Multi-Dimensional Weyl Modules and Symmetric Functions

By Boris Feigin, S. Loktev. 2002

97 citations recorded by the source corpus.

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 slr. 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).

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d3122b39a1334a80335098f457239934a234af29

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In Discrete Mathematics and Combinatorics

1. Tanusree Khandai
2. Ghislain Fourier
3. Vyjayanthi Chari

In Geometry and Topology

1. Ghislain Fourier
2. Vyjayanthi Chari
3. Tanusree Khandai
4. Michael Lau
5. D. Hernandez
6. R. Venkatesh
7. P. Littelmann
8. Alexander S. Kleshchev

In Mathematical Physics

1. Vyjayanthi Chari
2. Ghislain Fourier
3. Tanusree Khandai
4. Michael Lau
5. R. Venkatesh
6. D. Hernandez

Citations in the corpus

• Irreducible finite-dimensional representations of equivariant map algebras (2009, unknown): “We denote by M(X, g) the Lie algebra of regular functions on X with values in g (equivalently, the set of morphisms of schemes X → g), called the untwisted map algebra or the Lie algebra of currents ([FL04]).”
• A categorical approach to Weyl modules (2009, methodology): “In [ FL ] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained.”