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COEFFICIENT QUIVERS,
${\mathbb {F}}_1$-REPRESENTATIONS, AND EULER CHARACTERISTICS OF QUIVER GRASSMANNIANS
Published online by Cambridge University Press: 13 December 2023
Abstract
A quiver representation assigns a vector space to each vertex, and a linear map to each arrow of a quiver. When one considers the category $\mathrm {Vect}(\mathbb {F}_1)$ of vector spaces “over
$\mathbb {F}_1$” (the field with one element), one obtains
$\mathbb {F}_1$-representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficient quivers. To be precise, we prove that the category
$\mathrm {Rep}(Q,\mathbb {F}_1)$ is equivalent to the (suitably defined) category of coefficient quivers over Q. This provides a conceptual way to see Euler characteristics of a class of quiver Grassmannians as the number of “
$\mathbb {F}_1$-rational points” of quiver Grassmannians. We generalize techniques originally developed for string and band modules to compute the Euler characteristics of quiver Grassmannians associated with
$\mathbb {F}_1$-representations. These techniques apply to a large class of
$\mathbb {F}_1$-representations, which we call the
$\mathbb {F}_1$-representations with finite nice length: we prove sufficient conditions for an
$\mathbb {F}_1$-representation to have finite nice length, and classify such representations for certain families of quivers. Finally, we explore the Hall algebras associated with
$\mathbb {F}_1$-representations of quivers. We answer the question of how a change in orientation affects the Hall algebra of nilpotent
$\mathbb {F}_1$-representations of a quiver with bounded representation type. We also discuss Hall algebras associated with representations with finite nice length, and compute them for certain families of quivers.
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal
References
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