# Multi-Dimensional Weyl Modules and Symmetric Functions

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DOI: 10.1007/s00220-004-1166-8

- Cite this article as:
- Feigin, B. & Loktev, S. Commun. Math. Phys. (2004) 251: 427. doi:10.1007/s00220-004-1166-8

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## 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).