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no. 6
A charge monomial basis of the Garsia-Procesi ring
Hanada, Mitsuki1
1 University of California, Berkeley, Department of Mathematics
Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1617-1649

We construct a basis of the Garsia-Procesi ring using the catabolizability type of standard Young tableaux and the charge statistic. This basis turns out to be equal to the descent basis defined in [3]. Our new construction connects the combinatorics of the basis with the well-known combinatorial formula for the modified Hall-Littlewood polynomials $\tilde{H}_\mu [X;q]$, due to Lascoux, which expresses the polynomials as a sum over standard tableaux that satisfy a catabolizability condition. In addition, we prove that identifying a basis for the antisymmetric part of $R_{\mu }$ with respect to a Young subgroup $S_\gamma $ is equivalent to finding pairs of standard tableaux that satisfy conditions regarding catabolizability and descents. This gives an elementary proof of the fact that the graded Frobenius character of $R_{\mu }$ is given by the catabolizability formula for $\tilde{H}_\mu [X;q]$.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.458
Classification: 05E10, 05E05, 05A30
Keywords: Garsia-Procesi ring, Hall-Littlewood polynomials, charge, catabolism

Hanada, Mitsuki 1

1 University of California, Berkeley, Department of Mathematics
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Hanada, Mitsuki. A charge monomial basis of the Garsia-Procesi ring. Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1617-1649. doi: 10.5802/alco.458

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