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Some observations of plethysms. (English) Zbl 0695.20013
Plethysms [\(\alpha\) ]\(\odot [\beta]\) of ordinary irreducible representations [\(\alpha\) ] of \(S_ m\) and [\(\beta\) ] of \(S_ n\) are representations of symmetric groups \(S_{m\cdot n}\) which are induced from certain irreducible representations (\(\alpha\) ;\(\beta)\) of wreath products \(S_ m\wr S_ n\). The decomposition of plethysms is one of the open problems in representation theory of symmetric groups, and it is an important problem since it arises in many applications.
The present author shows in particular that their multiplicities stabilize, which means that the sequence of numbers \[ z_ i:=([\alpha_ 1+i,\alpha_ 2,...]\odot [\beta],[\lambda_ 1+ni,\lambda_ 2,...]) \] becomes constant, after a while, a suitable i is derived. Moreover he conjectures, that the character of the particular plethysm [m]\(\odot [n]\) is defined by its restriction to the subset \(S_{mn-1}\cup \{(1,...,mn)\}\) of \(S_{mn}\). He provides a proof for the case \(n=3\).
Reviewer: A.Kerber

20C30 Representations of finite symmetric groups
Full Text: DOI
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