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The unipotent modules of \(\mathrm{GL}_{n}(\mathbb {F}_{q})\) via tableaux. (English) Zbl 1392.05111
In [Bull. Lond. Math. Soc. 8, 229–232 (1976; Zbl 0358.20019)], G. D. James used the action of the symmetric group \(\mathbb{S}_n\) on Young tableaux to construct the irreducible representations of \(\mathbb{S}_n\). In a subsequent paper [Representations of general linear groups. Cambridge etc.: Cambridge University Press (1984; Zbl 0541.20025)], G. D. James constructs the unipotent modules for the finite linear group \(\mathrm{GL}_n(\mathbb{F}_q)\) over any field containing a non-trivial \(p\)-th root of the unity, with \(p\) the characteristic of \(\mathbb{F}_q\). However, this construction is very different from the tableaux approach used in the symmetric group, and in particular the proofs do not translate to proofs for the symmetric group. James then asks whether a construction exists that is more similar to that of the symmetric group.
In the paper under review, the author answers James’ question with a construction of the irreducible unipotent modules for \(\mathrm{GL}_n(\mathbb{F}_q)\) which is a natural analogue of that for the symmetric group. This construction uses the action of \(\mathrm{GL}_n(\mathbb{F}_q)\) on Young diagrams whose boxes are labeled with elements of \(\mathbb{F}_q^n\) and is based on the generalized Gelfand-Graev representations of N. Kawanaka [Adv. Stud. Pure Math. 6, 175–206 (1985; Zbl 0573.20038)].

05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
20C20 Modular representations and characters
20C33 Representations of finite groups of Lie type
Full Text: DOI
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[8] Kawanaka, N.: Generalized Gelfand-Graev representations and Ennola duality. In: Algebraic Groups and Related Topics (Kyoto/Nagoya. 1983), Advanced Studies in Pure Mathematics, vol. 6, pp. 175-206. North-Holland, Amsterdam (1985) · Zbl 0068.25605
[9] Macdonald, I.G.: Symmetric Functions and Hall Polynomials Oxford Mathematical Monographs, 2nd edn. The Clarendon Press, New York (1995). (With contributions by A. Zelevinsky Oxford Science Publications)
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