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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.

MSC:

20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
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