Let L denote a semisimple Lie algebra over the complex numbers with a fixed Cartan subalgebra H. We may assume that L is generated (as a Lie algebra) by a finite number of strictly upper triangular matrices and their transposes. It is convenient, for various reasons, to denote such a set of generators by A = {Xa | a in D}, where D is a finite set, and the set of transpose matrices by {X-a | a in D}. Here D is a set of simple positive roots of L and the Xa are the corresponding root vectors.
Let (s,V) denote an irreducible finite dimensional representation of L, s : L --> End(V). The restriction of (s,V) to the Cartan subalgebra H decomposes into a direct sum of 1-dimensional representations of H. These are the weight spaces of (s,V). We shall need to assume that this decomposition is multiplicity free. This representation is uniquely determined by the action of the simple root vectors on the lowest weight vector of V. For example, the first several highest weight vectors of the Lie algebra A2 and the dimensions of the corresponding irreducible representations looks like: weight system of A2
We associate to (s,V) a digraph G as follows: the vertices of G are indexed by the weight spaces of (s,V) and we draw an edge from vertex W to vertex W' labeled with Xa (a in D) if Xa(W)=W'. This is the crystal graph of (s,V).
Example: Let L=B5 and let (s,V) denote the standard 11-dimensional fundamental representation associated to the weight e5 (in the notation of Stembridge's coxeter package). The crystal graph looks like: crystal graph of the standard 11-dimensional of B5. The lowest weight vector is the left-most vertex and the highest weight vector is the right-most vertex. (The edges are left-to-right, though the directions are omitted in the graph pictured.)
Example: Let L=C5 and let (s,V) denote the standard 10-dimensional fundamental representation associated to the weight e5 (in the notation of the coxeter package). The crystal graph looks like: crystal graph of the standard 10-dimensional of C5
The tensor product of two irreducible representations s1, s2 of L is another representation s1xs2 of L, typically not irreducible. Thanks to a theorem of Kashiwara, the crystal graph of the tensor product s1xs2 can be obtained from those of s1, s2 (resp.) by means of a simple algorithm.
Example: Let L=A_2. Let (s1,V1) denote the standard 2-dimensional representation having highest weight (-2/3)*e1+(1/3)*e2+(1/3)*e3 (in the notation of coxeter). The crystal graph looks like: crystal graph of the standard 2-dimensional of A2. It may also be plotting in weight space: crystal graph of the standard 2-dimensional of A2 in weight space. Let (s2,V2) denote the contragredient of s1 with highest weight (-1/3)*e1+(-1/3)*e2+(-2/3)*e3. The crystal graph looks like: crystal graph of the standard 2-dimensional of A2 The crystal graph of the tensor product s1xs2 has lowest weight vector in the lower left-hand corner and the highest weight vector in the upper right-hand corner. (The edges are pointing up and right, though the directions are omitted in the graph pictured.) There are two connected components in this graph corresponding to the two irreducible components of the tensor product. The connected component of the tensor product which contains the highest weight vector is called the Cartan product of s1 and s2, denoted s1*s2. The linearization of s1*s2 is 8-dimensional. This 8-dimensional representation of A2 corresponds to an 8-dimensional representation of SU(3) which is sometimes referred to as the the "eight-fold way" in particle physics. The tensor product of this representation with itself has a crystal graph with 64 vertices.
The crystal graph of the tensor product s1xs1 decomposes into a six dimensional irreducible representation of A2 and s2. The restriction of the 6 dimensional representation to the subalgebra of A2 of type A1 decomposes into three irreducibles.
The pictures above were produced using the crystal and dynkin MAPLE packages.
Last updated 4-5-97