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no. 6
Higher Specht polynomials under the diagonal action
Gillespie, Maria1
1 Colorado State University Department of Mathematics 1874 Campus Delivery Fort Collins, CO 80523 (USA)
Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1731-1750.

We introduce higher Specht polynomials - analogs of Specht polynomials in higher degrees - in two sets of variables x1,...,xn and y1,...,yn under the diagonal action of the symmetric group Sn. This generalizes the classical Specht polynomial construction in one set of variables, as well as the higher Specht basis for the coinvariant ring Rn due to Ariki, Terasoma, and Yamada, which has the advantage of respecting the decomposition into irreducibles.

As our main application of the general theory, we provide a higher Specht basis for the hook shape Garsia–Haiman modules. In the process, we obtain a new formula for their doubly graded Frobenius series in terms of new generalized cocharge statistics on tableaux.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.396
Classification: 05E10, 05E05, 05E40, 20C30
Keywords: Diagonal harmonics, diagonal coinvariants, representation theory of the symmetric group, Young tableaux, Young symmetrizers, Garsia-Haiman modules, polynomial rings

Gillespie, Maria 1

1 Colorado State University Department of Mathematics 1874 Campus Delivery Fort Collins, CO 80523 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Gillespie, Maria. Higher Specht polynomials under the diagonal action. Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1731-1750. doi : 10.5802/alco.396. https://alco.centre-mersenne.org/articles/10.5802/alco.396/

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