Plethysms of symmetric functions and representations of SL 2 (C)
Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 27-68.

Let λ denote the Schur functor labelled by the partition λ and let E be the natural representation of SL 2 (C). We make a systematic study of when there is an isomorphism λ Sym E μ Sym m E of representations of SL 2 (C). Generalizing earlier results of King and Manivel, we classify all such isomorphisms when λ and μ are conjugate partitions and when one of λ or μ is a rectangle. We give a complete classification when λ and μ each have at most two rows or columns or is a hook partition and a partial classification when =m. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when λ Sym E 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 q-binomial identity in this setting.

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DOI: https://doi.org/10.5802/alco.150
Classification: 05E05,  05E10,  20C30,  22E46,  22E47
Keywords: Plethysm, Hermite Reciprocity, Hook Content Formula
@article{ALCO_2021__4_1_27_0,
     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},
     publisher = {MathOA foundation},
     volume = {4},
     number = {1},
     year = {2021},
     doi = {10.5802/alco.150},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.150/}
}
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. https://alco.centre-mersenne.org/articles/10.5802/alco.150/

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