Mutating seeds: types 𝔸 and 𝔸 ˜ .

Ibrahim Assem[1]; Christophe Reutenauer[2]

  • [1] Département de mathématiques, Université de Sherbrooke Sherbrooke (Québec) J1K2R1,Canada
  • [2] Laboratoire de combinatoire et d’informatique mathématique, Université du Québec à Montréal Case postale 8888, succursale Centre-ville Montréal (Québec) H3C 3P8, Canada

Annales mathématiques Blaise Pascal (2012)

  • Volume: 19, Issue: 1, page 29-73
  • ISSN: 1259-1734

Abstract

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In the cases 𝔸 and 𝔸 ˜ , we describe the seeds obtained by sequences of mutations from an initial seed. In the 𝔸 ˜ case, we deduce a linear representation of the group of mutations which contains as matrix entries all cluster variables obtained after an arbitrary sequence of mutations (this sequence is an element of the group). Nontransjective variables correspond to certain subgroups of finite index. A noncommutative rational series is constructed, which contains all this information.

How to cite

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Assem, Ibrahim, and Reutenauer, Christophe. "Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.." Annales mathématiques Blaise Pascal 19.1 (2012): 29-73. <http://eudml.org/doc/251110>.

@article{Assem2012,
abstract = {In the cases $\mathbb\{A\}$ and $\widetilde\{\mathbb\{A\}\}$, we describe the seeds obtained by sequences of mutations from an initial seed. In the $\widetilde\{\mathbb\{A\}\}$ case, we deduce a linear representation of the group of mutations which contains as matrix entries all cluster variables obtained after an arbitrary sequence of mutations (this sequence is an element of the group). Nontransjective variables correspond to certain subgroups of finite index. A noncommutative rational series is constructed, which contains all this information.},
affiliation = {Département de mathématiques, Université de Sherbrooke Sherbrooke (Québec) J1K2R1,Canada; Laboratoire de combinatoire et d’informatique mathématique, Université du Québec à Montréal Case postale 8888, succursale Centre-ville Montréal (Québec) H3C 3P8, Canada},
author = {Assem, Ibrahim, Reutenauer, Christophe},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Cluster algebras; mutations; seeds; quivers; cluster algebras},
language = {eng},
month = {1},
number = {1},
pages = {29-73},
publisher = {Annales mathématiques Blaise Pascal},
title = {Mutating seeds: types $\mathbb\{A\}$ and $\widetilde\{\mathbb\{A\}\}$.},
url = {http://eudml.org/doc/251110},
volume = {19},
year = {2012},
}

TY - JOUR
AU - Assem, Ibrahim
AU - Reutenauer, Christophe
TI - Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.
JO - Annales mathématiques Blaise Pascal
DA - 2012/1//
PB - Annales mathématiques Blaise Pascal
VL - 19
IS - 1
SP - 29
EP - 73
AB - In the cases $\mathbb{A}$ and $\widetilde{\mathbb{A}}$, we describe the seeds obtained by sequences of mutations from an initial seed. In the $\widetilde{\mathbb{A}}$ case, we deduce a linear representation of the group of mutations which contains as matrix entries all cluster variables obtained after an arbitrary sequence of mutations (this sequence is an element of the group). Nontransjective variables correspond to certain subgroups of finite index. A noncommutative rational series is constructed, which contains all this information.
LA - eng
KW - Cluster algebras; mutations; seeds; quivers; cluster algebras
UR - http://eudml.org/doc/251110
ER -

References

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