Geometric and combinatorial realizations of crystal graphs

Details Geometric and combinatorial realizations of crystal graphs Geometric and combinatorial realizations of crystal graphs

2003-10-20

arxiv.org/abs/math/0310314v4

Algebr. Represent. Theory 9 (2006), no. 2, 161-199

Mathematics

Representation Theory

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

Scientific paper

10.1007/s10468-005-0565-7

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.

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