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Quantum groups were introduced independently by Drinfeld and Jimbo in their study of two-dimensional lattice models. Since then quantum groups have turned out to be the fundamental algebraic structure behind many branches of mathematics, such as topological invariant theory of links and knots, representation theory of Kac-Moody algebras and topological quantum field theory. Crystal bases, developed independently by Kashiwara and Lusztig, provide a powerful combinatorial tool to study the representations of quantum groups.

There are two main categories of affine crystals:
(1) crystals of infinite dimensional integrable highest weight modules;
(2) crystals of finite dimensional modules.
Whereas the first category is well understood, the second one is still far from well understood. Recently, in collaboration with Okado and Shimozono, we have introduced virtual crystals which are based on certain embedding of affine algebras which conjecturally describe the combinatorics of the affine finite dimensional crystals. As a tidbit we also show that the same ideas can be used to find an efficient algorithm to compute fermionic formulas. For the special case of simply-laced algebras this is Kleber's algorithm.