Hall algebras of cyclic quivers and q-deformed Fock spaces

Bangming Deng Tsinghua University Jie Xiao Tsinghua University

Quantum Algebra Rings and Algebras mathscidoc:1610.29002

Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel--Hall algebra ${\mathscr D}(Q)$ of a quiver $Q$ as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation $L(\Lambda_0)$ of ${\mathscr D}(\Delta_n)$ of the cyclic quiver $\Delta_n$ provides a realization of the $q$-deformed Fock space $\bigwedge^\infty$ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of $\Delta_n$, we obtain a decomposition of the basic representation $L(\Lambda_0)$ which induces the Kashiwara--Miwa--Stern decomposition of $\bigwedge^\infty$ and a construction of the canonical basis of $\bigwedge^\infty$ defined by Leclerc and Thibon in terms of certain monomial basis elements in ${\mathscr D}(\Delta_n)$.
cyclic quiver, Ringel-Hall algebra, quantum group, Fock space
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  title={Hall algebras of cyclic quivers and q-deformed Fock spaces},
  author={Bangming Deng, and Jie Xiao},
Bangming Deng, and Jie Xiao. Hall algebras of cyclic quivers and q-deformed Fock spaces. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161006200156270235091.
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