ar X iv : m at h / 01 05 25 3 v 3 [ m at h . Q A ] 9 O ct 2 00 3 NONCOMMUTATIVE DIFFERENTIALS AND YANG - MILLS ON PERMUTATION GROUPS S

We study noncommutative differential structures on the group of permutations S N , defined by conjugacy classes. The 2-cycles class defines an exterior algebra Λ N which is a super analogue of the Fomin-Kirillov algebra E N for Schubert calculus on the cohomology of the GL N flag variety. Noncom-mutative 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 E N. We also construct Λ N and E N as braided groups in the category of S N-crossed modules, giving a new approach to the latter that makes sense for all flag varieties.