Multivariate diagonal harmonics: character polynomials and proofs of two conjectures of Bergeron
Abstract
We prove Bergeron's Hook-Components and Coefficient-Length conjectures for the Vandermonde polarization module and full diagonal harmonics in every rank and number of alphabets. Hook-label characters are elementary skewings of the alternating character, and the maximal Schur length in label μ is n−μ1. We compute character and dimension polynomials of diagonal coinvariants in every bosonic--fermionic superspace for D≥3, n≥max{D,2D−6}, answering Church--Ellenberg--Farb's character and dimension questions. For augmented commutative algebras in characteristic zero, we construct natural two-Schur presentations and filtered free resolutions with one Schur generator in each homological degree. For polynomial algebras, specialization to orthogonal idempotents identifies the first unstable correction for every nonempty tail τ and degree D>|τ|. The stable threshold n≥ D+τ1 is sharp when the row dimension is at least ℓ(τ). First parabolic algebras recover finite jets, their multiplication and their maps. Selected-label lifts give uniform Vandermonde generation through degree D for D≥3, n≥2D−3, and disjoint-label products generate every first occurrence for positive integral row weights. Constructive degree reflection determines all two-column moment homology and generates its first syzygies from degree n+1; two-column Vandermonde generation reduces to degree n−1. Invariants and harmonics separate through the character range and through degree n in two alphabets. The hook identities also give the fully graded sign component of the t=0, (1,2)-superspace Theta conjecture.
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BibTeX
@misc{ercan2026multivariate,
title = {{Multivariate diagonal harmonics: character polynomials and proofs of two conjectures of Bergeron}},
author = {Eren Ercan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00629},
eprint = {2611.00629},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00629},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multivariate diagonal harmonics: character polynomials and proofs of two conjectures of Bergeron
AU - Eren Ercan
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00629
UR - https://arxiv.org/abs/2611.00629
AB - We prove Bergeron's Hook-Components and Coefficient-Length conjectures for the Vandermonde polarization module and full diagonal harmonics in every rank and number of alphabets. Hook-label characters are elementary skewings of the alternating character, and the maximal Schur length in label μ is n−μ_1. We compute character and dimension polynomials of diagonal coinvariants in every bosonic--fermionic superspace for D≥3, n≥max{D,2D−6}, answering Church--Ellenberg--Farb's character and dimension questions. For augmented commutative algebras in characteristic zero, we construct natural two-Schur presentations and filtered free resolutions with one Schur generator in each homological degree. For polynomial algebras, specialization to orthogonal idempotents identifies the first unstable correction for every nonempty tail τ and degree D>|τ|. The stable threshold n≥ D+τ_1 is sharp when the row dimension is at least ℓ(τ). First parabolic algebras recover finite jets, their multiplication and their maps. Selected-label lifts give uniform Vandermonde generation through degree D for D≥3, n≥2D−3, and disjoint-label products generate every first occurrence for positive integral row weights. Constructive degree reflection determines all two-column moment homology and generates its first syzygies from degree n+1; two-column Vandermonde generation reduces to degree n−1. Invariants and harmonics separate through the character range and through degree n in two alphabets. The hook identities also give the fully graded sign component of the t=0, (1,2)-superspace Theta conjecture.
ER -