Nonvanishing higher Specht polynomials and a construction for three row and hook shape Garsia--Procesi modules
Abstract
Given a polynomial ring quotient R=ℂ[x1,…,xn]/I with an action of the symmetric group induced by permuting the variables, a higher Specht basis is a collection of bases for each irreducible Sn-module in its decomposition that mimics the behavior of the classical Specht polynomial construction in the lowest degrees. Higher Specht bases have now been constructed for the coinvariant ring, the full polynomial ring, the rings Rn,k appearing in the t=0 Delta conjecture, the hook shape Garsia-Haiman modules, and the two-row Garsia-Procesi modules, and more. We establish a general theory for determining when a higher Specht polynomial is nonzero, and give a proof of a conjectural higher Specht basis for all Garsia-Procesi modules in the cases of three row shapes and hook shapes.
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BibTeX
@misc{chou2026nonvanishing,
title = {{Nonvanishing higher Specht polynomials and a construction for three row and hook shape Garsia--Procesi modules}},
author = {Raymond Chou and Maria Gillespie and Mitsuki Hanada},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12255},
eprint = {2609.12255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12255},
}
RIS
TY - UNPB
TI - Nonvanishing higher Specht polynomials and a construction for three row and hook shape Garsia--Procesi modules
AU - Raymond Chou
AU - Maria Gillespie
AU - Mitsuki Hanada
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12255
UR - https://arxiv.org/abs/2609.12255
AB - Given a polynomial ring quotient R=ℂ[x_1,…,x_n]/I with an action of the symmetric group induced by permuting the variables, a higher Specht basis is a collection of bases for each irreducible S_n-module in its decomposition that mimics the behavior of the classical Specht polynomial construction in the lowest degrees. Higher Specht bases have now been constructed for the coinvariant ring, the full polynomial ring, the rings R_n,k appearing in the t=0 Delta conjecture, the hook shape Garsia-Haiman modules, and the two-row Garsia-Procesi modules, and more. We establish a general theory for determining when a higher Specht polynomial is nonzero, and give a proof of a conjectural higher Specht basis for all Garsia-Procesi modules in the cases of three row shapes and hook shapes.
ER -