The Colomo-Pronko conjecture for frozen-corner alternating sign matrices
Abstract
We prove the Colomo-Pronko conjecture for alternating sign matrices with a prescribed square of zeros at a corner, for all matrix sizes and freezing parameters. A known multiple-integral formula for the frozen-corner count yields determinant representations built from fixed polynomial kernels. We relate these kernels to the conjectured determinant through an inverse identity for the commutator of a signed Pascal matrix with reversal. In odd dimension, the comparison uses the one-dimensional nullspace and projection along it to eliminate the central coordinate. Combined with the asymptotic analysis of Colomo and Pronko, our result removes the conjectural assumption from their GUE Tracy-Widom fluctuation theorem for the intersection of the frozen boundary with the main diagonal in uniformly random alternating sign matrices. The finite-dimensional algebraic core of the proof has been formalized in Lean 4.
Related
- Limiting shape of alternating sign matrices
- A note on near alternating sign matrices with prescribed row and column weights
- The Airy line ensemble at the edge of uniform alternating sign matrices
- Dense alternating sign matrices and imprecise copulas
- The Robust Thompson's Conjecture for Alternating Groups
BibTeX
@misc{li2026colomopronko,
title = {{The Colomo-Pronko conjecture for frozen-corner alternating sign matrices}},
author = {Yinjie Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14351},
eprint = {2609.14351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14351},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Colomo-Pronko conjecture for frozen-corner alternating sign matrices
AU - Yinjie Li
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14351
UR - https://arxiv.org/abs/2609.14351
AB - We prove the Colomo-Pronko conjecture for alternating sign matrices with a prescribed square of zeros at a corner, for all matrix sizes and freezing parameters. A known multiple-integral formula for the frozen-corner count yields determinant representations built from fixed polynomial kernels. We relate these kernels to the conjectured determinant through an inverse identity for the commutator of a signed Pascal matrix with reversal. In odd dimension, the comparison uses the one-dimensional nullspace and projection along it to eliminate the central coordinate. Combined with the asymptotic analysis of Colomo and Pronko, our result removes the conjectural assumption from their GUE Tracy-Widom fluctuation theorem for the intersection of the frozen boundary with the main diagonal in uniformly random alternating sign matrices. The finite-dimensional algebraic core of the proof has been formalized in Lean 4.
ER -