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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2024, Volume 326, Pages 382–397
DOI: https://doi.org/10.4213/tm4403 (Mi tm4403) 		 
Newton–Okounkov Polytopes of Type $A$ Flag Varieties of Small Ranks Arising from Cluster Structures

Yunhyung Choa, Naoki Fujitab, Akihiro Higashitanic, Eunjeong Leed

a Department of Mathematics Education, Sungkyunkwan University, Seoul 03063, Republic of Korea
b Faculty of Advanced Science and Technology, Kumamoto University, 2-39-1 Kurokami, Chuo-ku, Kumamoto 860-8555, Japan
c Graduate School of Information Science and Technology, Osaka University, Osaka 565-0871, Japan
d Department of Mathematics, Chungbuk National University, Cheongju 28644, Republic of Korea
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DOI: https://doi.org/10.4213/tm4403
Abstract: A flag variety is a smooth projective homogeneous variety. In this paper, we study Newton–Okounkov polytopes of the flag variety $\mathrm {Fl}(\mathbb C^4)$ arising from its cluster structure. More precisely, we present defining inequalities of such Newton–Okounkov polytopes of $\mathrm {Fl}(\mathbb C^4)$. Moreover, we classify these polytopes, establishing their equivalence under unimodular transformations.
Keywords: flag varieties, Newton–Okounkov bodies, cluster algebras, toric varieties.
Funding agency Grant number
National Research Foundation of Korea 2020R1C1C1A01010972
2020R1A5A1016126
RS-2023-00239947
Japan Society for the Promotion of Science 20K14281
MEXT Japan Leading Initiative for Excellent Young Researchers (LEADER) Project
Y. Cho was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP; Ministry of Science, ICT and Future Planning), nos. 2020R1C1C1A01010972 and 2020R1A5A1016126. N. Fujita was supported by the JSPS Grant-in-Aid for Early-Career Scientists (no. 20K14281) and by the MEXT Japan Leading Initiative for Excellent Young Researchers (LEADER) project. E. Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT), no. RS-2023-00239947.
Received: September 29, 2023
Revised: February 27, 2024
Accepted: June 7, 2024
English version:
Proceedings of the Steklov Institute of Mathematics, 2024, Volume 326, Pages 352–366
DOI: https://doi.org/10.1134/S0081543824040175
Document Type: Article
Language: Russian
Citation: Yunhyung Cho, Naoki Fujita, Akihiro Higashitani, Eunjeong Lee, “Newton–Okounkov Polytopes of Type $A$ Flag Varieties of Small Ranks Arising from Cluster Structures”, Topology, Geometry, Combinatorics, and Mathematical Physics, Collected papers. Dedicated to Victor Matveevich Buchstaber on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 326, Steklov Math. Inst., Moscow, 2024, 382–397; Proc. Steklov Inst. Math., 326 (2024), 352–366
Citation in format AMSBIB
\Bibitem{ChoFujHig24}
\by Yunhyung~Cho, Naoki~Fujita, Akihiro~Higashitani, Eunjeong~Lee
\paper Newton--Okounkov Polytopes of Type $A$ Flag Varieties of Small Ranks Arising from Cluster Structures
\inbook Topology, Geometry, Combinatorics, and Mathematical Physics
\bookinfo Collected papers. Dedicated to Victor Matveevich Buchstaber on the occasion of his 80th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2024
\vol 326
\pages 382--397
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4403}
\crossref{https://doi.org/10.4213/tm4403}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2024
\vol 326
\pages 352--366
\crossref{https://doi.org/10.1134/S0081543824040175}
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    		Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
     
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