Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties

Details Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties

2011-03-22

arxiv.org/abs/1103.4207v1

Mathematics

Representation Theory

20 pages

Scientific paper

We prove that a Weyl module for the current Lie algebra associated with a simple Lie algebra of type $ADE$ is rigid, that is, it has a unique Loewy series. Further we use this result to prove that the grading on a Weyl module defined by the degree of currents coincides with another grading which comes from the degree of the homology group of the quiver variety. As a corollary we obtain a formula for the Poincar\'{e} polynomials of quiver varieties of type $ADE$ in terms of the energy functions defined on the crystals for tensor products of level-zero fundamental representations of the corresponding quantum affine algebras.

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