Universal Verma modules and the Misra-Miwa Fock spaceAlgebra & Discrete Mathematics
|Peter Tingley, MIT
|Fri, Jan 8 2010, 1:10PM
The Misra-Miwa v-deformed Fock space is a representation of the quantized affine algebra of type A. It has a standard basis indexed by partitions, and the non-zero matrix entries of the action of the Chevalley generators with respect to this basis are powers of v. Partitions also index the polynomial Weyl modules for the integral quantum group associated to gl(N), as N tends to infinity. We explain how the powers of v which appear in the Misra-Miwa Fock space also appear naturally in the context of Weyl modules. The main tool we use is the Shapovalov determinant for a universal Verma module.
We will begin with a combinatorial construction of Fock space, and motivate our work with a simple relationship between undeformed Fock space and the category of representations of gl(N). The trickery comes in understanding the v-deformation. We will give a precise statement of how this is done, and explain the appearance of `universal' Verma modules, without presenting all the details of the proof. This is joint work with Arun Ram.