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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Symmetric polynomials and $U_q(\widehat {sl}_2)$
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by Naihuan Jing PDF
Represent. Theory 4 (2000), 46-63 Request permission


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
  • Email:
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: Represent. Theory 4 (2000), 46-63
  • MSC (2000): Primary 17B; Secondary 5E
  • DOI:
  • MathSciNet review: 1740180