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Counting Lie ideals of niltriangular matrices
Abstract
We give an explicit finite sum for the number of ideals of the Lie algebra of strictly lower triangular n× n matrices over 𝔽q, valid for every prime power q. The sum runs over integer compositions, with weights expressed using ordinary and Gaussian binomial coefficients. A contraction bijection transforms Gagnon's configuration sum into a weighted enumeration of nonnesting partitions, with antichains of intervals chosen independently in each block. We derive a Stieltjes continued fraction for the block weights. Lagrange inversion and the coefficient formula for Stieltjes--Rogers polynomials then yield the explicit sum.
Related
BibTeX
@misc{khodyunya2026counting,
title = {{Counting Lie ideals of niltriangular matrices}},
author = {N. D. Khodyunya},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09514},
eprint = {2609.09514},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09514},
}
RIS
TY - UNPB
TI - Counting Lie ideals of niltriangular matrices
AU - N. D. Khodyunya
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.09514
UR - https://arxiv.org/abs/2609.09514
AB - We give an explicit finite sum for the number of ideals of the Lie algebra of strictly lower triangular n× n matrices over 𝔽_q, valid for every prime power q. The sum runs over integer compositions, with weights expressed using ordinary and Gaussian binomial coefficients. A contraction bijection transforms Gagnon's configuration sum into a weighted enumeration of nonnesting partitions, with antichains of intervals chosen independently in each block. We derive a Stieltjes continued fraction for the block weights. Lagrange inversion and the coefficient formula for Stieltjes--Rogers polynomials then yield the explicit sum.
ER -