Crystal graphs were introduced by Kashiwara in connection with his theory of crystal bases [1]; see also [2] for a textbook providing an introduction to the subject and references therein. Broadly speaking, a "crystal" consists of a set

(the vertices of the crystal graph) and a number of maps, the so-called Kashiwara operators

, where

ranges over some finite index set. You draw a directed edge

of color

between two vertices

provided that

.

The tensor product of two crystals is computed using the rule

, where

is again the Kashiwara operator belonging to an edge of color

and

measure the maximal distances from a fixed vertex along edges of color

in the crystal graph and its reverse graph, respectively.

The main point of this Demonstration is to show that

*Mathematica* can be used for the decomposition of tensor products of Kirillov–Reshetikhin (KR) modules of the affine quantum algebra

into modules of the finite quantum algebra

. KR modules were first introduced in the context of Yangians [3] but have been generalized to

-deformed enveloping algebras. They are distinguished among the finite-dimensional representations of the quantum affine algebra since they are known to possess crystal bases and, hence, crystal graphs. Moreover, KR crystals are "perfect crystals"; they themselves as well as their tensor products are connected.

The displayed crystal graphs are the tensor products

, where

denotes the KR crystal graph associated with a single column of height

. The partition

can be varied within a fixed range; the parts of

have to be strictly smaller than

. Note that the graphs grow rapidly in size and can quickly consume the available memory and CPU time of your computer, so it is best to start with small examples!

Deleting all edges of color

gives a disconnected graph where each connected component corresponds to a crystal graph of

. This tests the famous Kirillov–Reshetikhin conjecture [4][5] on simple examples.

[1] M. Kashiwara, "On Crystal Bases, Representations of Groups," in

*CMS Conference Proceedings 16*, Providence, RI: American Mathematical Society, 1995 pp. 155–197.

[2] J. Hong and S-J. Kang,

*Introduction to Quantum Groups and Crystal Bases*, Providence, RI: American Mathematical Society, 2002.

[3] A. N. Kirillov and N. Yu. Reshetikhin, "Representations of Yangians and Multiplicities of the Inclusions of the Irreducible Components of the Tensor Product of Representations of Simple Lie Algebras,"

*Journal of Soviet Mathematics*,

**52**(3), 1990 pp. 3156–3164.

[4] H. Nakajima, "t-Analogs of q-Characters of Kirillov–Reshetikhin Modules of Quantum Affine Algebras,"

* Representation Theory*,

**7**, 2003 pp. 259–274.

[5] D. Hernandez, "Kirillov–Reshetikhin Conjecture: The General Case,"

*International Mathematics Research Notices*,

**2010**(1), 20101 pp. 149–193.

[6] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Z. Tsuboi, "Paths, Crystals and Fermionic Formulae,"

*MathPhys Odyssey* *2001* (M. Kashiwara and T. Miwa, eds.), Boston, MA: Birkhäuser, 2002 pp. 205–272.

[7] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, "Remarks on Fermionic Formula,"

*Recent Developments in Quantum Affine Algebras and Related Topics* (N. Jing and K. C. Misra, eds.), Providence, RI: American Mathematical Society, 1999 pp. 243–291.

[8] A. Nakayashiki and Y. Yamada, "Kostka Polynomials and Energy Functions in Solvable Lattice Models,"

*Selecta Mathematics, New Series*,

**3**(4), 1997 pp. 547–599.