By G. D. James
Publisher: Cambridge University Press
Print Publication Year: 1984
Online Publication Date:May 2010
Online ISBN:9780511661921
Paperback ISBN:9780521269810
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511661921.001
Subjects: Algebra
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This essay concerns the unipotent representations of the finite general linear groups GLn (q). An irreducible unipotent representation is, by definition, a composition factor of the permutation representation of GLn (q) on a Borel subgroup, and the ordinary irreducible unipotent representations may be indexed by partitions λ of n, as may the ordinary irreducible representations of the symmetric group. The remarkable feature is that the representation theory of over an arbitrary field appears to be the case “q = 1” of the subject we study here.
The most important results are undoubtedly the Submodule Theorem (Chapter 11) and the Kernel Intersection Theorem (Chapter 15), but there seems to have been no previous work on the representation modules for the unipotent representations of GLn (q), so we claim originality for all the results apart from those whose source is quoted or which are obviously known (Chapters 3 – 8).
Chapters 1 and 2 set the scene, by outlining the connection between and representations of GLn (q) over fields of characteristic dividing q, and by giving examples of the situation to be considered later. The preliminary results which we need are derived in Chapters 3 – 8. Thereafter, we assume that the characteristic of our ground field K does not divide q, but otherwise K is arbitrary.
pp. i-iv
pp. v-vi
pp. vii-ix
pp. x-xii
pp. 1-6
pp. 7-17
3 - Gaussian polynomials : Read PDF
pp. 18-19
4 - Compositions of n : Read PDF
pp. 20-22
5 - Root subgroups of Gn : Read PDF
pp. 23-27
6 - Subgroups of Gn associated with compositions : Read PDF
pp. 28-29
7 - Coset representatives : Read PDF
pp. 30-34
8 - Subgroups of Gn used for induction : Read PDF
pp. 35-39
9 - Some idempotent elements of K̄Gn : Read PDF
pp. 40-46
10 - The permutation module Mλ : Read PDF
pp. 47-55
11 - The Submodule Theorem : Read PDF
pp. 56-66
12 - A lower bound for the dimension of Sμ : Read PDF
pp. 67-74
13 - The Kernel Intersection Theorem for S(n−m,m) : Read PDF
pp. 75-79
14 - Reordering the parts of λ : Read PDF
pp. 80-83
15 - The Kernel Intersection Theorem : Read PDF
pp. 84-99
16 - Consequences of the Kernel Intersection Theorem : Read PDF
pp. 100-108
17 - Removing the first column from [λ] : Read PDF
pp. 109-113
18 - Isotropic spaces : Read PDF
pp. 114-123
19 - The prime divisors of Gaussian polynomials : Read PDF
pp. 124-135
20 - The composition factors of S(n-m,m) : Read PDF
pp. 136-144
pp. 145-145
pp. 146-147