PreprintCC BY 4.0arXiv
Length-generating functions for twisted involutions of Coxeter groups
Abstract
For a Coxeter system (W,S) of finite rank and an involutive automorphism ∗:W→ W which preserves S, we prove that the length-generating function ∑z∈IW,∗qℓ(z) of the set of ∗-twisted involutions is rational. We derive recurrence relations analogous to the well-known recurrence relations expressing the growth series W(q):=∑w∈ Wqℓ(w) in terms of the growth series WI(q) of the parabolic subgroups of W. Our result extends to any subset 𝒞 of ∗-twisted involutions that is closed under ∗-twisted conjugation. We use these recurrence relations to give an alternative proof of a power-series identity for twisted involutions due to Lusztig.
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BibTeX
@misc{man2026lengthgenerating,
title = {{Length-generating functions for twisted involutions of Coxeter groups}},
author = {Ronald de Man},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08000},
eprint = {2609.08000},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08000},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Length-generating functions for twisted involutions of Coxeter groups
AU - Ronald de Man
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.08000
UR - https://arxiv.org/abs/2609.08000
AB - For a Coxeter system (W,S) of finite rank and an involutive automorphism ∗:W→ W which preserves S, we prove that the length-generating function ∑_z∈I_W,∗q^ℓ(z) of the set of ∗-twisted involutions is rational. We derive recurrence relations analogous to the well-known recurrence relations expressing the growth series W(q):=∑_w∈ Wq^ℓ(w) in terms of the growth series W_I(q) of the parabolic subgroups of W. Our result extends to any subset 𝒞 of ∗-twisted involutions that is closed under ∗-twisted conjugation. We use these recurrence relations to give an alternative proof of a power-series identity for twisted involutions due to Lusztig.
ER -