Accumulation points of real Schur roots

Abstract

Let k be an algebraically closed field and Q be an acyclic quiver with n vertices. Consider the category rep(Q) of finite dimensional representations of Q over k. The exceptional representations of Q, that is, the indecomposable objects of rep(Q) without self-extensions, correspond to the so-called real Schur roots of the usual root system attached to Q. These roots are special elements of the Grothendieck group Z n of rep(Q). When we identify the dimension vectors of the representations (that is, the non-negative vectors of Z n ) up to positive multiple, we see that the real Schur roots can accumulate in some directions of R n ⊃ Z n . This paper is devoted to the study of these accumulation points. After giving new properties of the canonical decomposition of dimension vectors, we show how to use this decomposition to describe the rational accumulation points. Finally, we study the irrational accumulation points and we give a complete description of them in case Q is of weakly hyperbolic type.

Isotropic Schur Roots

“…There are clearly only finitely many starting roots δ 0 , but the choices of the r i and X i yield, in general, infinitely many possible isotropic Schur roots. As observed in [15], when Q is wild connected with more than 3 vertices, there are infinitely many τ -orbit of isotropic Schur roots (provided there is at least one isotropic Schur root). An interesting question would be to describe the minimal root types of the orbits of E under B n−1 .…”

“…Our aim is to describe all simple objects in A(δ) or, equivalently, all σ δ -stable objects. We start with the following proposition; see [15].…”

“…Proof. As shown in [15], there is an exceptional sequence E = (V, W ) of length two such that δ is a root in C(E). Since E has length two, C(E) is equivalent to the category of representations of an acyclic quiver Q E with two vertices.…”

Infinitesimal semi‐invariant pictures and co‐amalgamation

“…When an algebra admits infinitely many $$\tau$$‐tilting pairs, the walls of the semi‐invariant picture can accumulate (see, e.g., [43]). In the case of a tame hereditary algebra or tame cluster‐tilted algebra, there is a unique “limiting wall” corresponding to the null root.…”

Infinitesimal semi-invariant pictures and co-amalgamation

“…When an algebra admits infinitely many τ -tilting pairs, the walls of the semi-invariant picture can accumulate (see e.g. [Paq16]). In the case of a tame hereditary algebra or tame cluster-tilted algebra, there is a unique "limiting wall" corresponding to the null root.…”