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COEFFICIENT QUIVERS, ${\mathbb {F}}_1$-REPRESENTATIONS, AND EULER CHARACTERISTICS OF QUIVER GRASSMANNIANS

Part of: Algebraic combinatorics Hopf algebras, quantum groups and related topics Lie algebras and Lie superalgebras Representation theory of rings and algebras

Published online by Cambridge University Press:  13 December 2023

JAIUNG JUN Open the ORCID record for JAIUNG JUN [Opens in a new window]  and
ALEXANDER SISTKO Open the ORCID record for ALEXANDER SISTKO [Opens in a new window]
JAIUNG JUN
Affiliation:
Department of Mathematics State University of New York at New Paltz 1 Hawk Drive New Paltz, New York 12561 United States junj@newpaltz.edu
ALEXANDER SISTKO*
Affiliation:
Department of Mathematics Manhattan College 4513 Manhattan College Parkway Riverdale, New York 10471 United States
*
asistko01@manhattan.edu
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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.

Keywords

the field with one elementrepresentations of quiversHall algebraEuler characteristicquiver Grassmannian

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets
Secondary: 05E10: Combinatorial aspects of representation theory 16T30: Connections with combinatorics 17B35: Universal enveloping (super)algebras
Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 57
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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