Structure coefficients for quantum groups
Yixin Lan, Yumeng Wu, Jie Xiao
TL;DR
The paper addresses refining the structure of quantum groups of finite type by studying structure coefficients through Hall algebras of quivers with automorphisms. It develops a geometric framework based on Lusztig's sheaves with a periodic functor to realize coefficients between the PBW basis and the canonical basis via modified Grothendieck groups, unifying Lusztig's and Caldero–Reineke's perspectives when the automorphism is trivial. A key contribution is an alternative geometric proof of the existence of Hall polynomials and a slight generalization of the Caldero–Reineke expression for the bar involution in symmetrizable finite types, with explicit polynomial dependence on $v$ and the Frobenius parameter $q$. The results yield a polynomial description of the PBW-to-canonical basis transition and connect these coefficients to Hall polynomials upon specialization, thereby deepening the link between geometric representation theory of quivers and the combinatorics of quantum groups.
Abstract
According to the Hall algebras of quivers with automorphisms under Lusztig's construction, the polynominal forms of several structure coefficients for quantum groups of all finite types are presented in this note. We first provide a geometric realization of the coefficients between PBW basis and the canonical basis via standard sheaves on quiver moduli spaces with admissible automorphisms. This realization is constructed through Lusztig sheaves equipped with periodic functors and their modified Grothendieck groups. Second, within this geometric framework, we present an alternative proof for the existence of Hall polynomials originally due to Ringel. Finally, we give a slight generalization of the Reineke-Caldero expression for the bar involution of PBW basis elements in symmetrizable cases. When the periodic functor $\mathbf{a}^*$ is taken $\operatorname{id}$, our results are the same as Lusztig's and Caldero-Reineke's.