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

2011/03/22 by Ryosuke Kodera, Katsuyuki Naoi, Kodera, Ryosuke +1 · 2 citations
Mathematics · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.1103.4207

Abstract

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 Poincaré 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.

Citations

• Geometry of the Moment Map for Representations of Quivers

Cited by

• Trace and categorical sl(n) representations
• Macdonald Polynomials and BGG reciprocity for current algebras

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