Spectral Rigidity in Toroidal Grothendieck Rings and Multiparameter Quantum Groups

@misc{tang2026spectral,
  title = {{Spectral Rigidity in Toroidal Grothendieck Rings and Multiparameter Quantum Groups}},
  author = {Xiaomin Tang and Yu Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04189},
  eprint = {2610.04189},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04189},
  note = {CC0},
}

TY  - UNPB
TI  - Spectral Rigidity in Toroidal Grothendieck Rings and Multiparameter Quantum Groups
AU  - Xiaomin Tang
AU  - Yu Zhang
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04189
UR  - https://arxiv.org/abs/2610.04189
AB  - Let Q be a bipartite Dynkin quiver of finite simply-laced type and 𝒞_Q the corresponding Hernandez--Leclerc category. We study a two-parameter specialization of the Fedele--Hernandez toroidal Grothendieck ring and identify its generic positive part with a Cartan-type multiparameter quantum group. The structural input is the identity 2D^(2)−D_0=ω_Q∘(°_Q,°_Q), where D_0,D^(2) are the two effective toroidal commutation forms and ω_Q is the antisymmetric Euler form of Q. It realizes the specialized character algebra, after an auxiliary scalar extension, as a graded bicharacter twist of the one-parameter algebra at v=t_0t^1/2. This yields the required root grading and PBW graded dimensions. For every source--sink edge r→ s, these dimensions combine with the local A--Y commutation formula to give a quadratic annihilator for Ad_Z_r. Consequently Spec_Z_r(Z_s)={t_0t,t_0^−1}, without thinness or type-specific character formulas. The two spectral values yield the oriented multiparameter Serre relations. With q_ii=t_0^2t, q_ij=t_0^−1, q_ji=t_0^−1t^−1 (i→ j), and q_ij=1 on nonedges, we obtain U_q^+(g) ≃ 𝒦_∞(𝒞_Q)⊗_RK for g of type A_n,D_n,E_6,E_7,E_8.
ER  -