Speaker
淺井聡太 (大阪大学情報科学研究科)
Date
October 6, 15:45-16:35
Title
Tame elements in the Grothendieck groups of special biserial algebras (slide)
Abstract
This talk is based on joint work with Laurent Demonet and Osamu Iyama in progress. I will explain a numerical way to construct some tame element (if exists) in the Grothendieck group \(K_0(\operatorname{proj} A)\) of a special biserial algebra \(A=KQ/I\). We consider the presentation space \(\operatorname{PHom}(\theta):=\operatorname{Hom}_A(P_1,P_0)\) for each element \(\theta=[P_0]-[P_1] \in K_0(\operatorname{proj} A)\), where \(P_0\) and \(P_1\) have no nonzero common direct summands. Every \(f \in \operatorname{PHom}(\theta)\) defines a 2-term complex \(P_f\). As in Derksen-Fei, we say that \(\theta\) is indecomposable if \(P_f\) is indecomposable for a general \(f\). An indecomposable element \(\theta\) is said to be rigid if some \(f\) satisfies \(\operatorname{Hom}(P_f,P_f[1])=0\), and \(\theta\) is said to be tame if \(\theta\) is not rigid and \(\operatorname{Hom}(P_f,P_g[1])=0\) for a general pair \((f,g)\). Since \(A\) is a special biserial algebra, every element is rigid or tame, so we would like to find some tame element (if exists). In our study, we obtained a combinatorial solution to this problem, in terms of paths in the quiver \(Q\) admitted in \(A\) and stability conditions introduced by King.