×

zbMATH — the first resource for mathematics

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

MSC:
20C30 Representations of finite symmetric groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chen, Y.M; Garsia, A.M; Remmel, J, Algorithms for plethysm, (), 109-154 · Zbl 0556.20013
[2] Duncan, D.G, Note on a formula by Todd, J. London math. soc., 27, 235-236, (1952) · Zbl 0046.01601
[3] James, G; Kerber, A, The representation theory of the symmetric group, ()
[4] Littlewood, D.E, The theory of group characters, (1950), Oxford Univ. Press Oxford · Zbl 0011.25001
[5] Newell, M.J, A theorem on the plethysm of S-functions, Quart. J. math., 2, 161-166, (1951) · Zbl 0043.26003
[6] Robinson, G.de B, Induced representations and invariants, Canad. J. math., 2, 334-343, (1950) · Zbl 0039.02002
[7] Thrall, R.M, On symmetrized Kronecker products and the structure of the free Lie ring, Amer. J. math., 64, 71-388, (1942) · Zbl 0061.04201
[8] Todd, J.A, A note on the algebra of S-functions, (), 328-334 · Zbl 0034.16002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.