Mathematics – Representation Theory
Scientific paper
2004-11-04
J. Algebra 305 (2006), 664-686
Mathematics
Representation Theory
20 pages, 1 figure; v2: Minor changes to match published version
Scientific paper
10.1016/j.jalgebra.2006.07.026
Ginzburg and Nakajima have given two different geometric constructions of quotients of the universal enveloping algebra of sl_n and its irreducible finite-dimensional highest weight representations using the convolution product in the Borel-Moore homology of flag varieties and quiver varieties respectively. The purpose of this paper is to explain the precise relationship between the two constructions. In particular, we show that while the two yield different quotients of the universal enveloping algebra, they produce the same representations and the natural bases which arise in both constructions are the same. We also examine how this relationship can be used to translate the crystal structure on irreducible components of quiver varieties, defined by Kashiwara and Saito, to a crystal structure on the varieties appearing in Ginzburg's construction, thus recovering results of Malkin.
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