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The Coxeter transformation on cominuscule posets. (English) Zbl 07075207
Summary: Let $$J(C)$$ be the poset of order ideals of a cominuscule poset $$C$$ where $$C$$ comes from two of the three infinite families of cominuscule posets or the exceptional cases. We show that the Auslander-Reiten translation $$\tau$$ on the Grothendieck group of the bounded derived category for the incidence algebra of the poset $$J(C)$$, which is called the Coxeter transformation in this context, has finite order. Specifically, we show that $$\tau^{h+1}=\pm id$$ where $$h$$ is the Coxeter number for the relevant root system.

##### MSC:
 16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 05E10 Combinatorial aspects of representation theory 16G20 Representations of quivers and partially ordered sets 16G60 Representation type (finite, tame, wild, etc.) of associative algebras 18G10 Resolutions; derived functors (category-theoretic aspects) 06A07 Combinatorics of partially ordered sets
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