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A simple classification of the special matchings in the symmetric group
Abstract
We establish a simple characterization of special matchings of all Bruhat intervals in the symmetric group, yielding an explicit classification that depends only on the one-line notations of the endpoints. We also study extensions of graded posets by partial special matchings, proving the existence and uniqueness of such extensions and providing an explicit construction. Combining these results, we obtain a combinatorial algorithm for computing Kazhdan--Lusztig polynomials for an isomorphism-closed family of Bruhat intervals that includes all intervals whose bottom element has length at most one.
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BibTeX
@misc{caselli2026simple,
title = {{A simple classification of the special matchings in the symmetric group}},
author = {Fabrizio Caselli and Mario Marietti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21647},
eprint = {2610.21647},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21647},
}
RIS
TY - UNPB
TI - A simple classification of the special matchings in the symmetric group
AU - Fabrizio Caselli
AU - Mario Marietti
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21647
UR - https://arxiv.org/abs/2610.21647
AB - We establish a simple characterization of special matchings of all Bruhat intervals in the symmetric group, yielding an explicit classification that depends only on the one-line notations of the endpoints. We also study extensions of graded posets by partial special matchings, proving the existence and uniqueness of such extensions and providing an explicit construction. Combining these results, we obtain a combinatorial algorithm for computing Kazhdan--Lusztig polynomials for an isomorphism-closed family of Bruhat intervals that includes all intervals whose bottom element has length at most one.
ER -