Polynomial solutions of the Knizhnik–Zamolodchikov equations and Schur–Weyl duality

An integral formula for the solutions of Knizhnik-Zamolodchikov (KZ) equation with values in an arbitrary irreducible representation of the symmetric group SN is presented for integer values of the parameter. The corresponding integrals can be computed effectively as certain iterated residues determined by a given Young diagram and give polynomials with integer coefficients. The derivation is based on Schur-Weyl duality and the results of Matsuo on the original SU(n) KZ equation. The duality between the spaces of solutions with parameters m and −m is discussed in relation with the intersection pairing in the corresponding homology groups.