Reduced representations of rooted trees

Let k be a field and Q a quiver. The category of finite dimensional representations of Q will be denoted by repk(Q). We can define the pointwise tensor product on the category repk(Q), i.e. for two representations N, M the representation N⊗M is defined at a vertex x∈Q0 as (N⊗M)x=Nx⊗Mx and for an arrow α∈Q1 we define (N⊗M)α=Nα⊗Mα. This tensor product has been investigated in recent work of Herschend and Kinser [1-3]. In [3] Kinser investigated the representation ring of rooted trees. A rooted tree (Q,σ) is a quiver Q, whose underlying graph is tree and which has a unique sink σ. The representation ring of Q is the free abelian group generated by the isomorphism classes of indecomposable objects in repk(Q). It is endowed with a multiplication induced by the pointwise tensor product. See [3] for details. For these rooted trees, Kinser constructed reduced representations, which are defined inductively. These reduced representations then lead to a set of orthogonal primitive idempotents in the representation ring of Q. On the other hand, there is an inductive construction of so-called radiation modules for a tree with arbitrary orientation in [5]. In the special case of a rooted tree quiver, this construction results in the reduced representations introduced by Kinser. The idea used in the construction of the radiation modules then allows us to prove a new characterization of reduced representations which isTheoremLet Q be a rooted tree quiver. An indecomposable representation V∈repk(Q)is reduced if and only if V is a direct summand of V⊗V.This theorem then allows us to prove further results for the special case, where Q is a rooted tree quiver, whose underlying graph is of Dynkin type. In that case V being reduced is also equivalent to the condition that the vector space of V at the sink σ is one dimensional.