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Symmetric polynomials and $U_q(\widehat{sl}_2)$

Author(s): Naihuan Jing
Journal: Represent. Theory 4 (2000), 46-63.
MSC (2000): Primary 17B; Secondary 5E
Posted: February 7, 2000
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Abstract:

We study the explicit formula of Lusztig's integral forms of the level one quantum affine algebra $U_q(\widehat{sl}_2)$ in the endomorphism ring of symmetric functions in infinitely many variables tensored with the group algebra of $\mathbb Z$. Schur functions are realized as certain orthonormal basis vectors in the vertex representation associated to the standard Heisenberg algebra. In this picture the Littlewood-Richardson rule is expressed by integral formulas, and is used to define the action of Lusztig's $\mathbb Z[q, q^{-1}]$-form of $U_q(\widehat{sl}_2)$ on Schur polynomials. As a result the $\mathbb Z[q, q^{-1}]$-lattice of Schur functions tensored with the group algebra contains Lusztig's integral lattice.

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Additional Information:

Naihuan Jing
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205
Email: jing@math.ncsu.edu

DOI: 10.1090/S1088-4165-00-00065-0
PII: S 1088-4165(00)00065-0
Keywords: Symmetric functions, vertex operators, quantum affine algebras, Littlewood-Richardson rule
Received by editor(s): February 17, 1999 and, in revised form, December 10, 1999
Posted: February 7, 2000
Additional Notes: Research supported in part by NSA grant MDA904-97-1-0062 and Mathematical Sciences Research Institute.
Copyright of article: Copyright 2000, American Mathematical Society

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