ALGEBRAIC COMBINATORICS

$\left({\mathrm{GL}}_{k}×{\mathrm{Sym}}_{n}\right)$-modules and Nabla of hook-indexed Schur functions
Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 1033-1051.

The aim of this paper is to describe structural properties of spaces of diagonal rectangular harmonic polynomials in several sets (say $k$) of $n$ variables, both as ${\mathrm{GL}}_{k}$-modules and ${\mathrm{Sym}}_{n}$-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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.236
Classification: 05E05,  05E10,  20C30
Keywords: Nabla operator, Macdonald polynomials, Schur functions
Bergeron, François 1

1 Université du Québec à Montréal Dépt. de Mathématiques C.P. 8888, Succ. Centre-Ville Montréal H3C 3P8 (Canada)
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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, Volume 5 (2022) no. 5, pp. 1033-1051. doi : 10.5802/alco.236. https://alco.centre-mersenne.org/articles/10.5802/alco.236/

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