---
_id: '14930'
abstract:
- lang: eng
text: In this paper we investigate locally free representations of a quiver Q over
a commutative Frobenius algebra R by arithmetic Fourier transform. When the base
field is finite we prove that the number of isomorphism classes of absolutely
indecomposable locally free representations of fixed rank is independent of the
orientation of Q. We also prove that the number of isomorphism classes of locally
free absolutely indecomposable representations of the preprojective algebra of
Q over R equals the number of isomorphism classes of locally free absolutely indecomposable
representations of Q over R[t]/(t2). Using these results together with results
of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification
of pairs (Q, R) such that the set of isomorphism classes of indecomposable locally
free representations of Q over R is finite. Finally when the representation is
free of rank 1 at each vertex of Q, we study the function that counts the number
of isomorphism classes of absolutely indecomposable locally free representations
of Q over the Frobenius algebra Fq[t]/(tr). We prove that they are polynomial
in q and their generating function is rational and satisfies a functional equation.
acknowledgement: Special thanks go to Christof Geiss, Bernard Leclerc and Jan Schröer
for explaining their work but also for sharing some unpublished results with us.
We also thank the referee for many useful suggestions. We would like to thank Tommaso
Scognamiglio for pointing out a mistake in the proof of Proposition 5.17 in an earlier
version of the paper. We would like also to thank Alexander Beilinson, Bill Crawley-Boevey,
Joel Kamnitzer, and Peng Shan for useful discussions.
article_number: '20'
article_processing_charge: No
article_type: original
author:
- first_name: Tamás
full_name: Hausel, Tamás
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Emmanuel
full_name: Letellier, Emmanuel
last_name: Letellier
- first_name: Fernando
full_name: Rodriguez-Villegas, Fernando
last_name: Rodriguez-Villegas
citation:
ama: Hausel T, Letellier E, Rodriguez-Villegas F. Locally free representations of
quivers over commutative Frobenius algebras. Selecta Mathematica. 2024;30(2).
doi:10.1007/s00029-023-00914-2
apa: Hausel, T., Letellier, E., & Rodriguez-Villegas, F. (2024). Locally free
representations of quivers over commutative Frobenius algebras. Selecta Mathematica.
Springer Nature. https://doi.org/10.1007/s00029-023-00914-2
chicago: Hausel, Tamás, Emmanuel Letellier, and Fernando Rodriguez-Villegas. “Locally
Free Representations of Quivers over Commutative Frobenius Algebras.” Selecta
Mathematica. Springer Nature, 2024. https://doi.org/10.1007/s00029-023-00914-2.
ieee: T. Hausel, E. Letellier, and F. Rodriguez-Villegas, “Locally free representations
of quivers over commutative Frobenius algebras,” Selecta Mathematica, vol.
30, no. 2. Springer Nature, 2024.
ista: Hausel T, Letellier E, Rodriguez-Villegas F. 2024. Locally free representations
of quivers over commutative Frobenius algebras. Selecta Mathematica. 30(2), 20.
mla: Hausel, Tamás, et al. “Locally Free Representations of Quivers over Commutative
Frobenius Algebras.” Selecta Mathematica, vol. 30, no. 2, 20, Springer
Nature, 2024, doi:10.1007/s00029-023-00914-2.
short: T. Hausel, E. Letellier, F. Rodriguez-Villegas, Selecta Mathematica 30 (2024).
date_created: 2024-02-04T23:00:53Z
date_published: 2024-01-27T00:00:00Z
date_updated: 2024-02-05T12:58:21Z
day: '27'
department:
- _id: TaHa
doi: 10.1007/s00029-023-00914-2
intvolume: ' 30'
issue: '2'
language:
- iso: eng
month: '01'
oa_version: None
publication: Selecta Mathematica
publication_identifier:
eissn:
- 1420-9020
issn:
- 1022-1824
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Locally free representations of quivers over commutative Frobenius algebras
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 30
year: '2024'
...
