Hall polynomials for tame type

TL;DR

This paper proves the existence of Hall polynomials for coherent sheaves on tame weighted projective lines and tame quivers, enabling the construction of associated Hall algebras and confirming related conjectures.

Contribution

It establishes the existence of Hall polynomials for tame type structures, extending previous results and confirming conjectures in the field.

Findings

Hall polynomials exist for coherent sheaves on tame weighted projective lines

Hall polynomials also exist for tame quivers, refining previous results

Construction of generic Ringel--Hall algebra and its Drinfeld double for these structures

Abstract

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.