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

TL;DR

This paper demonstrates that the basic representation of the Drinfeld double Ringel--Hall algebra of a cyclic quiver realizes the $q$-deformed Fock space, connecting algebraic structures with combinatorial bases and decompositions.

Contribution

It extends the construction of the $q$-deformed Fock space realization via the Drinfeld double Ringel--Hall algebra of cyclic quivers, providing explicit decompositions and basis constructions.

Findings

Realization of the $q$-deformed Fock space as a basic representation.

Decomposition of the representation aligns with Kashiwara--Miwa--Stern decomposition.

Construction of the canonical basis in terms of monomial basis elements.

Abstract

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 ${\mathscr D}(\Delta_n)$.