Representations of regular trees and invariants of AR-components for generalized Kronecker quivers

2017/02/14 by Bissinger, Daniel
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1702.04206

Abstract

We investigate the generalized Kronecker algebra K_r = kΓ_r with r ≥ 3 arrows. Given a regular component C of the Auslander-Reiten quiver of K_r, we show that the quasi-rank rk(C) ∈ ℤ_{≤ 1} can be described almost exactly as the distance W(C) ∈ ℕ₀ between two non-intersecting cones in C, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality -W(C) ≤ rk(C) ≤ - W(C) + 3. Utilizing covering theory, we construct for each n ∈ ℕ₀ a bijection φ_n between the field k and \ C | C regular component, W(C) = n \. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.

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• Shift orbits for elementary representations of Kronecker quivers
• AR-components for generalized Beilinson algebras
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