W. M. and Seitz, G., Some results on 2-transitive groups, Inventiones Math. 13 (1971), 125–142.Google Scholar
7. Richen, F., Modular representations of split (B, N)-pairs, Trans. Amer. Math. Soc. 140 (1969), 435–460.Google Scholar
8. Sawada, H., A characterization of the modular representations of finite groups with split (B, N)-pairs, Math. Z. 155 (1977), 29–41.Google Scholar
9. Tinberg, N., Some indecomposable modules of groups with split (B, N)-pairs, J. Alg. 61 (1979), 508–526.Google Scholar