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Communications in Mathematical Physics

, Volume 251, Issue 3, pp 427–445 | Cite as

Multi-Dimensional Weyl Modules and Symmetric Functions

  • B. Feigin
  • S. Loktev
Article

Abstract

The Weyl modules in the sense of V. Chari and A. Pressley ([CP]) over the current Lie algebra on an affine variety are studied. We show that local Weyl modules are finite-dimensional and generalize the tensor product decomposition theorem from [CP]. More explicit results are stated for currents on a non-singular affine variety of dimension d with coefficients in the Lie algebra sl r . The Weyl modules with highest weights proportional to the vector representation one are related to the multi-dimensional analogs of harmonic functions. The dimensions of such local Weyl modules are calculated in the following cases. For d=1 we show that the dimensions are equal to powers of r. For d=2 we show that the dimensions are given by products of the higher Catalan numbers (the usual Catalan numbers for r=2).

Keywords

Neural Network Nonlinear Dynamics Tensor Product Harmonic Function High Weight 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Landau institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Institute for Theoretical and Experimental PhysicsMoscowRussia
  3. 3.Independent University of MoscowMoscowRussia

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