On the number of $\tau $-tilting modules over the Auslander algebras of radical square zero Nakayama algebras

Hanpeng Gao, Zongzhen Xie, Zhaoyong Huang Colloquium Mathematicum MSC: Primary 16G10; Secondary 05E10, 05A19. DOI: 10.4064/cm8474-7-2021 Published online: 7 April 2022


Let $\Lambda _n$ be a radical square zero Nakayama algebra with $n$ simple modules and $\Gamma _n$ the Auslander algebra of $\Lambda _n$. We calculate the number $|\tau \text {-tilt}\,\Gamma _n|$ of $\tau $-tilting modules and the number $|{\rm s}\tau \text {-tilt}\,\Gamma _n|$ of support $\tau $-tilting modules over $\Gamma _n$. In particular, we prove the recurrence relations $$|\tau \text {-tilt}\,\Gamma _n|=3|\tau \text {-tilt}\,\Gamma _{n-1}|+|\tau \text {-tilt}\,\Gamma _{n-2}|,$$ $$|{\rm s}\tau \text {-tilt}\,\Gamma _n|=6|{\rm s}\tau \text {-tilt}\,\Gamma _{n-1}|+3|{\rm s}\tau \text {-tilt}\,\Gamma _{n-2}|,$$ from which the exact values of $|\tau \text {-tilt}\,\Gamma _n|$ and $|{\rm s}\tau \text {-tilt}\,\Gamma _n|$ are derived.


  • Hanpeng GaoSchool of Mathematical Sciences
    Anhui University
    230601 Hefei, Anhui Province, P.R. China
  • Zongzhen XieDepartment of Mathematics
    and Computer Science
    School of Biomedical Engineering
    and Informatics
    Nanjing Medical University
    211166 Nanjing, Jiangsu Province, P.R. China
  • Zhaoyong HuangDepartment of Mathematics
    Nanjing University
    210093 Nanjing, Jiangsu Province, P.R. China

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