• The q-character [FR99] is an injective homomorphism from Grothendieck ring K(C) of the category C of finite-dimensional modules over a quantum loop algebra into a Laurent polynomial ring in many variables, which is compatible with the ordinary character over the underlying simple Lie algebra. Hernandez and Leclerc [HL10, HL16] provide an algorithm for computing the q-characters of Kirillov-Reshetikhin modules by using cluster algebra structure of C. In summary, a certain sequence of quiver mutations makes a Laurent polynomial attached to a cluster variable (in the sense of changing variables), which goes to the (truncated) q-character of a Kirillov-Reshetikhin module.
  
  

• The code in [Link] computes the q-characters of KR modules in all untwisted types following Hernandez-Leclerc's cluster algebra algorithm, where the code uses the cluster algebra package in SageMath developed by Musiker and Stump [MS11] and the optional package [slabbe] to visualize quivers.
  
  

• REFERENCE:
  [FR99] E. Frenkel and N. Yu. Reshetikhin, The q-characters of representations of quantum affine algebras and deformations of W-algebras, Recent developments in quantum affine algebras and related topics, Contemp. Math. 248 (1999), 163--205.
  [FM01] E. Frenkel and E. Mukhin,  Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras,  Comm. Math. Phys. 216 (2001), no. 1, 23--57.
  [HL10] D. Hernandez and B. Leclerc,  Cluster algebras and quantum affine algebras,  Duke Math. J. 154 (2010),  no. 2, 265--341
  [HL16] D. Hernandez and B. Leclerc,  A cluster algebra approach to q-characters of Kirillov-Reshetikhin modules,   J. Eur. Math. Soc. (JEMS) 18 (2016), no. 5, 1113--1159.
  [MS11] G. Musiker, C. Stump,  A Compendium on the cluster algebra and quiver package in Sage, preprint (2011),  arXiv:1102.4844 (Sem. Lothar. Combin. 67, 67 pages, 2011). 

EXAMPLES (see [Jupyter Notebook])