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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
Software:
SageMath
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References:
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