Polytopal invariance for Lusztig's q-weight multiplicities
Abstract
Let Φ be an irreducible root system normalized so that long roots have length √2, and let μ≤λ be dominant weights. We associate to (Φ,μ,λ) the convex polytope PΦμ,λ = (μ+C)∩(λ−C)∩(D̅−ρ), where C=ℝ≥ 0Φ+ and D̅ is the closed dominant Weyl chamber. We prove that if PΦμ,λ and PΦ'μ',λ' are isometric, then KΦλμ(q)=KΦ'λ'μ'(q), where KΦλμ(q) is Lusztig's q-analogue of the μ-weight multiplicity in the irreducible highest-weight representation V(λ), often called the generalized Kostka-Foulkes polynomial. Moreover, we construct the polynomial directly from the metric shape of the polytope, and we classify when two such polytopes are isometric. Such coincidences are abundant and can occur in different Dynkin types, and in different Dynkin families. By the Lusztig-Kato theorem, the main theorem gives a metric analogue, in the spherical setting, of the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.
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BibTeX
@misc{libedinsky2026polytopal,
title = {{Polytopal invariance for Lusztig's $q$-weight multiplicities}},
author = {Nicolás Libedinsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14275},
eprint = {2610.14275},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14275},
}
RIS
TY - UNPB
TI - Polytopal invariance for Lusztig's q-weight multiplicities
AU - Nicolás Libedinsky
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14275
UR - https://arxiv.org/abs/2610.14275
AB - Let Φ be an irreducible root system normalized so that long roots have length √2, and let μ≤λ be dominant weights. We associate to (Φ,μ,λ) the convex polytope P^Φ_μ,λ = (μ+C)∩(λ−C)∩(D̅−ρ), where C=ℝ_≥ 0Φ^+ and D̅ is the closed dominant Weyl chamber. We prove that if P^Φ_μ,λ and P^Φ'_μ',λ' are isometric, then K^Φ_λμ(q)=K^Φ'_λ'μ'(q), where K^Φ_λμ(q) is Lusztig's q-analogue of the μ-weight multiplicity in the irreducible highest-weight representation V(λ), often called the generalized Kostka-Foulkes polynomial. Moreover, we construct the polynomial directly from the metric shape of the polytope, and we classify when two such polytopes are isometric. Such coincidences are abundant and can occur in different Dynkin types, and in different Dynkin families. By the Lusztig-Kato theorem, the main theorem gives a metric analogue, in the spherical setting, of the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.
ER -