Noncommutative Differentials and Yang-Mills on Permutation Groups S_N
Abstract
We study noncommutative differential structures on the group of permutations S_N, defined by conjugacy classes. The 2-cycles class defines an exterior algebra \Lambda_N which is a super analogue of the Fomin-Kirillov algebra \CE_N for Schubert calculus on the cohomology of the GL_N 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 \CE_N. We also construct \Lambda_N and \CE_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.