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
@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 = {http://www.numdam.org/articles/10.5802/alco.150/} }
TY - JOUR AU - Paget, Rowena AU - Wildon, Mark TI - Plethysms of symmetric functions and representations of $\protect \mathrm{SL}_2({\protect \bf C})$ JO - Algebraic Combinatorics PY - 2021 SP - 27 EP - 68 VL - 4 IS - 1 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.150/ DO - 10.5802/alco.150 LA - en ID - ALCO_2021__4_1_27_0 ER -
%0 Journal Article %A Paget, Rowena %A Wildon, Mark %T Plethysms of symmetric functions and representations of $\protect \mathrm{SL}_2({\protect \bf C})$ %J Algebraic Combinatorics %D 2021 %P 27-68 %V 4 %N 1 %I MathOA foundation %U http://www.numdam.org/articles/10.5802/alco.150/ %R 10.5802/alco.150 %G en %F ALCO_2021__4_1_27_0
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. http://www.numdam.org/articles/10.5802/alco.150/
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