Symmetric polynomials and 𝑈_{𝑞}(̂𝑠𝑙₂)

Jing, Naihuan

doi:10.1090/S1088-4165-00-00065-0

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

Represent. Theory 4 (2000), 46-63

DOI: https://doi.org/10.1090/S1088-4165-00-00065-0

Published electronically: 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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