PreprintCC BY 4.0arXiv
Triple parabolic cosets have unique Bruhat minima
Abstract
A folklore result says that any parabolic double coset WI w WJ in a Coxeter group W is an interval of the Bruhat order, provided that WI and WJ are finite. We include a self-contained proof, without assuming that W is finite. Furthermore, we prove that each "triple coset" WI u WJ v WK has a Bruhat minimum and -- if WI, WJ, WK are finite -- a Bruhat maximum, but (in general) is not a Bruhat interval. This proves and extends a conjecture on MathOverflow #472072.
Related
BibTeX
@misc{becht2026triple,
title = {{Triple parabolic cosets have unique Bruhat minima}},
author = {Mikey Becht and Darij Grinberg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31602},
eprint = {2610.31602},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31602},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Triple parabolic cosets have unique Bruhat minima
AU - Mikey Becht
AU - Darij Grinberg
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31602
UR - https://arxiv.org/abs/2610.31602
AB - A folklore result says that any parabolic double coset W_I w W_J in a Coxeter group W is an interval of the Bruhat order, provided that W_I and W_J are finite. We include a self-contained proof, without assuming that W is finite. Furthermore, we prove that each "triple coset" W_I u W_J v W_K has a Bruhat minimum and -- if W_I, W_J, W_K are finite -- a Bruhat maximum, but (in general) is not a Bruhat interval. This proves and extends a conjecture on MathOverflow #472072.
ER -