Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 060, 46 pages      arXiv:1409.3610      https://doi.org/10.3842/SIGMA.2015.060
Contribution to the Special Issue on New Directions in Lie Theory

$T$-Path Formula and Atomic Bases for Cluster Algebras of Type $D$

Emily Gunawan and Gregg Musiker
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Received September 24, 2014, in final form July 09, 2015; Published online July 28, 2015

Abstract
We extend a $T$-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type $D$.

Key words: cluster algebra; triangulated surface; atomic basis.

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