Commutators of signed n-cycles

@misc{bader2026commutators,
  title = {{Commutators of signed $n$-cycles}},
  author = {Philipp Bader and Bernhard Böhmler and Patrick Wegener},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24518},
  eprint = {2609.24518},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24518},
}

TY  - UNPB
TI  - Commutators of signed n-cycles
AU  - Philipp Bader
AU  - Bernhard Böhmler
AU  - Patrick Wegener
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24518
UR  - https://arxiv.org/abs/2609.24518
AB  - We show that for n ≥ 6 each element of the commutator subgroup in the symmetric group S_n resp. in the signed symmetric group (ℤ/2ℤ)^nrtimesS_n is the commutator of two n-cycles resp. the commutator of two n-cycles with a negative sign product; with one exception. If n ≡ 2 mod 4, the element −id of (ℤ/2ℤ)^nrtimesS_n is not such a commutator. In the language of Coxeter groups, this yields a description of commutators of Coxeter elements in types A and B.
ER  -