PreprintCC BY 4.0arXiv
Min-plus Bruhat decomposition
Abstract
We study min-plus Bruhat cells Btrop(n) ⊙ p⊙ Btrop(n), where Btrop(n) denotes the set of all upper-triangular matrices over RR ∪ {∞}, and p is a permutation. It is proved that dim (Btrop(n) ⊙ p ⊙ Btrop(n))= (n(n+1))/2 + l(p), where l(p) is the length of p in Coxeter generators. Also we provide an explicit description of the cell Btrop(n)(RR) ⊙ p0⊙ Btrop(n)(RR) (up to a subset of dimension less than n2) for the highest in Bruhat order permutation p0. Finally, it is shown that for a tropically non-singular matrix A for at most one permutation pA it can hold A∈ Btrop(n)⊙ pA⊙ Btrop(n).
Related
BibTeX
@misc{grigoriev2026minplus,
title = {{Min-plus Bruhat decomposition}},
author = {Dima Grigoriev},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03896},
eprint = {2611.03896},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03896},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Min-plus Bruhat decomposition
AU - Dima Grigoriev
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03896
UR - https://arxiv.org/abs/2611.03896
AB - We study min-plus Bruhat cells B_trop^(n) ⊙ p⊙ B_trop^(n), where B_trop^(n) denotes the set of all upper-triangular matrices over RR ∪ {∞}, and p is a permutation. It is proved that dim (B_trop^(n) ⊙ p ⊙ B_trop^(n))= (n(n+1))/2 + l(p), where l(p) is the length of p in Coxeter generators. Also we provide an explicit description of the cell B_trop^(n)(RR) ⊙ p_0⊙ B_trop^(n)(RR) (up to a subset of dimension less than n^2) for the highest in Bruhat order permutation p_0. Finally, it is shown that for a tropically non-singular matrix A for at most one permutation p_A it can hold A∈ B_trop^(n)⊙ p_A⊙ B_trop^(n).
ER -