This project is maintained by ysykimura
2022年4月22日 16:30~18:45
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Non-rigid regions of real Grothendieck groups
These talks are mostly based on the joint work with Osamu Iyama (arXiv:2112.14908).
For any finite dimensional algebra $A$ over an algebraically closed field, the real Grothendieck group $K_0(proj A)_R$ can be identified with an Euclidean space.
By using stability conditions of King, Brüstle-Smith-Treffinger and Bridgeland considered a wall-chamber structure on $K_0(proj A)_R$.
This structure is strongly related to TF equivalence, which is an equivalence relation on $K_0(proj A)_R$ given by numerical torsion pairs of Baumann-Kamnitzer-Tingley.
For example, the g-vector cone $C^+(U)$ associated to each 2-term presilting complex $U$ is a TF equivalence class by results of Yurikusa and Brüstle-Smith-Treffinger.
On the other hand, these cones do not cover $K_0(proj A)_R$ if $A$ is not $\tau$-tilting finite, so we define the non-rigid region of $K_0(proj A)_R$ as the region where the g-vector cones do not exist.
I would like to talk about two approaches to study the non-rigid region.
In the first talk, I will explain our result that the cone defined by the canonical decomposition of each element in $K_0(proj A)$ of Derksen-Fei is a TF equivalence class if $A$ belongs to the class called E-tame algebras.
In the second talk, I will deal with the description of the non-rigid region in terms of 2-term presilting complexes and a certain closed subset called the purely non-rigid region of $K_0(proj A)_R$.
Moreover, I will also talk on the algorithm to determine the purely non-rigid regions of special biserial algebras obtained in my sole work (arXiv:2201.09543).