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Representation Theory

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

Author: Naihuan Jing
Journal: Represent. Theory 4 (2000), 46-63
MSC (2000): Primary 17B; Secondary 5E
Published electronically: February 7, 2000
MathSciNet review: 1740180
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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
MR Author ID: 232836

Keywords: Symmetric functions, vertex operators, quantum affine algebras, Littlewood-Richardson rule
Received by editor(s): February 17, 1999
Received by editor(s) in revised form: December 10, 1999
Published electronically: February 7, 2000
Additional Notes: Research supported in part by NSA grant MDA904-97-1-0062 and Mathematical Sciences Research Institute.
Article copyright: © Copyright 2000 American Mathematical Society