On a relation between the basic representation of the affine Lie algebra $\widehat{\text{sl}}$ and a Schur--Weyl representation of the infinite symmetric group

We prove that there is a natural grading-preserving isomorphism of $\text{sl}$-modules between the basic module of the affine Lie algebra $\widehat{\text{sl}}$ (with the homogeneous grading) and a Schur–Weyl module of the infinite symmetric group $\text{sinf}$ with a grading defined through the combinatorial notion of the major index of a Young tableau, and study the properties of this isomorphism. The results reveal new and deep interrelations between the representation theory of $\widehat{\text{sl}}$ and the Virasoro algebra on the one hand, and the representation theory of $\text{sinf}$ and the related combinatorics on the other hand.