turkmath.org

Let P be a cominuscule poset which can be thought of as a parabolic analogue of the poset of positive roots of a finite root system. Let J(P) be the poset of order ideals of P. In this talk, we will investigate the periodicity of the Coxeter transformation on the poset J(P), and show that the Coxeter transformation has finite order for two of the three infinite families of cominuscule posets, and the exceptional cases. Our motivation comes from a conjecture by Chapoton which states that the Coxeter transformation has finite order on the poset J(R) when R is the poset of positive roots of a finite root system. Our solution is formulated in representation theory of finite dimensional algebras, and we will further discuss the results within the same context.