On the Canonical Decomposition of Quiver Representations

By Harm Derksen, Jerzy Weyman. Compositio Mathematica, 2002

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Cluster complexes via semi-invariants (2007, methodology): “For the vectors with nonnegative coordinates we recover the simplicial complex corresponding to generic decompositions of dimension vectors obtained in [ 5 ].”
A categorification of non-crossing partitions (2013, background): “It therefore follows as in [30, Theorem 16] that, in both cases, the Derksen–Weyman algorithm applied to the lar (pi) produces n − 1 real Schur roots, say p̄1, . . . , p̄n−1, other than ar , and that these are the projective roots for a⊥r .”
Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, background): “(X,X) = dimk EndH(X) > 0. We call d ∈ ∆re + a real Schur root if the corresponding X ∈ ind(modH) satisﬁes EndH(X) = k. We denote by ∆reS + the set of real Schur roots. For Schur roots, we refer to [60, 28] for example. Immediately, dim gives a bijection from the set of indecomposable rigid H-modu”
The combinatorics of quiver representations (2006, methodology): “Lemma 2.15 follows immediately from the correctness of the algorithm in [ 14 ], because the output of the algorithm has the desired property.”
General Presentations of Algebras (2009, methodology): “An efficient algorithm for finding the canonical decomposition of a dimension vector was given in [6].”