Symmetric-determinant apolarity in odd characteristic
Abstract
Let D_N be the determinant of the generic symmetric N x N matrix, and let differential operators act by ordinary differentiation over a field of odd characteristic p. We determine the apolar ideal of D_N in every size. In addition to the classical quadratic relations, it is generated by one (p-1) x (p-1) determinant of differential variables for each set of 2p-2 indices. These additional generators are minimal modulo the quadratic ideal. An exterior-algebra realization identifies the remaining apolar relations with radicals of invariant pairings for SL_2. Semisimplification gives a Hilbert series in terms of Dyck paths of height at most p-2, and the Steinberg symmetrizer gives the explicit generators of the entire radical. We also compute the ranks of powers of a general linear form. The weak Lefschetz property holds exactly when N <= 2p-2, and the strong Lefschetz property holds exactly when N < p.
Related
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BibTeX
@misc{xie2026symmetricdeterminant,
title = {{Symmetric-determinant apolarity in odd characteristic}},
author = {Ying Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38212},
eprint = {2609.38212},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38212},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symmetric-determinant apolarity in odd characteristic
AU - Ying Xie
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.38212
UR - https://arxiv.org/abs/2609.38212
AB - Let D_N be the determinant of the generic symmetric N x N matrix, and let differential operators act by ordinary differentiation over a field of odd characteristic p. We determine the apolar ideal of D_N in every size. In addition to the classical quadratic relations, it is generated by one (p-1) x (p-1) determinant of differential variables for each set of 2p-2 indices. These additional generators are minimal modulo the quadratic ideal. An exterior-algebra realization identifies the remaining apolar relations with radicals of invariant pairings for SL_2. Semisimplification gives a Hilbert series in terms of Dyck paths of height at most p-2, and the Steinberg symmetrizer gives the explicit generators of the entire radical. We also compute the ranks of powers of a general linear form. The weak Lefschetz property holds exactly when N <= 2p-2, and the strong Lefschetz property holds exactly when N < p.
ER -