Abstract
We examine [Formula: see text]-cluster theory from an elementary point of view using a generalization of [Formula: see text]-ary trees which we call [Formula: see text]-noncrossing trees. We show that these trees are in bijection with [Formula: see text]-clusters in the [Formula: see text]-cluster category of a quiver of type [Formula: see text]. Similar trees are in bijection with complete exceptional sequences. Most of this paper is expository, explaining definitions and known results about these topics in representation theory. One application of these trees is that the mutation formula for [Formula: see text]-clusters is derived from the more elementary mutation of trees. The main new result is that the natural map of an [Formula: see text]-noncrossing tree into the plane is an embedding. We also explain the relationship between [Formula: see text]-noncrossing trees and finite Harder–Narasimhan systems in the derived category of the module category of type [Formula: see text].
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