Kac polynomials for canonical algebras

TL;DR

This paper proves that the count of geometrically indecomposable representations of canonical and squid algebras over finite fields can be expressed as polynomials in the field size, linking to moduli stack volumes.

Contribution

It establishes that these counts are polynomial in q for canonical and squid algebras, extending known results to new algebra classes.

Findings

Number of indecomposable representations is given by a polynomial in q.

Expresses moduli stack volumes in terms of Kac polynomials.

Extends polynomial count results to almost concealed canonical algebras.

Abstract

We prove that the number of geometrically indecomposable representations of fixed dimension vector d of a canonical algebra C defined over a finite field Fq is given by a polynomial in q (depending on C and d). We prove a similar result for squid algebras (and for any almost concealed canonical algebra). Finally we express the volume of the moduli stacks of representations of these algebras of a fixed dimension vector in terms of the corresponding Kac polynomials.