Cluster Algebras for Bosonic Plethysm
Abstract
Let Bbbk be an algebraically closed field of characteristic zero, let V=Bbbkℓ and W=Bbbkm, and set ℛℓ,m=Sym(Sym2V⊗ W)UV. We construct an explicit skew-symmetrizable seed Σℓ,m by restricting and folding the determinantal seed for the flagged m-arrow Kronecker quiver. For every ℓ,m≥2, we have ℛℓ,m=𝒰(Σℓ,m), with polynomial frozen coefficients, and Σℓ,m admits a reddening sequence. The theta basis extends across the frozen boundary exactly for parameters in a rational polyhedral cone 𝒞ℓ,m. Its weight fibers count the multigraded highest-weight multiplicities of ℛℓ,m, and the Jacobi--Trudi identity expresses symmetric-square plethysm coefficients as finite alternating sums of these counts. Optimized frozens give an explicit finite system of inequalities for 𝒞ℓ,m.
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BibTeX
@misc{fan2026cluster,
title = {{Cluster Algebras for Bosonic Plethysm}},
author = {Yelin Fan and Jiarui Fei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.00963},
eprint = {2608.00963},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00963},
}
RIS
TY - UNPB
TI - Cluster Algebras for Bosonic Plethysm
AU - Yelin Fan
AU - Jiarui Fei
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2608.00963
UR - https://arxiv.org/abs/2608.00963
AB - Let Bbbk be an algebraically closed field of characteristic zero, let V=Bbbk^ℓ and W=Bbbk^m, and set ℛ_ℓ,m=Sym(Sym^2V⊗ W)^U_V. We construct an explicit skew-symmetrizable seed Σ_ℓ,m by restricting and folding the determinantal seed for the flagged m-arrow Kronecker quiver. For every ℓ,m≥2, we have ℛ_ℓ,m=𝒰(Σ_ℓ,m), with polynomial frozen coefficients, and Σ_ℓ,m admits a reddening sequence. The theta basis extends across the frozen boundary exactly for parameters in a rational polyhedral cone 𝒞_ℓ,m. Its weight fibers count the multigraded highest-weight multiplicities of ℛ_ℓ,m, and the Jacobi--Trudi identity expresses symmetric-square plethysm coefficients as finite alternating sums of these counts. Optimized frozens give an explicit finite system of inequalities for 𝒞_ℓ,m.
ER -