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Universal Verma modules and the Misra-Miwa Fock spaceAlgebra & Discrete Mathematics
|Speaker: ||Peter Tingley, MIT|
|Location: ||1147 MSB|
|Start time: ||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.