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Universal Verma modules and the Misra-Miwa Fock space. (English) Zbl 1207.81042

Summary: The Misra-Miwa \(v\)-deformed Fock space is a representation of the quantized affine algebra \(U_v(\widehat{\mathfrak {sl}}_\ell\)). 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 \(U_q(\mathfrak{gl}_N)\) 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.

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81V70 Many-body theory; quantum Hall effect
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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