An explicit model for the complex representations of \(S_ n\). (English) Zbl 0695.20008

The authors provide the following elegant construction of the model character (the sum of all the irreducibles) for the symmetric group, in terms of plethysms and using the Littlewood-Richardson rule: \[ \sum_{\alpha \vdash n}[\alpha]=\sum_{0\leq k\leq n/2}([2]\odot [k])[1^{n-2k}]. \]
Reviewer: A.Kerber


20C30 Representations of finite symmetric groups
Full Text: DOI


[1] G. D.James and A.Kerber, The representation theory of the symmetric group. Encyclopedia Math. Appl.16, Reading, Mass. 1981. · Zbl 0491.20010
[2] A. A. Klja?ko, Models for the complex representations of the groupsGL(n,q) and Wey1 groups. Soviet Math. Dokl.24, 496-499 (1981).
[3] J. Saxl, On multiplicity-free permutation representations. In: Finite geometries and designs, London Math. Soc. Lecture Notes49, 337-353 (1981).
[4] J. G.Thompson, Fixed point free involutions and finite projective planes. In: Finite simple groups II (ed. M. J. Collins), New York-London 1980. · Zbl 0464.05015
[5] R. M. Thrall, On symmetrized Kronecker powers and the structure of the free Lie ring. Amer. J. Math.64, 371-388 (1942). · Zbl 0061.04201 · doi:10.2307/2371691
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.