Tensor product of crystals

The tensor product crystal uses the $i$-string data ${\epsilon }_i$ and ${\phi }_i$ to decide on which tensor factor a Kashiwara operator acts. Its connected components are highest-weight crystals and encode the decomposition of the tensor product representation.

This tensor-product operation combines one crystal basis with another, with factor selection governed by their string parameters.

For the defining representation of the special linear Lie algebra ${\mathfrak{sl}}_3$, the crystal basis is $1\stackrel{1}{\to }2\stackrel{2}{\to }3$, with weights ${\epsilon }_i$ restricted to trace-zero diagonal matrices. Its dual is $\bar{3}\stackrel{2}{\to }\bar{2}\stackrel{1}{\to }\bar{1}$, with weights $-{\epsilon }_i$. Using the crystal tensor-product rule with lowering on the first factor when ${\phi }_i(a)>{\epsilon }_i(b)$, the tensor square has highest vertices $1\otimes 1$ and $1\otimes 2$, of highest weights $2{\omega }_1,{\omega }_2$, generating components of six and three vertices. The defining tensor dual has highest vertices $1\otimes \bar{3}$ and $1\otimes \bar{1}$, generating components of eight and one vertices and weights ${\omega }_1+{\omega }_2,0$. These encode the symmetric square plus exterior square, and the trace-free endomorphisms plus scalar identity, respectively. A crystal vertex labelled by a tensor pair need not represent that same undeformed ordinary simple tensor; the actual invariant vector is ${\sum }_i{e}_i\otimes e_i^{\ast }$.

Write ${\epsilon }_i(b)$ and ${\phi }_i(b)$ for the numbers of raising and lowering steps of color $i$ available from a vertex of a crystal basis. One convention for the tensor product of crystals isThe raising operator uses the first factor for ${\phi }_i(a)\ge {\epsilon }_i(b)$ and the second otherwise. The different strictness makes raising and lowering inverse along every edge. An edge is absent if the chosen Kashiwara operator vanishes. Stating the convention is essential when identifying the highest-weight vectors as ordered tensor vertices.