Centralizers of maps as automorphism groups of Yang--Baxter solutions
Abstract
For a partition X = Y sqcup Z of a set X and an arbitrary map g : Y → HomSet (Z, Z), we construct a solution Rg : X × X → X × X of the braid equation. We characterize intrinsically the solutions which are isomorphic to some Rg and we give a necessary and sufficient condition for two solutions Rg and Rg' to be isomorphic. Consequently, the number of isomorphism classes of the solutions Rg on a set with n elements is 1 + ∑k = 1n−1 õ (n − k, k), where õ (m, k) is the number of orbits of the symmetric group Sm, acting by conjugation, on the set of multisets with k elements of maps {1, …, m} → {1, …, m}. This construction is then used to obtain a solution of the braid equation whose automorphism group is exactly the centralizer Aut (f1, …, fk) of a given family f1, …, fk : X → X of maps, k ≥ 1. For k = 2, combined with a result of Agore, Chirvasitu and the author, this gives another proof of the fact that any group is isomorphic to the automorphism group of a solution of the braid equation (equivalently, of the set-theoretic Yang--Baxter equation).
Related
- Dual Reflection Groups from Solutions to the Yang--Baxter Equation
- Groups satisfying the minimal condition on centralizers are complete co-analytic
- Every group is the automorphism group of a set-theoretic solution of the Yang--Baxter equation
- Centralizers of elements in the orthogonal, symplectic and unitary groups
- Enumeration of orientably-regular maps with automorphism group PGam(2,2p) for prime p
BibTeX
@misc{militaru2026centralizers,
title = {{Centralizers of maps as automorphism groups of Yang--Baxter solutions}},
author = {Gigel Militaru},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03380},
eprint = {2611.03380},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03380},
}
RIS
TY - UNPB
TI - Centralizers of maps as automorphism groups of Yang--Baxter solutions
AU - Gigel Militaru
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03380
UR - https://arxiv.org/abs/2611.03380
AB - For a partition X = Y sqcup Z of a set X and an arbitrary map g : Y → Hom_Set (Z, Z), we construct a solution R_g : X × X → X × X of the braid equation. We characterize intrinsically the solutions which are isomorphic to some R_g and we give a necessary and sufficient condition for two solutions R_g and R_g' to be isomorphic. Consequently, the number of isomorphism classes of the solutions R_g on a set with n elements is 1 + ∑_k = 1^n−1 õ (n − k, k), where õ (m, k) is the number of orbits of the symmetric group S_m, acting by conjugation, on the set of multisets with k elements of maps {1, …, m} → {1, …, m}. This construction is then used to obtain a solution of the braid equation whose automorphism group is exactly the centralizer Aut (f_1, …, f_k) of a given family f_1, …, f_k : X → X of maps, k ≥ 1. For k = 2, combined with a result of Agore, Chirvasitu and the author, this gives another proof of the fact that any group is isomorphic to the automorphism group of a solution of the braid equation (equivalently, of the set-theoretic Yang--Baxter equation).
ER -