Geodesic
EXPLICIT CONSTRUCTION OF COMPANION BASES
Glasgow mathematical journal, Tome 58 (2016) no. 2, pp. 357-384

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DOI

A companion basis for a quiver Γ mutation equivalent to a simply-laced Dynkin quiver is a subset of the associated root system which is a $\mathbb{Z}$-basis for the integral root lattice with the property that the non-zero inner products of pairs of its elements correspond to the edges in the underlying graph of Γ. It is known in type A (and conjectured for all simply-laced Dynkin cases) that any companion basis can be used to compute the dimension vectors of the finitely generated indecomposable modules over the associated cluster-tilted algebra. Here, we present a procedure for explicitly constructing a companion basis for any quiver of mutation type A or D. 
PARSONS, MARK JAMES. EXPLICIT CONSTRUCTION OF COMPANION BASES. Glasgow mathematical journal, Tome 58 (2016) no. 2, pp. 357-384. doi: 10.1017/S0017089515000233
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     author = {PARSONS, MARK JAMES},
     title = {EXPLICIT {CONSTRUCTION} {OF} {COMPANION} {BASES}},
     journal = {Glasgow mathematical journal},
     pages = {357--384},
     year = {2016},
     volume = {58},
     number = {2},
     doi = {10.1017/S0017089515000233},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089515000233/}
}
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