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The Permutation Modules for GL( n +1, F q ) Acting on P n (F q ) and F q n + 1
Author(s) -
Bardoe Matthew,
Sin Peter
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461079900839x
Subject(s) - mathematics , combinatorics , projective space , permutation group , permutation (music) , rank (graph theory) , group (periodic table) , space (punctuation) , incidence (geometry) , discrete mathematics , pure mathematics , projective test , physics , geometry , computer science , quantum mechanics , acoustics , operating system
The paper studies the permutation representations of a finite general linear group, first on finite projective space and then on the set of vectors of its standard module. In both cases the submodule lattices of the permutation modules are determined. In the case of projective space, the result leads to the solution of certain incidence problems in finite projective geometry, generalizing the rank formula of Hamada. In the other case, the results yield as a corollary the submodule structure of certain symmetric powers of the standard module for the finite general linear group, from which one obtains the submodule structure of all symmetric powers of the standard module of the ambient algebraic group.