Cluster algebra algorithm for computing q-characters
The q-character [FR99] is an injective homomorphism from Grothendieck ring of finite-dimensional modules over a quantum loop algebra into a Laurent polynomial ring in countably many variables, which is compatible with the ordinary character over the underlying simple Lie algebra. Recently, Hernandez and Leclerc [HL10, HL16] provided an algorithm for computing the q-characters of Kirillov-Reshetikhin (KR for short) modules by using cluster algebra. In summary, a certain sequence of quiver mutations makes a Laurent polynomial attached to a cluster variable (in the sense of a change of variables), which goes to the "truncated" q-character of a KR 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])
Type A3
Type D4