Canonical decomposition and quiver representations over finite field

ABSTRACT

Given a quiver Q, let M_Q(α,q) be the number of isomorphism classes of representations of Q over the finite field 𝔽_q with dimension vector α, and let $\alpha ={\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n$ be the canonical decomposition of α. We establish a relationship between polynomials M_Q(α,q) and ${\prod }_{i=1}^n{M}_Q({\alpha }_i,q)$ when Q is a tame quiver. This is based on some methods in the representation theory of quivers and algebraic combinatorics.

KEYWORDS:

• Canonical decomposition
• finite field
• leading term
• quiver representation

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16G20

Acknowledgements

The authors would like to express their sincere gratitude to Professor Bangming Deng for his encouragement and many valuable discussion. In addition, the authors would like to thank the referee for many helpful comments and suggestions.

Additional information

Funding

The first author is supported by the National Natural Science Foundation of China (Grant No. 11701090) and the Natural Science Foundation of Fujian Province of China (Grant No. 2017J05004).