---
_id: '9998'
abstract:
- lang: eng
text: We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss
type A in detail as well as its connections with quantum XXZ spin chains and trigonometric
Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic
version of results of Givental and Kim, connecting quantum geometry of flag varieties
and Toda lattice.
acknowledgement: 'First of all we would like to thank Andrei Okounkov for invaluable
discussions, advises and sharing with us his fantastic viewpoint on modern quantum
geometry. We are also grateful to D. Korb and Z. Zhou for their interest and comments.
The work of A. Smirnov was supported in part by RFBR Grants under Numbers 15-02-04175
and 15-01-04217 and in part by NSF Grant DMS–2054527. The work of P. Koroteev, A.M.
Zeitlin and A. Smirnov is supported in part by AMS Simons travel Grant. A. M. Zeitlin
is partially supported by Simons Collaboration Grant, Award ID: 578501. Open access
funding provided by Institute of Science and Technology (IST Austria).'
article_number: '87'
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Peter
full_name: Koroteev, Peter
last_name: Koroteev
- first_name: Petr
full_name: Pushkar, Petr
id: 151DCEB6-9EC3-11E9-8480-ABECE5697425
last_name: Pushkar
- first_name: Andrey V.
full_name: Smirnov, Andrey V.
last_name: Smirnov
- first_name: Anton M.
full_name: Zeitlin, Anton M.
last_name: Zeitlin
citation:
ama: Koroteev P, Pushkar P, Smirnov AV, Zeitlin AM. Quantum K-theory of quiver varieties
and many-body systems. Selecta Mathematica. 2021;27(5). doi:10.1007/s00029-021-00698-3
apa: Koroteev, P., Pushkar, P., Smirnov, A. V., & Zeitlin, A. M. (2021). Quantum
K-theory of quiver varieties and many-body systems. Selecta Mathematica.
Springer Nature. https://doi.org/10.1007/s00029-021-00698-3
chicago: Koroteev, Peter, Petr Pushkar, Andrey V. Smirnov, and Anton M. Zeitlin.
“Quantum K-Theory of Quiver Varieties and Many-Body Systems.” Selecta Mathematica.
Springer Nature, 2021. https://doi.org/10.1007/s00029-021-00698-3.
ieee: P. Koroteev, P. Pushkar, A. V. Smirnov, and A. M. Zeitlin, “Quantum K-theory
of quiver varieties and many-body systems,” Selecta Mathematica, vol. 27,
no. 5. Springer Nature, 2021.
ista: Koroteev P, Pushkar P, Smirnov AV, Zeitlin AM. 2021. Quantum K-theory of quiver
varieties and many-body systems. Selecta Mathematica. 27(5), 87.
mla: Koroteev, Peter, et al. “Quantum K-Theory of Quiver Varieties and Many-Body
Systems.” Selecta Mathematica, vol. 27, no. 5, 87, Springer Nature, 2021,
doi:10.1007/s00029-021-00698-3.
short: P. Koroteev, P. Pushkar, A.V. Smirnov, A.M. Zeitlin, Selecta Mathematica
27 (2021).
date_created: 2021-09-12T22:01:22Z
date_published: 2021-08-30T00:00:00Z
date_updated: 2023-08-14T06:34:14Z
day: '30'
ddc:
- '530'
department:
- _id: TaHa
doi: 10.1007/s00029-021-00698-3
external_id:
isi:
- '000692795200001'
file:
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checksum: beadc5a722ffb48190e1e63ee2dbfee5
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creator: cchlebak
date_created: 2021-09-13T11:31:34Z
date_updated: 2021-09-13T11:31:34Z
file_id: '10010'
file_name: 2021_SelectaMath_Koroteev.pdf
file_size: 584648
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intvolume: ' 27'
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project:
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name: IST Austria Open Access Fund
publication: Selecta Mathematica
publication_identifier:
eissn:
- 1420-9020
issn:
- 1022-1824
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Quantum K-theory of quiver varieties and many-body systems
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
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...