2022/12/22 by Fu, Changjian, Geng, Shengfei · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2212.11497
It is proved that a multiset of permissible arcs over a tiling is uniquely determined by its intersection vector under a mild condition. This generalizes a classical result over marked surfaces with triangulations. We apply this result to study τ-tilting theory of gentle algebras and denominator conjecture in cluster algebras. In the case of gentle algebras, it is proved that different τ-rigid A-modules over a gentle algebra A have different dimension vectors if and only if A has no even oriented cycle with full relations. For cluster algebras, the denominator conjecture has been established for cluster algebras of type \mathbbA\mathbbBℂ.