Abstract
In this paper, we report on the $\tau$-tilting finiteness of some classes of finite-dimensional algebras over an algebraically closed field, including symmetric algebras of polynomial growth, $0$-Hecke algebras and $0$-Schur algebras. Consequently, we find that derived equivalence preserves the $\tau$-tilting finiteness over symmetric algebras of polynomial growth, and self-injective cellular algebras of polynomial growth are $\tau$-tilting finite. Furthermore, the representation-finiteness and $\tau$-tilting finiteness over $0$-Hecke algebras and $0$-Schur algebras (with few exceptions) coincide.
Funding Statement
Miyamoto was partly supported by JSPS Grant-in-Aid for Early-Career Scientists (Grant No. 20K14302 and 24K16885), Grant-in-Aid for Scientific Research (A) (Grant No. 23H00479) and FY 2023 Research Project Expense Subsidy Program: Research Network Formation Project (National Institute of Technology, Japan). Wang is partially supported by JSPS Grant-in-Aid for JSPS Fellows (Grant No. 20J10492), National Key Research and Development Program of China (Grant No. 2020YFA0713000) and China Postdoctoral Science Foundation (Grant No. YJ20220119 and No. 2023M731988).
Acknowledgments
The authors would like to express their gratitude to Professor Susumu Ariki for introducing them to this area of research and for his valuable comments and suggestions. Additionally, the authors are deeply thankful to Ryoichi Kase for his insightful suggestions, this paper would not have been complete without his comments.
Citation
Kengo Miyamoto. Qi Wang. "On $\tau$-tilting Finiteness of Symmetric Algebras of Polynomial Growth." Taiwanese J. Math. 28 (6) 1073 - 1094, December, 2024. https://doi.org/10.11650/tjm/240708
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