Hall algebras of cyclic quivers and $q$-deformed Fock spaces

Abstract

Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel--Hall algebra $\mathcal{D}(Q)$ 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 $\mathcal{D}({\Delta }_n)$ 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 $\mathcal{D}({\Delta }_n)$.

References in corpus (1)

• Quivers of type A, flag varieties and representation theory