Geometric and Combinatorial Realizations of Crystals of Enveloping Algebras

Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig’s quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group associated to a simply-laced Kac-Moody algebra. Using an enumeration of the irreducible components of Lusztig’s quiver varieties in finite and affine type A by combinatorial data, we compute the geometrically defined crystal structure in terms of this combinatorics. We conclude by comparing the combinatorial realization of the crystal graph thus obtained with other combinatorial models involving Young tableaux and Young walls.