The aim of this paper is to describe structural properties of spaces of diagonal rectangular harmonic polynomials in several sets (say ) of variables, both as -modules and -modules. We construct explicit such modules associated to any hook shape partitions. For the two sets of variables case, we conjecture that the associated graded Frobenius characteristic corresponds to the effect of the operator Nabla on the corresponding hook-indexed Schur function, up to a usual renormalization. We prove identities that give indirect support to this conjecture, and show that its restriction to one set of variables holds. We further give indications on how the several sets context gives a better understanding of questions regarding the structures of these modules and the links between them.
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Mots clés : Nabla operator, Macdonald polynomials, Schur functions
@article{ALCO_2022__5_5_1033_0,
author = {Bergeron, Fran\c{c}ois},
title = {$(\protect \mathrm{GL}_k\times \protect \mathrm{Sym}_n)$-modules and {Nabla} of hook-indexed {Schur} functions},
journal = {Algebraic Combinatorics},
pages = {1033--1051},
publisher = {The Combinatorics Consortium},
volume = {5},
number = {5},
year = {2022},
doi = {10.5802/alco.236},
language = {en},
url = {http://www.numdam.org/articles/10.5802/alco.236/}
}
TY - JOUR
AU - Bergeron, François
TI - $(\protect \mathrm{GL}_k\times \protect \mathrm{Sym}_n)$-modules and Nabla of hook-indexed Schur functions
JO - Algebraic Combinatorics
PY - 2022
SP - 1033
EP - 1051
VL - 5
IS - 5
PB - The Combinatorics Consortium
UR - http://www.numdam.org/articles/10.5802/alco.236/
DO - 10.5802/alco.236
LA - en
ID - ALCO_2022__5_5_1033_0
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%P 1033-1051
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%I The Combinatorics Consortium
%U http://www.numdam.org/articles/10.5802/alco.236/
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Bergeron, François. $(\protect \mathrm{GL}_k\times \protect \mathrm{Sym}_n)$-modules and Nabla of hook-indexed Schur functions. Algebraic Combinatorics, Tome 5 (2022) no. 5, pp. 1033-1051. doi : 10.5802/alco.236. http://www.numdam.org/articles/10.5802/alco.236/
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