Geir Helleloid ; Fernando Rodriguez-Villegas - Counting Quiver Representations over Finite Fields Via Graph Enumeration

dmtcs:2742 - Discrete Mathematics & Theoretical Computer Science, January 1, 2009, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) - https://doi.org/10.46298/dmtcs.2742
Counting Quiver Representations over Finite Fields Via Graph EnumerationArticle

Authors: Geir Helleloid 1; Fernando Rodriguez-Villegas 1

  • 1 Department of Mathematics [Texas]

Let $\Gamma$ be a quiver on $n$ vertices $v_1, v_2, \ldots , v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua gave a formula for $A_{\Gamma}(\boldsymbol{\alpha}, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\mathbb{F}_q$ with dimension vector $\boldsymbol{\alpha}$. We use Hua's formula to show that the derivatives of $A_{\Gamma}(\boldsymbol{\alpha}, q)$ with respect to $q$, when evaluated at $q = 1$, are polynomials in the variables $g_{ij}$, and we can compute the highest degree terms in these polynomials. The formulas for these coefficients depend on the enumeration of certain families of connected graphs. This note simply gives an overview of these results; a complete account of this research is available on the arXiv and has been submitted for publication.


Volume: DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
Section: Proceedings
Published on: January 1, 2009
Imported on: January 31, 2017
Keywords: quiver representation,finite field,graph enumeration,absolutely indecomposable representation,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Periods and Special Values of L-functions; Funder: National Science Foundation; Code: 0200605

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