An explicit model for the complex representations of the finite general linear groups. (English) Zbl 0826.20038

A model for the complex representations of a finite group \(G\) is a set of characters \(\tau_1, \dots, \tau_k\) of \(G\) such that each \(\tau_i\) is induced from a linear character of a subgroup of \(G\) and such that \(\sum^k_{i = 1} \tau_i = \sum_{\chi \in \text{Irr} (G)} \chi\). Models for the finite symmetric and general linear groups were described by Kljačko. Inglis, Richardson and Saxl have shown that there is a model \(\tau_0, \tau_1, \dots, \tau_{[{n\over 2}]}\) for \(S_n\), with \(\tau_i\) a sum of all irreducible characters of \(S_n\) corresponding to partitions with precisely \(n - 2i\) odd parts. In this paper we show that a suitable analogy holds for a model of the general linear groups.


20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
Full Text: DOI


[1] R. W.Baddeley, Models and involution models for wreath products and certain Weyl groups. To appear in J. London Math. Soc. · Zbl 0757.20003
[2] E. Bannai, N. Kawanaka andS. Y. Song, The character table of the Hecke algebra \(H(GL_2 (\mathbb{F}_q ), Sp_{2n} (\mathbb{F}_q ))\) , 431-02. J. Algebra129, 320-366 (1990). · Zbl 0761.20013 · doi:10.1016/0021-8693(90)90224-C
[3] J. A. Green, The characters of the finite general linear groups. Trans. Amer. Math. Soc.80, 402-447 (1955). · Zbl 0068.25605 · doi:10.1090/S0002-9947-1955-0072878-2
[4] N. F. J. Inglis, R. W. Richardson andJ. Saxl, An explicit model for the complex representations ofS n . Arch. Math.54, 258-259 (1990). · Zbl 0695.20008 · doi:10.1007/BF01188521
[5] G. D.James and A.Kerber, The representation theory of the symmetric group. Encyclopedia Math. Appl.16, Cambridge-New York 1981. · Zbl 0491.20010
[6] A. A. Klja?ko, Models for complex representations of the groupsGL(n, q) and Weyl groups. Soviet Math. Dokl. (3)24, 496-499 (1981).
[7] A. A. Klyachko, Models for the complex representations of the groupsGL(n, q). Math. USSR Sb. (2)48, 365-379 (1984). · Zbl 0543.20026 · doi:10.1070/SM1984v048n02ABEH002680
[8] I. G. Macdonald, Symmetric functions and Hall polynomials (in Russian). Mir, Moscow 1985. · Zbl 0672.20007
[9] A.Zelevinsky, Representations of finite classical groups ?a Hopf algebra approach. LNM869, Berlin-Heidelberg-New York 1981. · Zbl 0465.20009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.