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In the present paper we prove that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line $\mathbb X$ over finite fields. These polynomials are then used to define the generic Ringel--Hall algebra of $\mathbb X$ as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein.

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

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