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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:

$\sum _{\alpha ⊢n}\left[\alpha \right]=\sum _{0\le k\le n/2}\left(\left[2\right]\odot \left[k\right]\right)\left[{1}^{n-2k}\right]·$

Reviewer: A.Kerber

##### MSC:
 20C30 Representations of finite symmetric groups
##### References:
 [1] G. D.James and A.Kerber, The representation theory of the symmetric group. Encyclopedia Math. Appl.16, Reading, Mass. 1981. [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. [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