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Tokyo-Nagoya Algebra Seminar
Seminar information archive ～04/18｜Next seminar｜Future seminars 04/19～
| Organizer(s) | Noriyuki Abe, Aaron Chan, Osamu Iyama, Yasuaki Gyoda, Hiroyuki Nakaoka, Ryo Takahashi |
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2021/12/16
16:45-18:15 Online
Please see the URL below for details on the online seminar.
Nicholas Williams (University of Cologne)
Cyclic polytopes and higher Auslander-Reiten theory (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Nicholas Williams (University of Cologne)
Cyclic polytopes and higher Auslander-Reiten theory (English)
[ Abstract ]
Oppermann and Thomas show that tilting modules over Iyama’s higher Auslander algebras of type A are in bijection with triangulations of even-dimensional cyclic polytopes. Triangulations of cyclic polytopes are partially ordered in two natural ways known as the higher Stasheff-Tamari orders, which were introduced in the 1990s by Kapranov, Voevodsky, Edelman, and Reiner as higher-dimensional generalisations of the Tamari lattice. These two partial orders were conjectured to be equal in 1996 by Edelman and Reiner, and we prove that this conjecture is true. We further show how the higher Stasheff-Tamari orders correspond in even dimensions to natural orders on tilting modules which were studied by Riedtmann, Schofield, Happel, and Unger. This then allows us to complete the picture of Oppermann and Thomas by showing that triangulations of odd-dimensional cyclic polytopes correspond to equivalence classes of d-maximal green sequences, which we introduce as higher-dimensional analogues of Keller’s maximal green sequences. We show that the higher Stasheff-Tamari orders correspond to natural orders on equivalence classes of d-maximal green sequences, which relate to the no-gap conjecture of Brustle, Dupont, and Perotin. The equality of the higher Stasheff-Tamari orders then implies that these algebraic orders on tilting modules and d-maximal green sequences are equal. If time permits, we will also discuss some results on mutation of cluster-tilting objects and triangulations.
[ Reference URL ]Oppermann and Thomas show that tilting modules over Iyama’s higher Auslander algebras of type A are in bijection with triangulations of even-dimensional cyclic polytopes. Triangulations of cyclic polytopes are partially ordered in two natural ways known as the higher Stasheff-Tamari orders, which were introduced in the 1990s by Kapranov, Voevodsky, Edelman, and Reiner as higher-dimensional generalisations of the Tamari lattice. These two partial orders were conjectured to be equal in 1996 by Edelman and Reiner, and we prove that this conjecture is true. We further show how the higher Stasheff-Tamari orders correspond in even dimensions to natural orders on tilting modules which were studied by Riedtmann, Schofield, Happel, and Unger. This then allows us to complete the picture of Oppermann and Thomas by showing that triangulations of odd-dimensional cyclic polytopes correspond to equivalence classes of d-maximal green sequences, which we introduce as higher-dimensional analogues of Keller’s maximal green sequences. We show that the higher Stasheff-Tamari orders correspond to natural orders on equivalence classes of d-maximal green sequences, which relate to the no-gap conjecture of Brustle, Dupont, and Perotin. The equality of the higher Stasheff-Tamari orders then implies that these algebraic orders on tilting modules and d-maximal green sequences are equal. If time permits, we will also discuss some results on mutation of cluster-tilting objects and triangulations.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
