Symmetric polynomials and 𝑈_{𝑞}(̂𝑠𝑙₂)

We study the explicit formula of Lusztig’s integral forms of the level one quantum affine algebra U q ( s l ^ 2 ) U_q(\widehat {sl}_2) in the endomorphism ring of symmetric functions in infinitely many variables tensored with the group algebra of Z \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 Z [ q , q − 1 ] \mathbb Z[q, q^{-1}] -form of U q ( s l ^ 2 ) U_q(\widehat {sl}_2) on Schur polynomials. As a result the Z [ q , q − 1 ] \mathbb Z[q, q^{-1}] -lattice of Schur functions tensored with the group algebra contains Lusztig’s integral lattice.