{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T02:04:57Z","timestamp":1768010697510,"version":"3.49.0"},"reference-count":11,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,2,6]],"date-time":"2021-02-06T00:00:00Z","timestamp":1612569600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,2,6]],"date-time":"2021-02-06T00:00:00Z","timestamp":1612569600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2021,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {C}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>C<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be a 2-Calabi\u2013Yau triangulated category with two cluster tilting subcategories <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {T}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>T<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {U}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>U<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. A result from J\u00f8rgensen and Yakimov (Sel Math (NS) 26:71\u201390, 2020) and Demonet et al. (Int Math Res Not 2019:852\u2013892, 2017) known as tropical duality says that the index with respect to <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {T}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>T<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> provides an isomorphism between the split Grothendieck groups of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {U}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>U<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {T}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>T<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We also have the notion of <jats:italic>c<\/jats:italic>-vectors, which using tropical duality have been proven to have sign coherence, and to be recoverable as dimension vectors of modules in a module category. The notion of triangulated categories extends to the notion of <jats:inline-formula><jats:alternatives><jats:tex-math>$$(d+2)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-angulated categories. Using a higher analogue of cluster tilting objects, this paper generalises tropical duality to higher dimensions. This implies that these basic cluster tilting objects have the same number of indecomposable summands. It also proves that under conditions of mutability, <jats:italic>c<\/jats:italic>-vectors in the <jats:inline-formula><jats:alternatives><jats:tex-math>$$(d+2)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-angulated case have sign coherence, and shows formulae for their computation. Finally, it proves that under the condition of mutability, the <jats:italic>c<\/jats:italic>-vectors are recoverable as dimension vectors of modules in a module category.<\/jats:p>","DOI":"10.1007\/s10485-020-09625-7","type":"journal-article","created":{"date-parts":[[2021,2,8]],"date-time":"2021-02-08T23:19:20Z","timestamp":1612826360000},"page":"529-545","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Tropical Duality in $$(d+2)$$-Angulated Categories"],"prefix":"10.1007","volume":"29","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5744-2438","authenticated-orcid":false,"given":"Joseph","family":"Reid","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,2,6]]},"reference":[{"key":"9625_CR1","first-page":"rnn029-17","volume":"2008","author":"R Dehy","year":"2008","unstructured":"Dehy, R., Keller, B.: On the combinatorics of rigid objects in 2-Calabi\u2013Yau categories. 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