English

Multi-dimensional Weyl Modules and Symmetric Functions

Quantum Algebra 2015-06-26 v4 Representation Theory

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 dd with coefficients in the Lie algebra slrsl_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=1d=1 we show that the dimensions are equal to powers of rr. For d=2d=2 we show that the dimensions are given by products of the higher Catalan numbers (the usual Catalan numbers for r=2r=2). We finally formulate a conjecture for an arbitrary dd and r=2r=2.

Keywords

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