Atomic Decompositions of Lie Characters and the Dominant Weight Poset
Abstract
Let g be a complex semisimple Lie algebra and, for a dominant integral weight λ, let Θλ be the multiplicity-free sum of the weights of the irreducible module V(λ). The Θμ form a ℤ-basis of the W-invariants, so there are unique integers a(μ,λ), the atomic numbers, with ch V(λ) = ∑μa(μ,λ)Θμ over dominant μ. We ask when they are nonnegative. They are the Moebius transform of the weight-multiplicity function on the dominant-weight poset, which the crosscut theorem turns into an alternating sum of at most 2rank Φ multiplicities indexed by subsets of the covers of μ in [μ,λ]. We prove that a(μ,λ)≥ 0 whenever every connected component of the Dynkin diagram is a path, and identify the coefficient with the dimension of an explicit weight space: it is cut out of V(λ) by alternately taking kernels of raising operators and cokernels of lowering operators, one per cover, in order along the path. Type D4 shows the hypothesis is necessary. Put β= α1 + 2α2 + 2α3 + 2α4; for every dominant μ with ⟨μ,α2∨⟩ = 1 and μ+β dominant, a(μ,μ+β) is −2 if ⟨μ,α1∨⟩ = 0 and −1 otherwise. Restriction to the support of λ−μ carries this family into every irreducible type with a trivalent node. Thus an irreducible finite root system has all atomic numbers nonnegative exactly when its Dynkin diagram is a path, namely in types An, Bn, Cn, F4, G2. Deep in the dominant chamber, in an explicit range, a(μ,λ) is the Kostant partition number of λ−μ for the nonsimple positive roots; negative atomic numbers are therefore confined to boundary slabs.
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BibTeX
@misc{hindi2026atomic,
title = {{Atomic Decompositions of Lie Characters and the Dominant Weight Poset}},
author = {Felipe de Mattos Chafik Hindi and Waldeck Schutzer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08024},
eprint = {2609.08024},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08024},
}
RIS
TY - UNPB
TI - Atomic Decompositions of Lie Characters and the Dominant Weight Poset
AU - Felipe de Mattos Chafik Hindi
AU - Waldeck Schutzer
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.08024
UR - https://arxiv.org/abs/2609.08024
AB - Let g be a complex semisimple Lie algebra and, for a dominant integral weight λ, let Θ_λ be the multiplicity-free sum of the weights of the irreducible module V(λ). The Θ_μ form a ℤ-basis of the W-invariants, so there are unique integers a(μ,λ), the atomic numbers, with ch V(λ) = ∑_μa(μ,λ)Θ_μ over dominant μ. We ask when they are nonnegative. They are the Moebius transform of the weight-multiplicity function on the dominant-weight poset, which the crosscut theorem turns into an alternating sum of at most 2^rank Φ multiplicities indexed by subsets of the covers of μ in [μ,λ]. We prove that a(μ,λ)≥ 0 whenever every connected component of the Dynkin diagram is a path, and identify the coefficient with the dimension of an explicit weight space: it is cut out of V(λ) by alternately taking kernels of raising operators and cokernels of lowering operators, one per cover, in order along the path. Type D_4 shows the hypothesis is necessary. Put β= α_1 + 2α_2 + 2α_3 + 2α_4; for every dominant μ with ⟨μ,α_2^∨⟩ = 1 and μ+β dominant, a(μ,μ+β) is −2 if ⟨μ,α_1^∨⟩ = 0 and −1 otherwise. Restriction to the support of λ−μ carries this family into every irreducible type with a trivalent node. Thus an irreducible finite root system has all atomic numbers nonnegative exactly when its Dynkin diagram is a path, namely in types A_n, B_n, C_n, F_4, G_2. Deep in the dominant chamber, in an explicit range, a(μ,λ) is the Kostant partition number of λ−μ for the nonsimple positive roots; negative atomic numbers are therefore confined to boundary slabs.
ER -