Restricting Specht Modules of Finite General Linear Groups to the Unitriangular Subgroup

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One of the most important problems in the representation theory of finite groups G is to find the irreducible complex representations of G. For the finite general linear groups G = GLn(q) of invertible matrices of order n over the finite field Fq with q elements this has been solved by J. A. Green. The main goal of this project is to construct standard integral bases of these representations over integral domains in which q is invertible. The main method used is to investigate the restriction of these representations to the unitriangular subgroups U of G consisting of all lower triangular matrices having only 1 as eigenvalue. This has been successfully applied to a special class of irreducible GLn(q)-modules by Q. Guo in her thesis. The Steinberg module is another special irreducible G-module. Restricting it to U gives the regular representation of U which contains all irreducible U-modules. Using this it is hoped to confirm longstanding conjectures of Higman and Lehrer and a more recent one due to Isaacs.