Multi-dimensional Weyl Modules and Symmetric Functions
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 with coefficients in the Lie algebra . 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 we show that the dimensions are equal to powers of . For we show that the dimensions are given by products of the higher Catalan numbers (the usual Catalan numbers for ). We finally formulate a conjecture for an arbitrary and .
Cite
@article{arxiv.math/0212001,
title = {Multi-dimensional Weyl Modules and Symmetric Functions},
author = {B. Feigin and S. Loktev},
journal= {arXiv preprint arXiv:math/0212001},
year = {2015}
}
Comments
LaTeX, 13 pages; more detail added. To appear at Comm. Math. Phys