Fusion Products, Cohomology of Gln Flag Manifolds and Kostka Polynomials

This paper explains the relation between the fusion product of symmetric power sln evaluation modules, as defined by Feigin and Loktev, and the graded coordinate ring Rμ which describes the cohomology ring of the flag variety Flμ′ of GLN . The graded multiplicity spaces appearing in the decomposition of the fusion product into irreducible sln-modules are identified with the multiplicity spaces of the Specht modules in Rμ. This proves that the Kostka polynomial gives the character of the fusion product in this case. In the case of the product of fundamental evaluation modules, we give the precise correspondence with the reduced wedge product, and thus the usual wedge space construction of irreducible level1 ŝln-modules in the limit N → ∞. The multiplicity spaces are W (sln)-algebra modules in this limit.