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A symmetry of silting quivers

Part of: Representation theory of rings and algebras

Published online by Cambridge University Press:  26 July 2023

Takuma Aihara Open the ORCID record for Takuma Aihara [Opens in a new window]  and
Qi Wang
Takuma Aihara
Affiliation:
Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan
Qi Wang*
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University, Hai Dian District, Beijing 100084, China
*
Corresponding author: Qi Wang; Email: infinite-wang@outlook.com
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Abstract

We investigate symmetry of the silting quiver of a given algebra which is induced by an anti-automorphism of the algebra. In particular, one shows that if there is a primitive idempotent fixed by the anti-automorphism, then the 2-silting quiver ($=$ the support $\tau$-tilting quiver) has a bisection. Consequently, in that case, we obtain that the cardinality of the 2-silting quiver is an even number (if it is finite).

Keywords

silting objectsilting mutationsilting quiversupport τ-tilting modulesupport τ-tilting quiveranti-automorphism

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets
Secondary: 16G60: Representation type (finite, tame, wild, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 65 , Issue 3 , September 2023 , pp. 687 - 696
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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Footnotes

TA was partly supported by JSPS Grant-in-Aid for Young Scientists 19K14497. QW was partly supported by JSPS Grant-in-Aid for Young Scientists 20J10492.

References

Adachi, T., The classification of $\tau$ -tilting modules over Nakayama algebras, J. Algebra 452 (2016), 227–262.Google Scholar
Adachi, T. and Aoki, T., The number of two-term tilting complexes over symmetric algebras with radical cube zero, Ann. Comb. 27(1) (2023), 149–167.CrossRefGoogle Scholar
Adachi, T., Iyama, O. and Reiten, I., $\tau$ -tilting theory, Compos. Math. 150(3) (2014), 415–452.Google Scholar
Aihara, T., Honma, T., Miyamoto, K. and Wang, Q., Report on the finiteness of silting objects, Proc. Edinb. Math. Soc. 2(2) (2021), 64–233.Google Scholar
Aihara, T. and Iyama, O., Silting mutation in triangulated categories, J. Lond. Math. Soc. 2(3) (2012), 85–668.Google Scholar
Alperin, J. L., Local representation theory, Cambridge Studies in Advanced Mathematics, vol. 11 (Cambridge University Press, Cambridge, 1986).Google Scholar
Asashiba, H., Mizuno, Y. and Nakashima, K., Simplicial complexes and tilting theory for Brauer tree algebras, J. Algebra 551 (2020), 119–153.Google Scholar
Ariki, S. and Speyer, L., Schurian-finiteness of blocks of type A Hecke algebras, Preprint (2022), arXiv: 2112.11148.Google Scholar
Eisele, F., Janssens, G. and Raedschelders, T., A reduction theorem for $\tau$ -rigid modules, Math. Z. 290(3-4) (2018), 1377–1413.Google Scholar
Graham, J. J. and Lehrer, G. I., Cellular algebras, Invent. Math. 123(1) (1996), 1–34.Google Scholar
Gelfand, I. M. and Ponomarev, V. A., Model algebras and representations of graphs, Funktsional. Anal. i Prilozhen. 13(3) (1979), 1–12.Google Scholar
Johnson, D. L., Presentations of groups, London Mathematical Society Student Texts, vol. 15 (Cambridge University Press, Cambridge, 1976).Google Scholar
Kulshammer, B., Crossed products and blocks with normal defect groups, Commun. Algebra 13(1) (1985), 147–168.Google Scholar
Konig, S. and Xi, C., On the structure of cellular algebras, in Algebra and modules, II, CMS Conf. Proc., vol. 24 (Amer. Math. Soc., Providence, RI, 1998), 365–386.Google Scholar
Mizuno, Y., Classifying $\tau$ -tilting modules over preprojective algebras of Dynkin type, Math. Z. 277(3-4) (2014), 665–690.Google Scholar
Nagao, H. and Tsushima, Y., Representations of finite groups (Academic Press, Inc., Boston, MA, 1989).Google Scholar
Wang, Q., On $\tau$ -tilting finiteness of the Schur algebra, J. Pure Appl. Algebra 226(1) (2022), 106818.Google Scholar
Xi, C., Quasi-hereditary algebras with a duality, J. Reine Angew. Math. 449 (1994), 201–215.Google Scholar