Cluster algebras are closely related to totally positivity in algebraic groups and canonical bases in quantum groups.

Is there some relation between cluster algebras and crystal graphs? Can the crystal graph on page 19 of the book be described using cluster variables (or cluster monomials)?

Thank you very much.


1 Answer 1


Yes, there are many relations between cluster algebras and crystal graphs. I am by no means an expert on these things, but let me mention one connection. Cluster algebras were originally discovered in the study of totally positive matrices and total positivity. This theory is ultimately interested in Lusztig's canonical basis $\mathcal{B}$ of the quantum group $U_q(\mathfrak{u})$, so crystal graphs are already not too far away.

In fact, Berenstein-Zelevinsky relate Lusztig's parametrization of the canonical basis with Kashiwara's string parametrization of the dual canonical basis by studying totally positive varieties. The key technique is to use generalized minors $\Delta_{\lambda,\gamma}$, which turn out to be cluster variables for the cluster algebra structure on the coordinate algebra of the unipotent radical $U$. This cluster algebra structure gives rise to many rational, subtraction-free expressions among functions of representation theoretic interest, enabling tropicalization. In fact, Berenstein-Kahzdan have developed the theory of geometric crystals, which are highly-structured varieties that may be tropicalized to given normal Kashiwara crystal. Generalized minors play an important role here as well.

More generally, double Bruhat cells $G^{u,v}=BuB\cap B_-vB_-$ are also cluster varieties: their coordinate rings $\mathbb{C}[G^{u,v}]$ each have the structure of a cluster algebra with cluster variables given by generalized minors. These also have connections to crystal graphs, but I know next to nothing about this more general set up. I think these examples served as a primary motivation development of cluster algebras by Fomin-Zelevinsky.

The crystal graph from the book you link to is a highest weight crystal graph of type $A_2$, and its basis vectors are being parametrized by string data. String data may be related to generalized minors (hence cluster variables) via the results of Berenstein-Zelevinsky and tropicalization.


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