Macdonald representations of complex reflection groups. (English) Zbl 1068.20039

Summary: I. G. Macdonald [Bull. Lond. Math. Soc. 4, 148-150 (1972; Zbl 0251.20043)] introduced a unified approach to give many irreducible representations of Weyl groups in terms of their root systems. This generalised to Weyl groups the earlier well known constructions based on Young tableaux due to W. Specht. These were interpreted in terms of positive systems of subsystems of root systems. A. M. Cohen [Ann. Sci. Éc. Norm. Supér., IV. Sér. 9, 379-436 (1976; Zbl 0359.20029)] extended the idea of root systems to complex reflection groups giving explicitly root systems for all dimensions greater than two. M. C. Hughes [Indag. Math. 47, 313-330 (1985; Zbl 0592.20056)] had further extended his ideas to generalise the concepts of subsystems and positive systems. These are now used to construct some irreducible representations of complex reflection groups.


20F55 Reflection and Coxeter groups (group-theoretic aspects)
20C15 Ordinary representations and characters
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