Convex orders in root systems and quantum tangent spaces
Abstract
We classify the reduced decompositions of the longest element w0 of the Weyl group of a semisimple Lie algebra g for which the corresponding Lusztig quantum root vectors span a quantum tangent space in Uq(g). We introduce a combinatorial condition on the convex order induced by a reduced decomposition, tangency, and prove that a reduced decomposition gives rise to a quantum tangent space if and only if it is tangent. The proof relies on a description of the coproduct of quantum root vectors in terms of the convex order, and on a projection in the right weak order attached to each positive root. Up to commutation, tangent reduced decompositions are obtained by concatenating the longest minimal coset representatives along a chain of cominuscule parabolic quotients; in particular there are none in types E8, F4 and G2.
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BibTeX
@misc{carotenuto2026convex,
title = {{Convex orders in root systems and quantum tangent spaces}},
author = {Alessandro Carotenuto and Christophe Hohlweg and Paolo Papi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35091},
eprint = {2610.35091},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35091},
}
RIS
TY - UNPB
TI - Convex orders in root systems and quantum tangent spaces
AU - Alessandro Carotenuto
AU - Christophe Hohlweg
AU - Paolo Papi
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35091
UR - https://arxiv.org/abs/2610.35091
AB - We classify the reduced decompositions of the longest element w_0 of the Weyl group of a semisimple Lie algebra g for which the corresponding Lusztig quantum root vectors span a quantum tangent space in U_q(g). We introduce a combinatorial condition on the convex order induced by a reduced decomposition, tangency, and prove that a reduced decomposition gives rise to a quantum tangent space if and only if it is tangent. The proof relies on a description of the coproduct of quantum root vectors in terms of the convex order, and on a projection in the right weak order attached to each positive root. Up to commutation, tangent reduced decompositions are obtained by concatenating the longest minimal coset representatives along a chain of cominuscule parabolic quotients; in particular there are none in types E_8, F_4 and G_2.
ER -