Symmetric Polynomials and $U_q(\hat{sl}_2)$

Details Symmetric Polynomials and $U_q(\hat{sl}_2)$ Symmetric Polynomials and $U_q(\hat{sl}_2)$

1999-02-18

arxiv.org/abs/math/9902109v2

Represent. Theory 4 (2000), 46-63.

Mathematics

Quantum Algebra

Revised version, 17 pages, AMSLaTex

Scientific paper

We study the explicit formula of Lusztig's integral forms of the level one quantum affine algebra $U_q(\hat{sl}_2)$ 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 $\mathbb Z[q, q]$-form of $U_q(\hat{sl}_2)$ on Schur polynomials.

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