Geometric and combinatorial realizations of crystal graphs

Abstract

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For affine type A, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids.

34 pages, 17 figures; v2: minor typos corrected; v3: corrections to section 8; v4: minor typos corrected

References in corpus (2)

• Bases of representations of type A affine Lie algebras via quiver varieties and statistical mechanics
• A Geometric Realization of Spin Representations and Young diagrams from Quiver Varieties

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• Mirković-Vilonen Polytopes and a quiver construction of crystal basis in type $A$
• Young tableaux, multi-segments, and PBW bases
• Heaps, crystals, and preprojective algebra modules
• Stabilization phenomena in Kac-Moody algebras and quiver varieties
• Quiver varieties and adjoint crystals of level $\ell$ for type $A_n^{(1)}$
• Crystal Structure on the Category of Modules over Colored Planar Rook Algebra
• Lectures on geometric realizations of crystals
• Affine crystals of type $D_n^{(1)}$ and Young walls