Let denote the Schur functor labelled by the partition and let be the natural representation of . We make a systematic study of when there is an isomorphism of representations of . Generalizing earlier results of King and Manivel, we classify all such isomorphisms when and are conjugate partitions and when one of or is a rectangle. We give a complete classification when and each have at most two rows or columns or is a hook partition and a partial classification when . As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon’s enumeration of plane partitions, and prove a new -binomial identity in this setting.
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Keywords: Plethysm, Hermite Reciprocity, Hook Content Formula
Paget, Rowena 1; Wildon, Mark 2
CC-BY 4.0
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author = {Paget, Rowena and Wildon, Mark},
title = {Plethysms of symmetric functions and representations of $\protect \mathrm{SL}_2({\protect \bf C})$},
journal = {Algebraic Combinatorics},
pages = {27--68},
year = {2021},
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Paget, Rowena; Wildon, Mark. Plethysms of symmetric functions and representations of $\protect \mathrm{SL}_2({\protect \bf C})$. Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 27-68. doi: 10.5802/alco.150
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