Hall多项式

Hall polynomials

作为Hall代数的结构常数, Hall数和Hall多项式与对称群的表示和量子群的结构有紧密联系.本文首先引入经典Hall代数和Hall多项式的概念,并阐述其与对称函数的联系.其次,定义有限域上代数的Ringel-Hall代数,并简述其与量子群的关系.最后,本文在Dynkin箭图和仿射箭图情形下,讨论Hall多项式的存在性.

As structure constants of Hall algebras, Hall numbers and Hall polynomials have deep connections with theories of representations of symmetric groups and quantum groups. In this survey, we first introduce notions of classical Hall algebras and Hall polynomials, and present their relationship with the ring of symmetric functions. We then define Ringel-Hall algebras of algebras over finite fields and state their connections with quantum groups. We also discuss the existence of Hall polynomials in the Dynkin and affine quiver cases.