Noncommutative Differentials and Yang-mills on Permutation Groups Sn

Abstract We study noncommutative differential structures on permutation groups SN , defined by conjugacy classes. The 2-cycles class defines an exterior algebra ΛN which is a super analogue of the quadratic algebra EN for Schubert calculus on the cohomology of the flag variety. Noncommutative de Rahm cohomology and moduli of flat connections are computed for N < 6. We find that flat connections of submaximal cardinality form a natural representation associated to each conjugacy class, often irreducible, and are analogues of the Dunkl elements in EN . We also construct ΛN and EN as braided groups in the category of SN -crossed modules, giving a new approach to the latter.