# The Decomposition Algorithm for Skew-Symmetrizable Exchange Matrices

@article{Gu2012TheDA, title={The Decomposition Algorithm for Skew-Symmetrizable Exchange Matrices}, author={Weiwen Gu}, journal={Electron. J. Comb.}, year={2012}, volume={19}, pages={54} }

Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admits unfoldings to skew-symmetric matrices. We develop an combinatorial algorithm that determines if a given skew-symmetrizable matrix is of such type. This algorithm generalizes the one in \cite{WG}. As a corollary, we use this algorithm to determine if a given skew-symmetrizable matrix has finite mutation type.

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## References

SHOWING 1-9 OF 9 REFERENCES

Cluster algebras of finite mutation type via unfoldings

- Mathematics
- 2010

We complete classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every mutation-finite…

Skew-symmetric cluster algebras of finite mutation type

- Mathematics
- 2008

In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster…

A Decomposition Algorithm for the Oriented Adjacency Graph of the Triangulations of a Bordered Surface with Marked Points

- MathematicsElectron. J. Comb.
- 2011

An algorithm that determines whether a given oriented graph is an oriented adjacency graph of a triangulation of bordered surfaces with marked points is developed and it is found that there are only finitely many oriented connected graphs with non-unique associated triangulations.

Cluster algebras and triangulated surfaces. Part I: Cluster complexes

- Mathematics
- 2006

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra…

Decomposition Algorithm for Median Graph of Triangulation of a Bordered 2D Surface

- Computer Science, MathematicsArXiv
- 2010

An algorithm is developed that identifies and decomposes a median graph of a triangulation of a 2-dimensional (2D) oriented bordered surface and in addition restores all corresponding triangulations whenever they exist and is more effective than previous known to determine quivers of finite mutation type of size greater than 10.

Cluster algebras II: Finite type classification

- Mathematics
- 2002

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many…

Introduction to Quantum Groups

- Mathematics
- 1998

We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra…

Cluster algebras I: Foundations

- Mathematics
- 2001

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

the electronic journal of combinatorics

- the electronic journal of combinatorics
- 2012