International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 326247, 19 pages
doi:10.1155/2010/326247
Research Article

Universal Verma Modules and the Misra-Miwa Fock Space

Arun Ram1 and Peter Tingley2

1Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia
2Department of Mathematics, MIT, 77 Massachusetts Ave, Cambridge, MA 02139, USA

Received 7 March 2010; Accepted 5 November 2010

Academic Editor: Alistair Savage

Copyright © 2010 Arun Ram and Peter Tingley. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Misra-Miwa 𝑣 -deformed Fock space is a representation of the quantized affine algebra 𝑈 𝑣 ( 𝔰 𝔩 ℓ ) . It has a standard basis indexed by partitions, and the nonzero 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 𝑈 𝑞 ( 𝔤 𝔩 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.