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

Voir la notice de l'article provenant de la source Cambridge University Press

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
@article{10_1017_S0017089515000233,
     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/}
}
TY  - JOUR
AU  - PARSONS, MARK JAMES
TI  - EXPLICIT CONSTRUCTION OF COMPANION BASES
JO  - Glasgow mathematical journal
PY  - 2016
SP  - 357
EP  - 384
VL  - 58
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089515000233/
DO  - 10.1017/S0017089515000233
ID  - 10_1017_S0017089515000233
ER  - 
%0 Journal Article
%A PARSONS, MARK JAMES
%T EXPLICIT CONSTRUCTION OF COMPANION BASES
%J Glasgow mathematical journal
%D 2016
%P 357-384
%V 58
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089515000233/
%R 10.1017/S0017089515000233
%F 10_1017_S0017089515000233

[1] 1.Barot, M., Geiss, C. and Zelevinsky, A., Cluster algebras of finite type and positive symmetrizable matrices, J. London Math. Soc. 73 (3) (2006), 545–564. Google Scholar | DOI

[2] 2.Barot, M. and Marsh, R., Reflection group presentations arising from cluster algebras, Trans. Amer. Math. Soc. 367 (2015), 1945–1967. Google Scholar

[3] 3.Bastian, J., Holm, T. and Ladkani, S., Derived equivalences for cluster-tilted algebras of Dynkin type D, J. Algebra. 410 (2014), 277–332. Google Scholar | DOI

[4] 4.Buan, A., Marsh, R. and Reiten, I., Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), 323–332. Google Scholar | DOI

[5] 5.Buan, A., Marsh, R. and Reiten, I., Cluster mutation via quiver representations, Comment. Math. Helv. 83 (2) (2008), 143–177. Google Scholar

[6] 6.Buan, A., Marsh, R. and Reiten, I., Cluster-tilted algebras of finite representation type, J. Algebr. 306 (2) (2006), 412–431. Google Scholar

[7] 7.Buan, A. and Vatne, D., Derived equivalence classification for cluster-tilted algebras of type A, J. Algebr. 319 (7) (2008), 2723–2738. Google Scholar

[8] 8.Butler, M. C. R. and Ringel, C. M., Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebr. 15 (1987), 145–179. Google Scholar

[9] 9.Caldero, P., Chapoton, F. and Schiffler, R., Quivers with relations arising from clusters (A case), Trans. Amer. Math. Soc. 358 (3) (2006), 1347–1364. Google Scholar

[10] 10.Caldero, P., Chapoton, F. and Schiffler, R., Quivers with relations and cluster-tilted algebras, Algebr. Represent. Theory 9 (4) (2006), 359–376. Google Scholar | DOI

[11] 11.Derksen, H., Weyman, J. and Zelevinsky, A., Quivers with potentials and their representations I: Mutations, Sel. Math. (N.S.) 14 (1) (2008), 59–119. Google Scholar

[12] 12.Fomin, S. and Zelevinsky, A., Cluster algebras II: Finite type classification, Invent. Math. 154 (1) (2003), 63–121. Google Scholar

[13] 13.Nájera, A.Chávez, C-vectors and dimension vectors for cluster-finite quivers, Bull. London Math. Soc. 45 (3) (2013), 1259–1266. Google Scholar

[14] 14.Nakanishi, T. and Stella, S., Diagrammatic description of c-vectors and d-vectors of cluster algebras of finite type, Electron. J. Combin. 21 (1) (2014), #P1.3. Google Scholar

[15] 15.Parsons, M. J., On indecomposable modules over cluster-tilted algebras of type A, PhD Thesis (University of Leicester, 2007). Google Scholar

[16] 16.Parsons, M. J., Companion bases for cluster-tilted algebras, Algebr. Represent. Theory 17 (3) (2014), 775–808. doi:10.1007/s10468-013-9418-y. Google Scholar

[17] 17.Ringel, C. M., Cluster-concealed algebras, Adv. Math. 226 (2) (2011), 1513–1537. Google Scholar

[18] 18.Vatne, D. F., The mutation class of D quivers, Comm. Algebr. 38 (3) (2010), 1137–1146. Google Scholar

Cité par Sources :