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PreprintCC BY 4.0arXiv

The q-Poincaré sum and separability in a Coxeter group

  • Ben Adenbaum
  • Darij Grinberg
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Abstract

Given a finite Coxeter group W and an element q of a commutative ring k, we define the element Lq :=∑w∈ Wqℓ( w) w of its group algebra $\mathbf{k}\left[ W\right] $. We say that an element w∈ W is W\emph{-length-symmetric} if its coefficient in the commutator $\left[ \mathbf{L}_{a},\mathbf{L}_{b}\right] $ is $0$ for all k and all a,b∈k. We prove several sufficient criteria for elements to be W-length-symmetric. In particular, we show that any separable permutation (i.e., permutation avoiding the patterns $2413$ and $3142$) in the symmetric group Sn is Sn-length-symmetric. More generally, we construct a notion of separable elements of a finite Coxeter group W (which, for finite Weyl groups, agrees with the eponymous notion introduced by Gaetz and Gao), and show that all such elements are W-length-symmetric. This is only a sufficient condition; other W-length-symmetric elements include (twisted) involutions and elements of dihedral parabolic subgroups.

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Cite

BibTeXRIS
BibTeX
@misc{adenbaum2026qpoincaré,
  title = {{The $q$-Poincaré sum and separability in a Coxeter group}},
  author = {Ben Adenbaum and Darij Grinberg},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07365},
  eprint = {2611.07365},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07365},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The q-Poincaré sum and separability in a Coxeter group
AU  - Ben Adenbaum
AU  - Darij Grinberg
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.07365
UR  - https://arxiv.org/abs/2611.07365
AB  - Given a finite Coxeter group W and an element q of a commutative ring k, we define the element L_q :=∑_w∈ Wq^ℓ( w) w of its group algebra $\mathbf{k}\left[ W\right] $. We say that an element w∈ W is W\emph{-length-symmetric} if its coefficient in the commutator $\left[ \mathbf{L}_{a},\mathbf{L}_{b}\right] $ is $0$ for all k and all a,b∈k. We prove several sufficient criteria for elements to be W-length-symmetric. In particular, we show that any separable permutation (i.e., permutation avoiding the patterns $2413$ and $3142$) in the symmetric group S_n is S_n-length-symmetric. More generally, we construct a notion of separable elements of a finite Coxeter group W (which, for finite Weyl groups, agrees with the eponymous notion introduced by Gaetz and Gao), and show that all such elements are W-length-symmetric. This is only a sufficient condition; other W-length-symmetric elements include (twisted) involutions and elements of dihedral parabolic subgroups.
ER  -